\(\int \frac {\sin ^5(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\) [226]

Optimal result

Integrand size = 24, antiderivative size = 313 \[ \int \frac {\sin ^5(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {\left (3 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{5/4} d}+\frac {\left (3 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{5/4} d}-\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (a^2-11 a b-2 b^2+2 b (2 a+b) \cos ^2(c+d x)\right )}{32 a (a-b)^2 b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]

[Out]

-1/8*cos(d*x+c)*(a+b-b*cos(d*x+c)^2)/(a-b)/b/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)^2+1/32*cos(d*x+c)*(a^2-11
*a*b-2*b^2+2*b*(2*a+b)*cos(d*x+c)^2)/a/(a-b)^2/b/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)+1/64*arctan(b^(1/4)*c
os(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))*(3*a+4*b-10*a^(1/2)*b^(1/2))/a^(3/2)/b^(5/4)/d/(a^(1/2)-b^(1/2))^(5/2)+1/64
*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*(3*a+4*b+10*a^(1/2)*b^(1/2))/a^(3/2)/b^(5/4)/d/(a^(1/2)+b
^(1/2))^(5/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3294, 1219, 1192, 1180, 211, 214} \[ \int \frac {\sin ^5(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {\left (-10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} b^{5/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\left (10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} b^{5/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\cos (c+d x) \left (a^2+2 b (2 a+b) \cos ^2(c+d x)-11 a b-2 b^2\right )}{32 a b d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

[In]

Int[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

((3*a - 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a]
- Sqrt[b])^(5/2)*b^(5/4)*d) + ((3*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] +
Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(5/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(8*(a
- b)*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (Cos[c + d*x]*(a^2 - 11*a*b - 2*b^2 + 2*b*(2*a
+ b)*Cos[c + d*x]^2))/(32*a*(a - b)^2*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1219

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x
^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2
*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c
*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {2 a (a-7 b)+10 a b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{16 a (a-b) b d} \\ & = -\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (a^2-11 a b-2 b^2+2 b (2 a+b) \cos ^2(c+d x)\right )}{32 a (a-b)^2 b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-4 a b \left (3 a^2-17 a b+2 b^2\right )-8 a b^2 (2 a+b) x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{128 a^2 (a-b)^2 b^2 d} \\ & = -\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (a^2-11 a b-2 b^2+2 b (2 a+b) \cos ^2(c+d x)\right )}{32 a (a-b)^2 b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {\left (3 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {b} d}+\frac {\left (3 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {b} d} \\ & = \frac {\left (3 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{5/4} d}+\frac {\left (3 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{5/4} d}-\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (a^2-11 a b-2 b^2+2 b (2 a+b) \cos ^2(c+d x)\right )}{32 a (a-b)^2 b d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 6.24 (sec) , antiderivative size = 786, normalized size of antiderivative = 2.51 \[ \int \frac {\sin ^5(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {\frac {32 \cos (c+d x) \left (a^2-9 a b-b^2+b (2 a+b) \cos (2 (c+d x))\right )}{a (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}-\frac {512 (a-b) \cos (c+d x) (2 a+b-b \cos (2 (c+d x)))}{(-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}+\frac {i \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {4 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-2 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+12 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-64 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+10 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+32 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-5 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-12 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+64 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-10 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+6 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-32 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+5 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-4 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-2 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+2 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6+i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a}}{128 (a-b)^2 b d} \]

[In]

Integrate[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

((32*Cos[c + d*x]*(a^2 - 9*a*b - b^2 + b*(2*a + b)*Cos[2*(c + d*x)]))/(a*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b
*Cos[4*(c + d*x)])) - (512*(a - b)*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d
*x)] + b*Cos[4*(c + d*x)])^2 + (I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (4*a*b*A
rcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (2*I)*a*b*Log[1 - 2
*Cos[c + d*x]*#1 + #1^2] - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 12*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x]
- #1)]*#1^2 - 64*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x]
- #1)]*#1^2 - (6*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (32*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*
#1^2 - (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 12*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4
+ 64*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4
+ (6*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (32*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (5*I
)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - 4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 2*b^2*Arc
Tan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + (2*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 + I*b^2*Log[1 -
2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/a)/(128*(a - b)^2*b*d
)

Maple [A] (verified)

Time = 5.97 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {-\frac {\frac {b \left (2 a +b \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{16 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (a^{2}-19 a b -6 b^{2}\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (5 a^{2}-14 a b -3 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{16 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (3 a^{2}+15 a b +2 b^{2}\right ) \cos \left (d x +c \right )}{32 \left (a -b \right ) a b}}{{\left (a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )\right )}^{2}}-\frac {\frac {\left (4 a \sqrt {a b}+2 \sqrt {a b}\, b -3 a^{2}+13 a b -4 b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (4 a \sqrt {a b}+2 \sqrt {a b}\, b +3 a^{2}-13 a b +4 b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}}{32 a \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(347\)
default	\(\frac {-\frac {\frac {b \left (2 a +b \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{16 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (a^{2}-19 a b -6 b^{2}\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (5 a^{2}-14 a b -3 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{16 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (3 a^{2}+15 a b +2 b^{2}\right ) \cos \left (d x +c \right )}{32 \left (a -b \right ) a b}}{{\left (a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )\right )}^{2}}-\frac {\frac {\left (4 a \sqrt {a b}+2 \sqrt {a b}\, b -3 a^{2}+13 a b -4 b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (4 a \sqrt {a b}+2 \sqrt {a b}\, b +3 a^{2}-13 a b +4 b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}}{32 a \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(347\)
risch	\(\text {Expression too large to display}\)	\(1459\)

[In]

int(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-(1/16*b*(2*a+b)/a/(a^2-2*a*b+b^2)*cos(d*x+c)^7+1/32*(a^2-19*a*b-6*b^2)/a/(a^2-2*a*b+b^2)*cos(d*x+c)^5-1/
16*(5*a^2-14*a*b-3*b^2)/a/(a^2-2*a*b+b^2)*cos(d*x+c)^3+1/32*(3*a^2+15*a*b+2*b^2)/(a-b)/a/b*cos(d*x+c))/(a-b+2*
b*cos(d*x+c)^2-b*cos(d*x+c)^4)^2-1/32/a/(a^2-2*a*b+b^2)*(1/2*(4*a*(a*b)^(1/2)+2*(a*b)^(1/2)*b-3*a^2+13*a*b-4*b
^2)/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-1/2*(4*a*(a*b)^(1/2)+
2*(a*b)^(1/2)*b+3*a^2-13*a*b+4*b^2)/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b
)*b)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4524 vs. \(2 (262) = 524\).

Time = 0.97 (sec) , antiderivative size = 4524, normalized size of antiderivative = 14.45 \[ \int \frac {\sin ^5(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

-1/128*(8*(2*a*b^2 + b^3)*cos(d*x + c)^7 + 4*(a^2*b - 19*a*b^2 - 6*b^3)*cos(d*x + c)^5 - 8*(5*a^2*b - 14*a*b^2
- 3*b^3)*cos(d*x + c)^3 + ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d
*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*
a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt((15*a^4 - 30*a
^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 + (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7
)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a
^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12
+ 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a
^3*b^7)*d^2))*log((81*a^5 - 1458*a^4*b + 9389*a^3*b^2 - 24972*a^2*b^3 + 10896*a*b^4 - 1280*b^5)*cos(d*x + c) +
((a^10*b^4 + 10*a^9*b^5 - 69*a^8*b^6 + 160*a^7*b^7 - 185*a^6*b^8 + 114*a^5*b^9 - 35*a^4*b^10 + 4*a^3*b^11)*d^
3*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*
b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45
*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)) - (27*a^7*b - 411*a^6*b^2 + 2383*a^5*b^3 - 5529*a^4*b^4 + 1962*a^3*b
^5 - 160*a^2*b^6)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 + (a^8*b^2 - 5*a^7*b^3 + 10*a^
6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 1043
61*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252
*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 +
10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))) - ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(
a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4
+ 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 +
a*b^5)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 - (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10
*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4
- 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10
+ 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^
4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))*log((81*a^5 - 1458*a^4*b + 9389*a^3*b^2 - 24972*a^2*b^3 + 10896*a*
b^4 - 1280*b^5)*cos(d*x + c) + ((a^10*b^4 + 10*a^9*b^5 - 69*a^8*b^6 + 160*a^7*b^7 - 185*a^6*b^8 + 114*a^5*b^9
- 35*a^4*b^10 + 4*a^3*b^11)*d^3*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 4
8160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 21
0*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)) + (27*a^7*b - 411*a^6*b^2 + 2383*a^5*b
^3 - 5529*a^4*b^4 + 1962*a^3*b^5 - 160*a^2*b^6)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4
- (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*
a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 12
0*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*
d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))) - ((a^3*b^3 - 2*a^2*b^4 +
a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b
^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4
*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 + (a^8*b^2
- 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 -
53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8
+ 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a
^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))*log(-(81*a^5 - 1458*a^4*b + 9389*a^3
*b^2 - 24972*a^2*b^3 + 10896*a*b^4 - 1280*b^5)*cos(d*x + c) + ((a^10*b^4 + 10*a^9*b^5 - 69*a^8*b^6 + 160*a^7*b
^7 - 185*a^6*b^8 + 114*a^5*b^9 - 35*a^4*b^10 + 4*a^3*b^11)*d^3*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 532
12*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 +
210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)) - (27*a
^7*b - 411*a^6*b^2 + 2383*a^5*b^3 - 5529*a^4*b^4 + 1962*a^3*b^5 - 160*a^2*b^6)*d)*sqrt((15*a^4 - 30*a^3*b - 22
9*a^2*b^2 + 116*a*b^3 - 16*b^4 + (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqr
t((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 -
10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*
b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d
^2))) + ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2
*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*
cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2
+ 116*a*b^3 - 16*b^4 - (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6
- 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b
^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*
a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))*log(-
(81*a^5 - 1458*a^4*b + 9389*a^3*b^2 - 24972*a^2*b^3 + 10896*a*b^4 - 1280*b^5)*cos(d*x + c) + ((a^10*b^4 + 10*a
^9*b^5 - 69*a^8*b^6 + 160*a^7*b^7 - 185*a^6*b^8 + 114*a^5*b^9 - 35*a^4*b^10 + 4*a^3*b^11)*d^3*sqrt((81*a^6 - 1
548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6
+ 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4
*b^14 + a^3*b^15)*d^4)) + (27*a^7*b - 411*a^6*b^2 + 2383*a^5*b^3 - 5529*a^4*b^4 + 1962*a^3*b^5 - 160*a^2*b^6)*
d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 - (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5
+ 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160
*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^
7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a
^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))) + 4*(3*a^3 + 12*a^2*b - 13*a*b^2 - 2*b^3)*cos(d*x + c))/((a^3*b^3 - 2*a^2
*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7
*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b -
4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^5(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(sin(d*x+c)**5/(a-b*sin(d*x+c)**4)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sin ^5(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{5}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]

[In]

integrate(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/8*(8*(2*a*b^4 + b^5)*cos(2*d*x + 2*c)*cos(d*x + c) + 8*(2*a^2*b^3 - 24*a*b^4 - 5*b^5)*sin(3*d*x + 3*c)*sin(2
*d*x + 2*c) + 8*(2*a*b^4 + b^5)*sin(2*d*x + 2*c)*sin(d*x + c) - ((2*a*b^4 + b^5)*cos(15*d*x + 15*c) + (2*a^2*b
^3 - 24*a*b^4 - 5*b^5)*cos(13*d*x + 13*c) - (70*a^2*b^3 - 76*a*b^4 - 9*b^5)*cos(11*d*x + 11*c) + (96*a^3*b^2 +
164*a^2*b^3 - 54*a*b^4 - 5*b^5)*cos(9*d*x + 9*c) + (96*a^3*b^2 + 164*a^2*b^3 - 54*a*b^4 - 5*b^5)*cos(7*d*x +
7*c) - (70*a^2*b^3 - 76*a*b^4 - 9*b^5)*cos(5*d*x + 5*c) + (2*a^2*b^3 - 24*a*b^4 - 5*b^5)*cos(3*d*x + 3*c) + (2
*a*b^4 + b^5)*cos(d*x + c))*cos(16*d*x + 16*c) - (2*a*b^4 + b^5 - 8*(2*a*b^4 + b^5)*cos(14*d*x + 14*c) - 4*(16
*a^2*b^3 - 6*a*b^4 - 7*b^5)*cos(12*d*x + 12*c) + 8*(32*a^2*b^3 + 2*a*b^4 - 7*b^5)*cos(10*d*x + 10*c) + 2*(256*
a^3*b^2 - 64*a^2*b^3 - 26*a*b^4 + 35*b^5)*cos(8*d*x + 8*c) + 8*(32*a^2*b^3 + 2*a*b^4 - 7*b^5)*cos(6*d*x + 6*c)
- 4*(16*a^2*b^3 - 6*a*b^4 - 7*b^5)*cos(4*d*x + 4*c) - 8*(2*a*b^4 + b^5)*cos(2*d*x + 2*c))*cos(15*d*x + 15*c)
+ 8*((2*a^2*b^3 - 24*a*b^4 - 5*b^5)*cos(13*d*x + 13*c) - (70*a^2*b^3 - 76*a*b^4 - 9*b^5)*cos(11*d*x + 11*c) +
(96*a^3*b^2 + 164*a^2*b^3 - 54*a*b^4 - 5*b^5)*cos(9*d*x + 9*c) + (96*a^3*b^2 + 164*a^2*b^3 - 54*a*b^4 - 5*b^5)
*cos(7*d*x + 7*c) - (70*a^2*b^3 - 76*a*b^4 - 9*b^5)*cos(5*d*x + 5*c) + (2*a^2*b^3 - 24*a*b^4 - 5*b^5)*cos(3*d*
x + 3*c) + (2*a*b^4 + b^5)*cos(d*x + c))*cos(14*d*x + 14*c) - (2*a^2*b^3 - 24*a*b^4 - 5*b^5 - 4*(16*a^3*b^2 -
206*a^2*b^3 + 128*a*b^4 + 35*b^5)*cos(12*d*x + 12*c) + 8*(32*a^3*b^2 - 398*a^2*b^3 + 88*a*b^4 + 35*b^5)*cos(10
*d*x + 10*c) + 2*(256*a^4*b - 3264*a^3*b^2 + 1734*a^2*b^3 - 360*a*b^4 - 175*b^5)*cos(8*d*x + 8*c) + 8*(32*a^3*
b^2 - 398*a^2*b^3 + 88*a*b^4 + 35*b^5)*cos(6*d*x + 6*c) - 4*(16*a^3*b^2 - 206*a^2*b^3 + 128*a*b^4 + 35*b^5)*co
s(4*d*x + 4*c) - 8*(2*a^2*b^3 - 24*a*b^4 - 5*b^5)*cos(2*d*x + 2*c))*cos(13*d*x + 13*c) - 4*((560*a^3*b^2 - 109
8*a^2*b^3 + 460*a*b^4 + 63*b^5)*cos(11*d*x + 11*c) - (768*a^4*b + 640*a^3*b^2 - 1580*a^2*b^3 + 338*a*b^4 + 35*
b^5)*cos(9*d*x + 9*c) - (768*a^4*b + 640*a^3*b^2 - 1580*a^2*b^3 + 338*a*b^4 + 35*b^5)*cos(7*d*x + 7*c) + (560*
a^3*b^2 - 1098*a^2*b^3 + 460*a*b^4 + 63*b^5)*cos(5*d*x + 5*c) - (16*a^3*b^2 - 206*a^2*b^3 + 128*a*b^4 + 35*b^5
)*cos(3*d*x + 3*c) - (16*a^2*b^3 - 6*a*b^4 - 7*b^5)*cos(d*x + c))*cos(12*d*x + 12*c) + (70*a^2*b^3 - 76*a*b^4
- 9*b^5 + 8*(1120*a^3*b^2 - 1706*a^2*b^3 + 388*a*b^4 + 63*b^5)*cos(10*d*x + 10*c) + 2*(8960*a^4*b - 16448*a^3*
b^2 + 8594*a^2*b^3 - 1796*a*b^4 - 315*b^5)*cos(8*d*x + 8*c) + 8*(1120*a^3*b^2 - 1706*a^2*b^3 + 388*a*b^4 + 63*
b^5)*cos(6*d*x + 6*c) - 4*(560*a^3*b^2 - 1098*a^2*b^3 + 460*a*b^4 + 63*b^5)*cos(4*d*x + 4*c) - 8*(70*a^2*b^3 -
76*a*b^4 - 9*b^5)*cos(2*d*x + 2*c))*cos(11*d*x + 11*c) - 8*((1536*a^4*b + 1952*a^3*b^2 - 2012*a^2*b^3 + 298*a
*b^4 + 35*b^5)*cos(9*d*x + 9*c) + (1536*a^4*b + 1952*a^3*b^2 - 2012*a^2*b^3 + 298*a*b^4 + 35*b^5)*cos(7*d*x +
7*c) - (1120*a^3*b^2 - 1706*a^2*b^3 + 388*a*b^4 + 63*b^5)*cos(5*d*x + 5*c) + (32*a^3*b^2 - 398*a^2*b^3 + 88*a*
b^4 + 35*b^5)*cos(3*d*x + 3*c) + (32*a^2*b^3 + 2*a*b^4 - 7*b^5)*cos(d*x + c))*cos(10*d*x + 10*c) - (96*a^3*b^2
+ 164*a^2*b^3 - 54*a*b^4 - 5*b^5 + 2*(12288*a^5 + 11776*a^4*b - 19296*a^3*b^2 + 10284*a^2*b^3 - 1410*a*b^4 -
175*b^5)*cos(8*d*x + 8*c) + 8*(1536*a^4*b + 1952*a^3*b^2 - 2012*a^2*b^3 + 298*a*b^4 + 35*b^5)*cos(6*d*x + 6*c)
- 4*(768*a^4*b + 640*a^3*b^2 - 1580*a^2*b^3 + 338*a*b^4 + 35*b^5)*cos(4*d*x + 4*c) - 8*(96*a^3*b^2 + 164*a^2*
b^3 - 54*a*b^4 - 5*b^5)*cos(2*d*x + 2*c))*cos(9*d*x + 9*c) - 2*((12288*a^5 + 11776*a^4*b - 19296*a^3*b^2 + 102
84*a^2*b^3 - 1410*a*b^4 - 175*b^5)*cos(7*d*x + 7*c) - (8960*a^4*b - 16448*a^3*b^2 + 8594*a^2*b^3 - 1796*a*b^4
- 315*b^5)*cos(5*d*x + 5*c) + (256*a^4*b - 3264*a^3*b^2 + 1734*a^2*b^3 - 360*a*b^4 - 175*b^5)*cos(3*d*x + 3*c)
+ (256*a^3*b^2 - 64*a^2*b^3 - 26*a*b^4 + 35*b^5)*cos(d*x + c))*cos(8*d*x + 8*c) - (96*a^3*b^2 + 164*a^2*b^3 -
54*a*b^4 - 5*b^5 + 8*(1536*a^4*b + 1952*a^3*b^2 - 2012*a^2*b^3 + 298*a*b^4 + 35*b^5)*cos(6*d*x + 6*c) - 4*(76
8*a^4*b + 640*a^3*b^2 - 1580*a^2*b^3 + 338*a*b^4 + 35*b^5)*cos(4*d*x + 4*c) - 8*(96*a^3*b^2 + 164*a^2*b^3 - 54
*a*b^4 - 5*b^5)*cos(2*d*x + 2*c))*cos(7*d*x + 7*c) + 8*((1120*a^3*b^2 - 1706*a^2*b^3 + 388*a*b^4 + 63*b^5)*cos
(5*d*x + 5*c) - (32*a^3*b^2 - 398*a^2*b^3 + 88*a*b^4 + 35*b^5)*cos(3*d*x + 3*c) - (32*a^2*b^3 + 2*a*b^4 - 7*b^
5)*cos(d*x + c))*cos(6*d*x + 6*c) + (70*a^2*b^3 - 76*a*b^4 - 9*b^5 - 4*(560*a^3*b^2 - 1098*a^2*b^3 + 460*a*b^4
+ 63*b^5)*cos(4*d*x + 4*c) - 8*(70*a^2*b^3 - 76*a*b^4 - 9*b^5)*cos(2*d*x + 2*c))*cos(5*d*x + 5*c) + 4*((16*a^
3*b^2 - 206*a^2*b^3 + 128*a*b^4 + 35*b^5)*cos(3*d*x + 3*c) + (16*a^2*b^3 - 6*a*b^4 - 7*b^5)*cos(d*x + c))*cos(
4*d*x + 4*c) - (2*a^2*b^3 - 24*a*b^4 - 5*b^5 - 8*(2*a^2*b^3 - 24*a*b^4 - 5*b^5)*cos(2*d*x + 2*c))*cos(3*d*x +
3*c) - (2*a*b^4 + b^5)*cos(d*x + c) - 8*((a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(16*d*x + 16*c)^2 + 64*(a^3*b^5 -
2*a^2*b^6 + a*b^7)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^
7)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(10*d*x
+ 10*c)^2 + 4*(16384*a^7*b - 57344*a^6*b^2 + 83712*a^5*b^3 - 67648*a^4*b^4 + 32841*a^3*b^5 - 9170*a^2*b^6 + 1
225*a*b^7)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(
6*d*x + 6*c)^2 + 16*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c)^2 + 6
4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c)^2 + (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(16*d*x + 16*c)^2 +
64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(14*d*x + 14*c)^2 + 16*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2
*b^6 + 49*a*b^7)*d*sin(12*d*x + 12*c)^2 + 64*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7
)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^7*b - 57344*a^6*b^2 + 83712*a^5*b^3 - 67648*a^4*b^4 + 32841*a^3*b^5 - 91
70*a^2*b^6 + 1225*a*b^7)*d*sin(8*d*x + 8*c)^2 + 64*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49
*a*b^7)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*sin(4*d*
x + 4*c)^2 + 64*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^3*
b^5 - 2*a^2*b^6 + a*b^7)*d*sin(2*d*x + 2*c)^2 - 16*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c) + (a^3*b^5
- 2*a^2*b^6 + a*b^7)*d - 2*(8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(14*d*x + 14*c) + 4*(8*a^4*b^4 - 23*a^3*b^5
+ 22*a^2*b^6 - 7*a*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(10*d*x
+ 10*c) - 2*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^4
*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(6*d*x + 6*c) + 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^
7)*d*cos(4*d*x + 4*c) + 8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c) - (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d)*
cos(16*d*x + 16*c) + 16*(4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^4*b^
4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(10*d*x + 10*c) - 2*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166
*a^2*b^6 + 35*a*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(6*d*x + 6*c
) + 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(4*d*x + 4*c) + 8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*c
os(2*d*x + 2*c) - (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d)*cos(14*d*x + 14*c) - 8*(8*(128*a^5*b^3 - 424*a^4*b^4 + 513*
a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(10*d*x + 10*c) + 2*(1024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*
a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*cos(8*d*x + 8*c) + 8*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*
b^6 + 49*a*b^7)*d*cos(6*d*x + 6*c) - 4*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*cos
(4*d*x + 4*c) - 8*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c) + (8*a^4*b^4 - 23*a^3*b^5
+ 22*a^2*b^6 - 7*a*b^7)*d)*cos(12*d*x + 12*c) + 16*(2*(2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*
b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*cos(8*d*x + 8*c) + 8*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6
+ 49*a*b^7)*d*cos(6*d*x + 6*c) - 4*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(4*
d*x + 4*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c) + (16*a^4*b^4 - 39*a^3*b^5
+ 30*a^2*b^6 - 7*a*b^7)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^
5 + 1722*a^2*b^6 - 245*a*b^7)*d*cos(6*d*x + 6*c) - 4*(1024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^
5 + 1442*a^2*b^6 - 245*a*b^7)*d*cos(4*d*x + 4*c) - 8*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 +
35*a*b^7)*d*cos(2*d*x + 2*c) + (128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d)*cos(8*d*x
+ 8*c) - 16*(4*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c) + 8*(16*
a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c) - (16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*
b^7)*d)*cos(6*d*x + 6*c) + 8*(8*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c) - (8*a^4*b^
4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d)*cos(4*d*x + 4*c) - 4*(4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(14*d*x +
14*c) + 2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(12*d*x + 12*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 +
30*a^2*b^6 - 7*a*b^7)*d*sin(10*d*x + 10*c) - (128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^
7)*d*sin(8*d*x + 8*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^4 -
23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(4*d*x + 4*c) + 4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(2*d*x + 2*c))*s
in(16*d*x + 16*c) + 32*(2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(12*d*x + 12*c) - 4*(16*a^4*b^4
- 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(10*d*x + 10*c) - (128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^
2*b^6 + 35*a*b^7)*d*sin(8*d*x + 8*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(6*d*x + 6*c) +
2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(4*d*x + 4*c) + 4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(
2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*
sin(10*d*x + 10*c) + (1024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*
sin(8*d*x + 8*c) + 4*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*sin(6*d*x + 6*c) - 2
*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c) - 4*(8*a^4*b^4 - 23*a^3*
b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 32*((2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a
^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*sin(8*d*x + 8*c) + 4*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*
b^5 - 322*a^2*b^6 + 49*a*b^7)*d*sin(6*d*x + 6*c) - 2*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 +
49*a*b^7)*d*sin(4*d*x + 4*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2*c))*sin(10*d
*x + 10*c) + 16*(2*(2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*si
n(6*d*x + 6*c) - (1024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*sin(
4*d*x + 4*c) - 2*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*sin(2*d*x + 2*c))*sin(8*
d*x + 8*c) - 64*((128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c) + 2*(16
*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(1/8*(4*(6*a^2*b
- 32*a*b^2 + 5*b^3)*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) + 4*(2*a*b^2 + b^3)*cos(d*x + c)*sin(2*d*x + 2*c) - 4*(2
*a*b^2 + b^3)*cos(2*d*x + 2*c)*sin(d*x + c) - ((2*a*b^2 + b^3)*sin(7*d*x + 7*c) + (6*a^2*b - 32*a*b^2 + 5*b^3)
*sin(5*d*x + 5*c) - (6*a^2*b - 32*a*b^2 + 5*b^3)*sin(3*d*x + 3*c) - (2*a*b^2 + b^3)*sin(d*x + c))*cos(8*d*x +
8*c) - 2*(2*(2*a*b^2 + b^3)*sin(6*d*x + 6*c) + (16*a^2*b + 2*a*b^2 - 3*b^3)*sin(4*d*x + 4*c) + 2*(2*a*b^2 + b^
3)*sin(2*d*x + 2*c))*cos(7*d*x + 7*c) + 4*((6*a^2*b - 32*a*b^2 + 5*b^3)*sin(5*d*x + 5*c) - (6*a^2*b - 32*a*b^2
+ 5*b^3)*sin(3*d*x + 3*c) - (2*a*b^2 + b^3)*sin(d*x + c))*cos(6*d*x + 6*c) - 2*((48*a^3 - 274*a^2*b + 136*a*b
^2 - 15*b^3)*sin(4*d*x + 4*c) + 2*(6*a^2*b - 32*a*b^2 + 5*b^3)*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) - 2*((48*a^3
- 274*a^2*b + 136*a*b^2 - 15*b^3)*sin(3*d*x + 3*c) + (16*a^2*b + 2*a*b^2 - 3*b^3)*sin(d*x + c))*cos(4*d*x + 4
*c) + ((2*a*b^2 + b^3)*cos(7*d*x + 7*c) + (6*a^2*b - 32*a*b^2 + 5*b^3)*cos(5*d*x + 5*c) - (6*a^2*b - 32*a*b^2
+ 5*b^3)*cos(3*d*x + 3*c) - (2*a*b^2 + b^3)*cos(d*x + c))*sin(8*d*x + 8*c) - (2*a*b^2 + b^3 - 4*(2*a*b^2 + b^3
)*cos(6*d*x + 6*c) - 2*(16*a^2*b + 2*a*b^2 - 3*b^3)*cos(4*d*x + 4*c) - 4*(2*a*b^2 + b^3)*cos(2*d*x + 2*c))*sin
(7*d*x + 7*c) - 4*((6*a^2*b - 32*a*b^2 + 5*b^3)*cos(5*d*x + 5*c) - (6*a^2*b - 32*a*b^2 + 5*b^3)*cos(3*d*x + 3*
c) - (2*a*b^2 + b^3)*cos(d*x + c))*sin(6*d*x + 6*c) - (6*a^2*b - 32*a*b^2 + 5*b^3 - 2*(48*a^3 - 274*a^2*b + 13
6*a*b^2 - 15*b^3)*cos(4*d*x + 4*c) - 4*(6*a^2*b - 32*a*b^2 + 5*b^3)*cos(2*d*x + 2*c))*sin(5*d*x + 5*c) + 2*((4
8*a^3 - 274*a^2*b + 136*a*b^2 - 15*b^3)*cos(3*d*x + 3*c) + (16*a^2*b + 2*a*b^2 - 3*b^3)*cos(d*x + c))*sin(4*d*
x + 4*c) + (6*a^2*b - 32*a*b^2 + 5*b^3 - 4*(6*a^2*b - 32*a*b^2 + 5*b^3)*cos(2*d*x + 2*c))*sin(3*d*x + 3*c) + (
2*a*b^2 + b^3)*sin(d*x + c))/(a^3*b^3 - 2*a^2*b^4 + a*b^5 + (a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(8*d*x + 8*c)^2 +
16*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(6*d*x + 6*c)^2 + 4*(64*a^5*b - 176*a^4*b^2 + 169*a^3*b^3 - 66*a^2*b^4 +
9*a*b^5)*cos(4*d*x + 4*c)^2 + 16*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(2*d*x + 2*c)^2 + (a^3*b^3 - 2*a^2*b^4 + a*b
^5)*sin(8*d*x + 8*c)^2 + 16*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*sin(6*d*x + 6*c)^2 + 4*(64*a^5*b - 176*a^4*b^2 + 169
*a^3*b^3 - 66*a^2*b^4 + 9*a*b^5)*sin(4*d*x + 4*c)^2 + 16*(8*a^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5)*sin(4
*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*sin(2*d*x + 2*c)^2 + 2*(a^3*b^3 - 2*a^2*b^4 +
a*b^5 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(6*d*x + 6*c) - 2*(8*a^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5)*c
os(4*d*x + 4*c) - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a^3*b^3 - 2*a^2*b^4
+ a*b^5 - 2*(8*a^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5)*cos(4*d*x + 4*c) - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)
*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5 - 4*(8*a^4*b^2 - 19*a^3
*b^3 + 14*a^2*b^4 - 3*a*b^5)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(2*d*x +
2*c) - 4*(2*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*sin(6*d*x + 6*c) + (8*a^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5)*s
in(4*d*x + 4*c) + 2*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^4*b^2 - 19*a^3
*b^3 + 14*a^2*b^4 - 3*a*b^5)*sin(4*d*x + 4*c) + 2*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*sin(2*d*x + 2*c))*sin(6*d*x +
6*c)), x) - ((2*a*b^4 + b^5)*sin(15*d*x + 15*c) + (2*a^2*b^3 - 24*a*b^4 - 5*b^5)*sin(13*d*x + 13*c) - (70*a^2*
b^3 - 76*a*b^4 - 9*b^5)*sin(11*d*x + 11*c) + (96*a^3*b^2 + 164*a^2*b^3 - 54*a*b^4 - 5*b^5)*sin(9*d*x + 9*c) +
(96*a^3*b^2 + 164*a^2*b^3 - 54*a*b^4 - 5*b^5)*sin(7*d*x + 7*c) - (70*a^2*b^3 - 76*a*b^4 - 9*b^5)*sin(5*d*x + 5
*c) + (2*a^2*b^3 - 24*a*b^4 - 5*b^5)*sin(3*d*x + 3*c) + (2*a*b^4 + b^5)*sin(d*x + c))*sin(16*d*x + 16*c) + 2*(
4*(2*a*b^4 + b^5)*sin(14*d*x + 14*c) + 2*(16*a^2*b^3 - 6*a*b^4 - 7*b^5)*sin(12*d*x + 12*c) - 4*(32*a^2*b^3 + 2
*a*b^4 - 7*b^5)*sin(10*d*x + 10*c) - (256*a^3*b^2 - 64*a^2*b^3 - 26*a*b^4 + 35*b^5)*sin(8*d*x + 8*c) - 4*(32*a
^2*b^3 + 2*a*b^4 - 7*b^5)*sin(6*d*x + 6*c) + 2*(16*a^2*b^3 - 6*a*b^4 - 7*b^5)*sin(4*d*x + 4*c) + 4*(2*a*b^4 +
b^5)*sin(2*d*x + 2*c))*sin(15*d*x + 15*c) + 8*((2*a^2*b^3 - 24*a*b^4 - 5*b^5)*sin(13*d*x + 13*c) - (70*a^2*b^3
- 76*a*b^4 - 9*b^5)*sin(11*d*x + 11*c) + (96*a^3*b^2 + 164*a^2*b^3 - 54*a*b^4 - 5*b^5)*sin(9*d*x + 9*c) + (96
*a^3*b^2 + 164*a^2*b^3 - 54*a*b^4 - 5*b^5)*sin(7*d*x + 7*c) - (70*a^2*b^3 - 76*a*b^4 - 9*b^5)*sin(5*d*x + 5*c)
+ (2*a^2*b^3 - 24*a*b^4 - 5*b^5)*sin(3*d*x + 3*c) + (2*a*b^4 + b^5)*sin(d*x + c))*sin(14*d*x + 14*c) + 2*(2*(
16*a^3*b^2 - 206*a^2*b^3 + 128*a*b^4 + 35*b^5)*sin(12*d*x + 12*c) - 4*(32*a^3*b^2 - 398*a^2*b^3 + 88*a*b^4 + 3
5*b^5)*sin(10*d*x + 10*c) - (256*a^4*b - 3264*a^3*b^2 + 1734*a^2*b^3 - 360*a*b^4 - 175*b^5)*sin(8*d*x + 8*c) -
4*(32*a^3*b^2 - 398*a^2*b^3 + 88*a*b^4 + 35*b^5)*sin(6*d*x + 6*c) + 2*(16*a^3*b^2 - 206*a^2*b^3 + 128*a*b^4 +
35*b^5)*sin(4*d*x + 4*c) + 4*(2*a^2*b^3 - 24*a*b^4 - 5*b^5)*sin(2*d*x + 2*c))*sin(13*d*x + 13*c) - 4*((560*a^
3*b^2 - 1098*a^2*b^3 + 460*a*b^4 + 63*b^5)*sin(11*d*x + 11*c) - (768*a^4*b + 640*a^3*b^2 - 1580*a^2*b^3 + 338*
a*b^4 + 35*b^5)*sin(9*d*x + 9*c) - (768*a^4*b + 640*a^3*b^2 - 1580*a^2*b^3 + 338*a*b^4 + 35*b^5)*sin(7*d*x + 7
*c) + (560*a^3*b^2 - 1098*a^2*b^3 + 460*a*b^4 + 63*b^5)*sin(5*d*x + 5*c) - (16*a^3*b^2 - 206*a^2*b^3 + 128*a*b
^4 + 35*b^5)*sin(3*d*x + 3*c) - (16*a^2*b^3 - 6*a*b^4 - 7*b^5)*sin(d*x + c))*sin(12*d*x + 12*c) + 2*(4*(1120*a
^3*b^2 - 1706*a^2*b^3 + 388*a*b^4 + 63*b^5)*sin(10*d*x + 10*c) + (8960*a^4*b - 16448*a^3*b^2 + 8594*a^2*b^3 -
1796*a*b^4 - 315*b^5)*sin(8*d*x + 8*c) + 4*(1120*a^3*b^2 - 1706*a^2*b^3 + 388*a*b^4 + 63*b^5)*sin(6*d*x + 6*c)
- 2*(560*a^3*b^2 - 1098*a^2*b^3 + 460*a*b^4 + 63*b^5)*sin(4*d*x + 4*c) - 4*(70*a^2*b^3 - 76*a*b^4 - 9*b^5)*si
n(2*d*x + 2*c))*sin(11*d*x + 11*c) - 8*((1536*a^4*b + 1952*a^3*b^2 - 2012*a^2*b^3 + 298*a*b^4 + 35*b^5)*sin(9*
d*x + 9*c) + (1536*a^4*b + 1952*a^3*b^2 - 2012*a^2*b^3 + 298*a*b^4 + 35*b^5)*sin(7*d*x + 7*c) - (1120*a^3*b^2
- 1706*a^2*b^3 + 388*a*b^4 + 63*b^5)*sin(5*d*x + 5*c) + (32*a^3*b^2 - 398*a^2*b^3 + 88*a*b^4 + 35*b^5)*sin(3*d
*x + 3*c) + (32*a^2*b^3 + 2*a*b^4 - 7*b^5)*sin(d*x + c))*sin(10*d*x + 10*c) - 2*((12288*a^5 + 11776*a^4*b - 19
296*a^3*b^2 + 10284*a^2*b^3 - 1410*a*b^4 - 175*b^5)*sin(8*d*x + 8*c) + 4*(1536*a^4*b + 1952*a^3*b^2 - 2012*a^2
*b^3 + 298*a*b^4 + 35*b^5)*sin(6*d*x + 6*c) - 2*(768*a^4*b + 640*a^3*b^2 - 1580*a^2*b^3 + 338*a*b^4 + 35*b^5)*
sin(4*d*x + 4*c) - 4*(96*a^3*b^2 + 164*a^2*b^3 - 54*a*b^4 - 5*b^5)*sin(2*d*x + 2*c))*sin(9*d*x + 9*c) - 2*((12
288*a^5 + 11776*a^4*b - 19296*a^3*b^2 + 10284*a^2*b^3 - 1410*a*b^4 - 175*b^5)*sin(7*d*x + 7*c) - (8960*a^4*b -
16448*a^3*b^2 + 8594*a^2*b^3 - 1796*a*b^4 - 315*b^5)*sin(5*d*x + 5*c) + (256*a^4*b - 3264*a^3*b^2 + 1734*a^2*
b^3 - 360*a*b^4 - 175*b^5)*sin(3*d*x + 3*c) + (256*a^3*b^2 - 64*a^2*b^3 - 26*a*b^4 + 35*b^5)*sin(d*x + c))*sin
(8*d*x + 8*c) - 4*(2*(1536*a^4*b + 1952*a^3*b^2 - 2012*a^2*b^3 + 298*a*b^4 + 35*b^5)*sin(6*d*x + 6*c) - (768*a
^4*b + 640*a^3*b^2 - 1580*a^2*b^3 + 338*a*b^4 + 35*b^5)*sin(4*d*x + 4*c) - 2*(96*a^3*b^2 + 164*a^2*b^3 - 54*a*
b^4 - 5*b^5)*sin(2*d*x + 2*c))*sin(7*d*x + 7*c) + 8*((1120*a^3*b^2 - 1706*a^2*b^3 + 388*a*b^4 + 63*b^5)*sin(5*
d*x + 5*c) - (32*a^3*b^2 - 398*a^2*b^3 + 88*a*b^4 + 35*b^5)*sin(3*d*x + 3*c) - (32*a^2*b^3 + 2*a*b^4 - 7*b^5)*
sin(d*x + c))*sin(6*d*x + 6*c) - 4*((560*a^3*b^2 - 1098*a^2*b^3 + 460*a*b^4 + 63*b^5)*sin(4*d*x + 4*c) + 2*(70
*a^2*b^3 - 76*a*b^4 - 9*b^5)*sin(2*d*x + 2*c))*sin(5*d*x + 5*c) + 4*((16*a^3*b^2 - 206*a^2*b^3 + 128*a*b^4 + 3
5*b^5)*sin(3*d*x + 3*c) + (16*a^2*b^3 - 6*a*b^4 - 7*b^5)*sin(d*x + c))*sin(4*d*x + 4*c))/((a^3*b^5 - 2*a^2*b^6
+ a*b^7)*d*cos(16*d*x + 16*c)^2 + 64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^5*b^3 -
240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^5*b^3 - 736*a^4*b^4 + 7
53*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^7*b - 57344*a^6*b^2 + 83712*a^5*b^3 -
67648*a^4*b^4 + 32841*a^3*b^5 - 9170*a^2*b^6 + 1225*a*b^7)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^5*b^3 - 736*a^4*b
^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 -
210*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c)^2 + 64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c)^2 + (a^3*b
^5 - 2*a^2*b^6 + a*b^7)*d*sin(16*d*x + 16*c)^2 + 64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(14*d*x + 14*c)^2 + 16*
(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*sin(12*d*x + 12*c)^2 + 64*(256*a^5*b^3 - 7
36*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^7*b - 57344*a^6*b^2 + 8
3712*a^5*b^3 - 67648*a^4*b^4 + 32841*a^3*b^5 - 9170*a^2*b^6 + 1225*a*b^7)*d*sin(8*d*x + 8*c)^2 + 64*(256*a^5*b
^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^5*b^3 - 240*a^4*b^4 +
337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^
7)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(2*d*x + 2*c)^2 - 16*(a^3*b^5 -
2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c) + (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d - 2*(8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*
d*cos(14*d*x + 14*c) + 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^4*b^4
- 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(10*d*x + 10*c) - 2*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a
^2*b^6 + 35*a*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(6*d*x + 6*c)
+ 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(4*d*x + 4*c) + 8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos
(2*d*x + 2*c) - (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d)*cos(16*d*x + 16*c) + 16*(4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b
^6 - 7*a*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(10*d*x + 10*c) -
2*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^4*b^4 - 39*
a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(6*d*x + 6*c) + 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(4
*d*x + 4*c) + 8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c) - (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d)*cos(14*d*x
+ 14*c) - 8*(8*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(10*d*x + 10*c) + 2*(1
024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*cos(8*d*x + 8*c) + 8*(1
28*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(6*d*x + 6*c) - 4*(64*a^5*b^3 - 240*a^4*
b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c) - 8*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*
b^7)*d*cos(2*d*x + 2*c) + (8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d)*cos(12*d*x + 12*c) + 16*(2*(2048*
a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*cos(8*d*x + 8*c) + 8*(256*a
^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(6*d*x + 6*c) - 4*(128*a^5*b^3 - 424*a^4*b^4
+ 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^
7)*d*cos(2*d*x + 2*c) + (16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^
6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*cos(6*d*x + 6*c) - 4*(1024*a^
6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*cos(4*d*x + 4*c) - 8*(128*a^5
*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*cos(2*d*x + 2*c) + (128*a^5*b^3 - 352*a^4*b^4 + 3
55*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d)*cos(8*d*x + 8*c) - 16*(4*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 26
6*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c) + 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*
c) - (16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d)*cos(6*d*x + 6*c) + 8*(8*(8*a^4*b^4 - 23*a^3*b^5 + 22*
a^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c) - (8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d)*cos(4*d*x + 4*c) -
4*(4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(14*d*x + 14*c) + 2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*
sin(12*d*x + 12*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(10*d*x + 10*c) - (128*a^5*b^3 -
352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*sin(8*d*x + 8*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b
^6 - 7*a*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(4*d*x + 4*c) + 4*(a
^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(2*d*x + 2*c))*sin(16*d*x + 16*c) + 32*(2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^
6 - 7*a*b^7)*d*sin(12*d*x + 12*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(10*d*x + 10*c) -
(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*sin(8*d*x + 8*c) - 4*(16*a^4*b^4 - 39*a^3
*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(4*d*
x + 4*c) + 4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^5*b^3 - 424*a
^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*sin(10*d*x + 10*c) + (1024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*
b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*sin(8*d*x + 8*c) + 4*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5
- 266*a^2*b^6 + 49*a*b^7)*d*sin(6*d*x + 6*c) - 2*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a
*b^7)*d*sin(4*d*x + 4*c) - 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2*c))*sin(12*d*x +
12*c) + 32*((2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*sin(8*d*x
+ 8*c) + 4*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*sin(6*d*x + 6*c) - 2*(128*a^5
*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 3
0*a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^
4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*sin(6*d*x + 6*c) - (1024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4
- 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*sin(4*d*x + 4*c) - 2*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 1
66*a^2*b^6 + 35*a*b^7)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 2
66*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c) + 2*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2
*c))*sin(6*d*x + 6*c))

Giac [F]

\[ \int \frac {\sin ^5(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{5}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]

[In]

integrate(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 19.51 (sec) , antiderivative size = 6362, normalized size of antiderivative = 20.33 \[ \int \frac {\sin ^5(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

int(sin(c + d*x)^5/(a - b*sin(c + d*x)^4)^3,x)

[Out]

- ((cos(c + d*x)^3*(14*a*b - 5*a^2 + 3*b^2))/(16*a*(a - b)^2) - (cos(c + d*x)^5*(19*a*b - a^2 + 6*b^2))/(32*a*
(a^2 - 2*a*b + b^2)) + (b*cos(c + d*x)^7*(2*a + b))/(16*a*(a^2 - 2*a*b + b^2)) + (cos(c + d*x)*(15*a*b + 3*a^2
+ 2*b^2))/(32*a*b*(a - b)))/(d*(a^2 - 2*a*b + b^2 + cos(c + d*x)^2*(4*a*b - 4*b^2) - cos(c + d*x)^4*(2*a*b -
6*b^2) - 4*b^2*cos(c + d*x)^6 + b^2*cos(c + d*x)^8)) - (atan(((((16384*a^3*b^6 - 172032*a^4*b^5 + 319488*a^5*b
^4 - 188416*a^6*b^3 + 24576*a^7*b^2)/(16384*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) - (cos(c + d*x)
*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a
^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*
a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*(16384*a^3*b^8 - 65536*a^4*b^7 + 98304*a^5*b^6 - 65536*a^6*b^5 + 1638
4*a^7*b^4))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9*b^5
)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 86*a^
2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2) +
(cos(c + d*x)*(9*a^4*b - 100*a*b^4 + 16*b^5 + 209*a^2*b^3 - 62*a^3*b^2))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^
3*b^3 + 6*a^4*b^2)))*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5
+ 30*a^6*b^4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^
9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*1i - (((16384*a^3*b^6 - 172032*a^4*b^5 + 319488*a
^5*b^4 - 188416*a^6*b^3 + 24576*a^7*b^2)/(16384*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) + (cos(c +
d*x)*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 -
15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 -
10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*(16384*a^3*b^8 - 65536*a^4*b^7 + 98304*a^5*b^6 - 65536*a^6*b^5 +
16384*a^7*b^4))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9
*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 8
6*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/
2) - (cos(c + d*x)*(9*a^4*b - 100*a*b^4 + 16*b^5 + 209*a^2*b^3 - 62*a^3*b^2))/(256*(a^6 - 4*a^5*b + a^2*b^4 -
4*a^3*b^3 + 6*a^4*b^2)))*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5
*b^5 + 30*a^6*b^4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^
7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*1i)/((((16384*a^3*b^6 - 172032*a^4*b^5 + 3194
88*a^5*b^4 - 188416*a^6*b^3 + 24576*a^7*b^2)/(16384*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) - (cos(
c + d*x)*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^
4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b
^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*(16384*a^3*b^8 - 65536*a^4*b^7 + 98304*a^5*b^6 - 65536*a^6*b^
5 + 16384*a^7*b^4))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*
(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2)
+ 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))
^(1/2) + (cos(c + d*x)*(9*a^4*b - 100*a*b^4 + 16*b^5 + 209*a^2*b^3 - 62*a^3*b^2))/(256*(a^6 - 4*a^5*b + a^2*b^
4 - 4*a^3*b^3 + 6*a^4*b^2)))*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229
*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 -
5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2) + (((16384*a^3*b^6 - 172032*a^4*b^5 + 319
488*a^5*b^4 - 188416*a^6*b^3 + 24576*a^7*b^2)/(16384*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) + (cos
(c + d*x)*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b
^4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*
b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*(16384*a^3*b^8 - 65536*a^4*b^7 + 98304*a^5*b^6 - 65536*a^6*b
^5 + 16384*a^7*b^4))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3
*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2
) + 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5))
)^(1/2) - (cos(c + d*x)*(9*a^4*b - 100*a*b^4 + 16*b^5 + 209*a^2*b^3 - 62*a^3*b^2))/(256*(a^6 - 4*a^5*b + a^2*b
^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 22
9*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 -
5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2) + (44*a*b^2 + 143*a^2*b - 18*a^3 - 16*b^
3)/(8192*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2))))*((80*b^3*(a^9*b^5)^(1/2) - 9*a^3*(a^9*b^5)^(1/2)
+ 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 - 301*a*b^2*(a^9*b^5)^(1/2) + 86*a^2*b*(a^
9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*2i)/d -
(atan(((((16384*a^3*b^6 - 172032*a^4*b^5 + 319488*a^5*b^4 - 188416*a^6*b^3 + 24576*a^7*b^2)/(16384*(a^7 - 4*a^
6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) - (cos(c + d*x)*((9*a^3*(a^9*b^5)^(1/2) - 80*b^3*(a^9*b^5)^(1/2) + 16*
a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1/2) - 86*a^2*b*(a^9*b^5)
^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*(16384*a^3*b^8
- 65536*a^4*b^7 + 98304*a^5*b^6 - 65536*a^6*b^5 + 16384*a^7*b^4))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 +
6*a^4*b^2)))*((9*a^3*(a^9*b^5)^(1/2) - 80*b^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a
^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1/2) - 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*
a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2) + (cos(c + d*x)*(9*a^4*b - 100*a*b^4 + 16*b^5 + 209*a^2*
b^3 - 62*a^3*b^2))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((9*a^3*(a^9*b^5)^(1/2) - 80*b^3*(
a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1/2)
- 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^
(1/2)*1i - (((16384*a^3*b^6 - 172032*a^4*b^5 + 319488*a^5*b^4 - 188416*a^6*b^3 + 24576*a^7*b^2)/(16384*(a^7 -
4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) + (cos(c + d*x)*((9*a^3*(a^9*b^5)^(1/2) - 80*b^3*(a^9*b^5)^(1/2) +
16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1/2) - 86*a^2*b*(a^9*
b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*(16384*a^3
*b^8 - 65536*a^4*b^7 + 98304*a^5*b^6 - 65536*a^6*b^5 + 16384*a^7*b^4))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b
^3 + 6*a^4*b^2)))*((9*a^3*(a^9*b^5)^(1/2) - 80*b^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 +
30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1/2) - 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 +
10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2) - (cos(c + d*x)*(9*a^4*b - 100*a*b^4 + 16*b^5 + 209*
a^2*b^3 - 62*a^3*b^2))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((9*a^3*(a^9*b^5)^(1/2) - 80*b
^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1
/2) - 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5
)))^(1/2)*1i)/((((16384*a^3*b^6 - 172032*a^4*b^5 + 319488*a^5*b^4 - 188416*a^6*b^3 + 24576*a^7*b^2)/(16384*(a^
7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) - (cos(c + d*x)*((9*a^3*(a^9*b^5)^(1/2) - 80*b^3*(a^9*b^5)^(1/
2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1/2) - 86*a^2*b*(
a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*(16384
*a^3*b^8 - 65536*a^4*b^7 + 98304*a^5*b^6 - 65536*a^6*b^5 + 16384*a^7*b^4))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a
^3*b^3 + 6*a^4*b^2)))*((9*a^3*(a^9*b^5)^(1/2) - 80*b^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^
5 + 30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1/2) - 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b
^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2) + (cos(c + d*x)*(9*a^4*b - 100*a*b^4 + 16*b^5 +
209*a^2*b^3 - 62*a^3*b^2))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((9*a^3*(a^9*b^5)^(1/2) -
80*b^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5
)^(1/2) - 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11
*b^5)))^(1/2) + (((16384*a^3*b^6 - 172032*a^4*b^5 + 319488*a^5*b^4 - 188416*a^6*b^3 + 24576*a^7*b^2)/(16384*(a
^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) + (cos(c + d*x)*((9*a^3*(a^9*b^5)^(1/2) - 80*b^3*(a^9*b^5)^(1
/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1/2) - 86*a^2*b*
(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*(1638
4*a^3*b^8 - 65536*a^4*b^7 + 98304*a^5*b^6 - 65536*a^6*b^5 + 16384*a^7*b^4))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*
a^3*b^3 + 6*a^4*b^2)))*((9*a^3*(a^9*b^5)^(1/2) - 80*b^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b
^5 + 30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1/2) - 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*
b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2) - (cos(c + d*x)*(9*a^4*b - 100*a*b^4 + 16*b^5 +
209*a^2*b^3 - 62*a^3*b^2))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((9*a^3*(a^9*b^5)^(1/2) -
80*b^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4 - 15*a^7*b^3 + 301*a*b^2*(a^9*b^
5)^(1/2) - 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8 - 10*a^9*b^7 + 5*a^10*b^6 - a^1
1*b^5)))^(1/2) + (44*a*b^2 + 143*a^2*b - 18*a^3 - 16*b^3)/(8192*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b
^2))))*((9*a^3*(a^9*b^5)^(1/2) - 80*b^3*(a^9*b^5)^(1/2) + 16*a^3*b^7 - 116*a^4*b^6 + 229*a^5*b^5 + 30*a^6*b^4
- 15*a^7*b^3 + 301*a*b^2*(a^9*b^5)^(1/2) - 86*a^2*b*(a^9*b^5)^(1/2))/(16384*(a^6*b^10 - 5*a^7*b^9 + 10*a^8*b^8
- 10*a^9*b^7 + 5*a^10*b^6 - a^11*b^5)))^(1/2)*2i)/d