\(\int \frac {\sin ^6(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\) [231]

Optimal result

Integrand size = 24, antiderivative size = 343 \[ \int \frac {\sin ^6(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\left (4 a-10 \sqrt {a} \sqrt {b}+3 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/2} d}+\frac {\left (4 a+10 \sqrt {a} \sqrt {b}+3 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/2} d}-\frac {\tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\tan (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}+\frac {\left (2 a^2+15 a b+3 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]

[Out]

-1/64*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(4*a+3*b-10*a^(1/2)*b^(1/2))/a^(5/4)/b^(3/2)/d/(a^(1/
2)-b^(1/2))^(5/2)+1/64*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(4*a+3*b+10*a^(1/2)*b^(1/2))/a^(5/4)
/b^(3/2)/d/(a^(1/2)+b^(1/2))^(5/2)-1/8*tan(d*x+c)*(a*(a+3*b)+(a^2+6*a*b+b^2)*tan(d*x+c)^2)/(a-b)^3/d/(a+2*a*ta
n(d*x+c)^2+(a-b)*tan(d*x+c)^4)^2-1/32*tan(d*x+c)*(2*a*(a^2-a*b-8*b^2)/(a-b)^3+(2*a^2+15*a*b+3*b^2)*tan(d*x+c)^
2/(a-b)^2)/a/b/d/(a+2*a*tan(d*x+c)^2+(a-b)*tan(d*x+c)^4)

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3296, 1347, 1692, 1180, 211} \[ \int \frac {\sin ^6(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\left (-10 \sqrt {a} \sqrt {b}+4 a+3 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} b^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\left (10 \sqrt {a} \sqrt {b}+4 a+3 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} b^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\tan (c+d x) \left (\frac {\left (2 a^2+15 a b+3 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}\right )}{32 a b d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\tan (c+d x) \left (\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)+a (a+3 b)\right )}{8 d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2} \]

[In]

Int[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

-1/64*((4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(a^(5/4)*(Sqrt
[a] - Sqrt[b])^(5/2)*b^(3/2)*d) + ((4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*
x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) - (Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b + b
^2)*Tan[c + d*x]^2))/(8*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((2*a
*(a^2 - a*b - 8*b^2))/(a - b)^3 + ((2*a^2 + 15*a*b + 3*b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a*b*d*(a + 2*a*Tan
[c + d*x]^2 + (a - b)*Tan[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 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 1347

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coe
ff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(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)*Simp[ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(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], x]] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]

Rule 1692

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 +
c*x^4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*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)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 3296

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^6 \left (1+x^2\right )^2}{\left (a+2 a x^2+(a-b) x^4\right )^3} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {\tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-\frac {2 a^3 b (a+3 b)}{(a-b)^3}+\frac {2 a^2 b \left (5 a^2+6 a b-3 b^2\right ) x^2}{(a-b)^3}+\frac {32 a^2 b^2 x^4}{(a-b)^2}-\frac {16 a^2 b x^6}{a-b}}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{16 a^2 b d} \\ & = -\frac {\tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\tan (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}+\frac {\left (2 a^2+15 a b+3 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {\frac {8 a^4 b (a+2 b)}{(a-b)^2}-\frac {4 a^3 b \left (2 a^2-17 a b+3 b^2\right ) x^2}{(a-b)^2}}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{128 a^4 b^2 d} \\ & = -\frac {\tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\tan (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}+\frac {\left (2 a^2+15 a b+3 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) \left (4 a-10 \sqrt {a} \sqrt {b}+3 b\right )\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 a \left (\sqrt {a}-\sqrt {b}\right )^2 b^{3/2} d}+\frac {\left (-\frac {2 a^3 b \left (2 a^2-17 a b+3 b^2\right )}{(a-b)^2}+\frac {\frac {16 a^4 b (a+2 b)}{a-b}+\frac {8 a^4 b \left (2 a^2-17 a b+3 b^2\right )}{(a-b)^2}}{4 \sqrt {a} \sqrt {b}}\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{128 a^4 b^2 d} \\ & = -\frac {\left (4 a-10 \sqrt {a} \sqrt {b}+3 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/2} d}+\frac {\left (4 a+10 \sqrt {a} \sqrt {b}+3 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/2} d}-\frac {\tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\tan (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}+\frac {\left (2 a^2+15 a b+3 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.23 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^6(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {\frac {\left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {b} \left (4 a+10 \sqrt {a} \sqrt {b}+3 b\right ) \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{a \sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {b} \left (4 a-10 \sqrt {a} \sqrt {b}+3 b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{a \sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {4 b \left (4 a^2-19 a b-3 b^2+3 b (a+b) \cos (2 (c+d x))\right ) \sin (2 (c+d x))}{a (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}-\frac {128 (a-b) b (2 a+b-b \cos (2 (c+d x))) \sin (2 (c+d x))}{(-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}}{64 (a-b)^2 b^2 d} \]

[In]

Integrate[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

(((Sqrt[a] - Sqrt[b])^2*Sqrt[b]*(4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqr
t[a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[a + Sqrt[a]*Sqrt[b]]) + ((Sqrt[a] + Sqrt[b])^2*Sqrt[b]*(4*a - 10*Sqrt[a]*Sqrt
[b] + 3*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[-a + Sqrt[a]*Sqrt[b
]]) + (4*b*(4*a^2 - 19*a*b - 3*b^2 + 3*b*(a + b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(a*(8*a - 3*b + 4*b*Cos[2
*(c + d*x)] - b*Cos[4*(c + d*x)])) - (128*(a - b)*b*(2*a + b - b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(-8*a + 3
*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2)/(64*(a - b)^2*b^2*d)

Maple [A] (verified)

Time = 4.79 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.26

method	result	size
derivativedivides	\(\frac {\frac {-\frac {\left (2 a^{2}+15 a b +3 b^{2}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{32 \left (a -b \right ) a b}-\frac {\left (3 a^{2}+14 a b -5 b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (6 a^{2}+19 a b -b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (a +2 b \right ) \tan \left (d x +c \right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}}{{\left (\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right )}^{2}}+\frac {\left (a -b \right ) \left (\frac {\left (-2 a^{2} \sqrt {a b}+17 a b \sqrt {a b}-3 b^{2} \sqrt {a b}+4 a^{3}-15 a^{2} b -a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (-2 a^{2} \sqrt {a b}+17 a b \sqrt {a b}-3 b^{2} \sqrt {a b}-4 a^{3}+15 a^{2} b +a \,b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{32 a b \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(432\)
default	\(\frac {\frac {-\frac {\left (2 a^{2}+15 a b +3 b^{2}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{32 \left (a -b \right ) a b}-\frac {\left (3 a^{2}+14 a b -5 b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (6 a^{2}+19 a b -b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (a +2 b \right ) \tan \left (d x +c \right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}}{{\left (\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right )}^{2}}+\frac {\left (a -b \right ) \left (\frac {\left (-2 a^{2} \sqrt {a b}+17 a b \sqrt {a b}-3 b^{2} \sqrt {a b}+4 a^{3}-15 a^{2} b -a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (-2 a^{2} \sqrt {a b}+17 a b \sqrt {a b}-3 b^{2} \sqrt {a b}-4 a^{3}+15 a^{2} b +a \,b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{32 a b \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(432\)
risch	\(\text {Expression too large to display}\)	\(2377\)

[In]

int(sin(d*x+c)^6/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*((-1/32*(2*a^2+15*a*b+3*b^2)/(a-b)/a/b*tan(d*x+c)^7-1/16*(3*a^2+14*a*b-5*b^2)/b/(a^2-2*a*b+b^2)*tan(d*x+c)
^5-1/32*(6*a^2+19*a*b-b^2)/b/(a^2-2*a*b+b^2)*tan(d*x+c)^3-1/16*a*(a+2*b)/b/(a^2-2*a*b+b^2)*tan(d*x+c))/(tan(d*
x+c)^4*a-b*tan(d*x+c)^4+2*a*tan(d*x+c)^2+a)^2+1/32/a/b/(a^2-2*a*b+b^2)*(a-b)*(1/2*(-2*a^2*(a*b)^(1/2)+17*a*b*(
a*b)^(1/2)-3*b^2*(a*b)^(1/2)+4*a^3-15*a^2*b-a*b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a
+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+1/2*(-2*a^2*(a*b)^(1/2)+17*a*b*(a*b)^(1/2)-3*b^2*(a*b)^(1/2)-4*a
^3+15*a^2*b+a*b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a
-b))^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5961 vs. \(2 (291) = 582\).

Time = 2.03 (sec) , antiderivative size = 5961, normalized size of antiderivative = 17.38 \[ \int \frac {\sin ^6(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^6(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(sin(d*x+c)**6/(a-b*sin(d*x+c)**4)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sin ^6(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{6}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]

[In]

integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/16*(4*(32*a^3*b^2 - 84*a^2*b^3 - 83*a*b^4 + 21*b^5)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + ((4*a^2*b^3 - 13*a*
b^4 + 3*b^5)*sin(14*d*x + 14*c) - 3*(8*a^2*b^3 - 33*a*b^4 + 7*b^5)*sin(12*d*x + 12*c) + (64*a^3*b^2 + 68*a^2*b
^3 - 225*a*b^4 + 63*b^5)*sin(10*d*x + 10*c) - 3*(128*a^3*b^2 + 32*a^2*b^3 - 61*a*b^4 + 35*b^5)*sin(8*d*x + 8*c
) - (64*a^3*b^2 + 452*a^2*b^3 - 9*a*b^4 - 105*b^5)*sin(6*d*x + 6*c) + 3*(40*a^2*b^3 - 29*a*b^4 - 21*b^5)*sin(4
*d*x + 4*c) - (4*a^2*b^3 - 37*a*b^4 - 21*b^5)*sin(2*d*x + 2*c))*cos(16*d*x + 16*c) + 2*(2*(32*a^3*b^2 - 84*a^2
*b^3 - 83*a*b^4 + 21*b^5)*sin(12*d*x + 12*c) - 8*(64*a^3*b^2 - 84*a^2*b^3 - 43*a*b^4 + 21*b^5)*sin(10*d*x + 10
*c) - (512*a^4*b - 3584*a^3*b^2 + 1388*a^2*b^3 - 11*a*b^4 - 315*b^5)*sin(8*d*x + 8*c) + 16*(172*a^2*b^3 - 37*a
*b^4 - 21*b^5)*sin(6*d*x + 6*c) + 2*(32*a^3*b^2 - 372*a^2*b^3 + 289*a*b^4 + 105*b^5)*sin(4*d*x + 4*c) + 8*(4*a
^2*b^3 - 25*a*b^4 - 9*b^5)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) - 2*(2*(512*a^4*b - 672*a^3*b^2 + 1228*a^2*b^3
+ 21*a*b^4 - 147*b^5)*sin(10*d*x + 10*c) - 3*(3072*a^4*b - 6272*a^3*b^2 + 2920*a^2*b^3 - 413*a*b^4 - 245*b^5)
*sin(8*d*x + 8*c) - 2*(512*a^4*b + 3936*a^3*b^2 - 6740*a^2*b^3 + 1281*a*b^4 + 441*b^5)*sin(6*d*x + 6*c) + 12*(
192*a^3*b^2 - 416*a^2*b^3 + 161*a*b^4 + 49*b^5)*sin(4*d*x + 4*c) - 2*(32*a^3*b^2 - 372*a^2*b^3 + 289*a*b^4 + 1
05*b^5)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) - 2*((8192*a^5 + 27136*a^4*b - 37696*a^3*b^2 + 17644*a^2*b^3 - 20
79*a*b^4 - 735*b^5)*sin(8*d*x + 8*c) + 8*(1024*a^4*b + 3712*a^3*b^2 - 3692*a^2*b^3 + 483*a*b^4 + 147*b^5)*sin(
6*d*x + 6*c) - 2*(512*a^4*b + 3936*a^3*b^2 - 6740*a^2*b^3 + 1281*a*b^4 + 441*b^5)*sin(4*d*x + 4*c) - 16*(172*a
^2*b^3 - 37*a*b^4 - 21*b^5)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) - 2*((8192*a^5 + 27136*a^4*b - 37696*a^3*b^2
+ 17644*a^2*b^3 - 2079*a*b^4 - 735*b^5)*sin(6*d*x + 6*c) - 3*(3072*a^4*b - 6272*a^3*b^2 + 2920*a^2*b^3 - 413*a
*b^4 - 245*b^5)*sin(4*d*x + 4*c) + (512*a^4*b - 3584*a^3*b^2 + 1388*a^2*b^3 - 11*a*b^4 - 315*b^5)*sin(2*d*x +
2*c))*cos(8*d*x + 8*c) - 4*((512*a^4*b - 672*a^3*b^2 + 1228*a^2*b^3 + 21*a*b^4 - 147*b^5)*sin(4*d*x + 4*c) + 4
*(64*a^3*b^2 - 84*a^2*b^3 - 43*a*b^4 + 21*b^5)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 16*((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 - 24
0*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 - 6
7648*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 - 2
10*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*(6
4*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 + 837
12*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 + 3
37*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*(102
4*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*si
n(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 - 35
2*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) - (1
28*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*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 - 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*(4*a^2*b - 13*a*b^2 + 3*b^3)*cos(6*d*x + 6*c)^2 + 12*(56*a^2*b - 29*a*b
^2 + 3*b^3)*cos(4*d*x + 4*c)^2 + 4*(4*a^2*b - 13*a*b^2 + 3*b^3)*cos(2*d*x + 2*c)^2 + 4*(4*a^2*b - 13*a*b^2 + 3
*b^3)*sin(6*d*x + 6*c)^2 + 12*(56*a^2*b - 29*a*b^2 + 3*b^3)*sin(4*d*x + 4*c)^2 + 2*(32*a^3 - 116*a^2*b + 147*a
*b^2 - 21*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(4*a^2*b - 13*a*b^2 + 3*b^3)*sin(2*d*x + 2*c)^2 - ((4*a^2
*b - 13*a*b^2 + 3*b^3)*cos(6*d*x + 6*c) + 6*(7*a*b^2 - b^3)*cos(4*d*x + 4*c) + (4*a^2*b - 13*a*b^2 + 3*b^3)*co
s(2*d*x + 2*c))*cos(8*d*x + 8*c) - (4*a^2*b - 13*a*b^2 + 3*b^3 - 2*(32*a^3 - 116*a^2*b + 147*a*b^2 - 21*b^3)*c
os(4*d*x + 4*c) - 8*(4*a^2*b - 13*a*b^2 + 3*b^3)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 2*(21*a*b^2 - 3*b^3 - (3
2*a^3 - 116*a^2*b + 147*a*b^2 - 21*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (4*a^2*b - 13*a*b^2 + 3*b^3)*cos(
2*d*x + 2*c) - ((4*a^2*b - 13*a*b^2 + 3*b^3)*sin(6*d*x + 6*c) + 6*(7*a*b^2 - b^3)*sin(4*d*x + 4*c) + (4*a^2*b
- 13*a*b^2 + 3*b^3)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*((32*a^3 - 116*a^2*b + 147*a*b^2 - 21*b^3)*sin(4*d*
x + 4*c) + 4*(4*a^2*b - 13*a*b^2 + 3*b^3)*sin(2*d*x + 2*c))*sin(6*d*x + 6*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)*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(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)*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(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)*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(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) + (3*a*b^4 + 3*b^5 - (4*a^2*b^3 - 13*a*b^4 + 3*b^5)
*cos(14*d*x + 14*c) + 3*(8*a^2*b^3 - 33*a*b^4 + 7*b^5)*cos(12*d*x + 12*c) - (64*a^3*b^2 + 68*a^2*b^3 - 225*a*b
^4 + 63*b^5)*cos(10*d*x + 10*c) + 3*(128*a^3*b^2 + 32*a^2*b^3 - 61*a*b^4 + 35*b^5)*cos(8*d*x + 8*c) + (64*a^3*
b^2 + 452*a^2*b^3 - 9*a*b^4 - 105*b^5)*cos(6*d*x + 6*c) - 3*(40*a^2*b^3 - 29*a*b^4 - 21*b^5)*cos(4*d*x + 4*c)
+ (4*a^2*b^3 - 37*a*b^4 - 21*b^5)*cos(2*d*x + 2*c))*sin(16*d*x + 16*c) + (4*a^2*b^3 - 37*a*b^4 - 21*b^5 - 4*(3
2*a^3*b^2 - 84*a^2*b^3 - 83*a*b^4 + 21*b^5)*cos(12*d*x + 12*c) + 16*(64*a^3*b^2 - 84*a^2*b^3 - 43*a*b^4 + 21*b
^5)*cos(10*d*x + 10*c) + 2*(512*a^4*b - 3584*a^3*b^2 + 1388*a^2*b^3 - 11*a*b^4 - 315*b^5)*cos(8*d*x + 8*c) - 3
2*(172*a^2*b^3 - 37*a*b^4 - 21*b^5)*cos(6*d*x + 6*c) - 4*(32*a^3*b^2 - 372*a^2*b^3 + 289*a*b^4 + 105*b^5)*cos(
4*d*x + 4*c) - 16*(4*a^2*b^3 - 25*a*b^4 - 9*b^5)*cos(2*d*x + 2*c))*sin(14*d*x + 14*c) - (120*a^2*b^3 - 87*a*b^
4 - 63*b^5 - 4*(512*a^4*b - 672*a^3*b^2 + 1228*a^2*b^3 + 21*a*b^4 - 147*b^5)*cos(10*d*x + 10*c) + 6*(3072*a^4*
b - 6272*a^3*b^2 + 2920*a^2*b^3 - 413*a*b^4 - 245*b^5)*cos(8*d*x + 8*c) + 4*(512*a^4*b + 3936*a^3*b^2 - 6740*a
^2*b^3 + 1281*a*b^4 + 441*b^5)*cos(6*d*x + 6*c) - 24*(192*a^3*b^2 - 416*a^2*b^3 + 161*a*b^4 + 49*b^5)*cos(4*d*
x + 4*c) + 4*(32*a^3*b^2 - 372*a^2*b^3 + 289*a*b^4 + 105*b^5)*cos(2*d*x + 2*c))*sin(12*d*x + 12*c) + (64*a^3*b
^2 + 452*a^2*b^3 - 9*a*b^4 - 105*b^5 + 2*(8192*a^5 + 27136*a^4*b - 37696*a^3*b^2 + 17644*a^2*b^3 - 2079*a*b^4
- 735*b^5)*cos(8*d*x + 8*c) + 16*(1024*a^4*b + 3712*a^3*b^2 - 3692*a^2*b^3 + 483*a*b^4 + 147*b^5)*cos(6*d*x +
6*c) - 4*(512*a^4*b + 3936*a^3*b^2 - 6740*a^2*b^3 + 1281*a*b^4 + 441*b^5)*cos(4*d*x + 4*c) - 32*(172*a^2*b^3 -
37*a*b^4 - 21*b^5)*cos(2*d*x + 2*c))*sin(10*d*x + 10*c) + (384*a^3*b^2 + 96*a^2*b^3 - 183*a*b^4 + 105*b^5 + 2
*(8192*a^5 + 27136*a^4*b - 37696*a^3*b^2 + 17644*a^2*b^3 - 2079*a*b^4 - 735*b^5)*cos(6*d*x + 6*c) - 6*(3072*a^
4*b - 6272*a^3*b^2 + 2920*a^2*b^3 - 413*a*b^4 - 245*b^5)*cos(4*d*x + 4*c) + 2*(512*a^4*b - 3584*a^3*b^2 + 1388
*a^2*b^3 - 11*a*b^4 - 315*b^5)*cos(2*d*x + 2*c))*sin(8*d*x + 8*c) - (64*a^3*b^2 + 68*a^2*b^3 - 225*a*b^4 + 63*
b^5 - 4*(512*a^4*b - 672*a^3*b^2 + 1228*a^2*b^3 + 21*a*b^4 - 147*b^5)*cos(4*d*x + 4*c) - 16*(64*a^3*b^2 - 84*a
^2*b^3 - 43*a*b^4 + 21*b^5)*cos(2*d*x + 2*c))*sin(6*d*x + 6*c) + (24*a^2*b^3 - 99*a*b^4 + 21*b^5 - 4*(32*a^3*b
^2 - 84*a^2*b^3 - 83*a*b^4 + 21*b^5)*cos(2*d*x + 2*c))*sin(4*d*x + 4*c) - (4*a^2*b^3 - 13*a*b^4 + 3*b^5)*sin(2
*d*x + 2*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 + 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 + 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 - 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 - 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*c
os(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*(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 - 24
5*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*si
n(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 + 172
2*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*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 - 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))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2231 vs. \(2 (291) = 582\).

Time = 1.40 (sec) , antiderivative size = 2231, normalized size of antiderivative = 6.50 \[ \int \frac {\sin ^6(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/64*(((6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4 - 63*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)
*a^3*b + 109*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(
a*b)*a*b^3 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3*b - 2*a^2*b^2 + a*b^3)^2*abs(-a + b) +
2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^8*b - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^7*b^2 - 4*sqrt(a^2 -
a*b + sqrt(a*b)*(a - b))*a^6*b^3 + 34*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^4 - 33*sqrt(a^2 - a*b + sqrt(a
*b)*(a - b))*a^4*b^5 + 7*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^6 + 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a
^2*b^7)*abs(a^3*b - 2*a^2*b^2 + a*b^3)*abs(-a + b) - (12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^11*b
- 117*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^2 + 431*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*
b)*a^9*b^3 - 773*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^4 + 703*sqrt(a^2 - a*b + sqrt(a*b)*(a - b
))*sqrt(a*b)*a^7*b^5 - 279*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^6 + 5*sqrt(a^2 - a*b + sqrt(a*b
)*(a - b))*sqrt(a*b)*a^5*b^7 + 17*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^8 + sqrt(a^2 - a*b + sqr
t(a*b)*(a - b))*sqrt(a*b)*a^3*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4*
b - 2*a^3*b^2 + a^2*b^3 + sqrt((a^4*b - 2*a^3*b^2 + a^2*b^3)^2 - (a^4*b - 2*a^3*b^2 + a^2*b^3)*(a^4*b - 3*a^3*
b^2 + 3*a^2*b^3 - a*b^4)))/(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4))))/((3*a^12*b^2 - 27*a^11*b^3 + 104*a^10*b^
4 - 224*a^9*b^5 + 294*a^8*b^6 - 238*a^7*b^7 + 112*a^6*b^8 - 24*a^5*b^9 - a^4*b^10 + a^3*b^11)*abs(a^3*b - 2*a^
2*b^2 + a*b^3)) - ((6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4 - 63*sqrt(a^2 - a*b - sqrt(a*b)*(a - b
))*sqrt(a*b)*a^3*b + 109*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a
- b))*sqrt(a*b)*a*b^3 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3*b - 2*a^2*b^2 + a*b^3)^2*ab
s(-a + b) - 2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^8*b - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^7*b^2 - 4
*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b^3 + 34*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^4 - 33*sqrt(a^2 -
a*b - sqrt(a*b)*(a - b))*a^4*b^5 + 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^6 + 2*sqrt(a^2 - a*b - sqrt(a*b
)*(a - b))*a^2*b^7)*abs(a^3*b - 2*a^2*b^2 + a*b^3)*abs(-a + b) - (12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(
a*b)*a^11*b - 117*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^2 + 431*sqrt(a^2 - a*b - sqrt(a*b)*(a -
b))*sqrt(a*b)*a^9*b^3 - 773*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^4 + 703*sqrt(a^2 - a*b - sqrt
(a*b)*(a - b))*sqrt(a*b)*a^7*b^5 - 279*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^6 + 5*sqrt(a^2 - a*
b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^7 + 17*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^8 + sqrt(a^2
- a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c
)/sqrt((a^4*b - 2*a^3*b^2 + a^2*b^3 - sqrt((a^4*b - 2*a^3*b^2 + a^2*b^3)^2 - (a^4*b - 2*a^3*b^2 + a^2*b^3)*(a^
4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)))/(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4))))/((3*a^12*b^2 - 27*a^11*b^3 +
104*a^10*b^4 - 224*a^9*b^5 + 294*a^8*b^6 - 238*a^7*b^7 + 112*a^6*b^8 - 24*a^5*b^9 - a^4*b^10 + a^3*b^11)*abs(
a^3*b - 2*a^2*b^2 + a*b^3)) - 2*(2*a^3*tan(d*x + c)^7 + 13*a^2*b*tan(d*x + c)^7 - 12*a*b^2*tan(d*x + c)^7 - 3*
b^3*tan(d*x + c)^7 + 6*a^3*tan(d*x + c)^5 + 28*a^2*b*tan(d*x + c)^5 - 10*a*b^2*tan(d*x + c)^5 + 6*a^3*tan(d*x
+ c)^3 + 19*a^2*b*tan(d*x + c)^3 - a*b^2*tan(d*x + c)^3 + 2*a^3*tan(d*x + c) + 4*a^2*b*tan(d*x + c))/((a*tan(d
*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)^2*(a^3*b - 2*a^2*b^2 + a*b^3)))/d

Mupad [B] (verification not implemented)

Time = 20.09 (sec) , antiderivative size = 6391, normalized size of antiderivative = 18.63 \[ \int \frac {\sin ^6(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

int(sin(c + d*x)^6/(a - b*sin(c + d*x)^4)^3,x)

[Out]

- ((tan(c + d*x)^3*(19*a*b + 6*a^2 - b^2))/(32*(a^2*b - 2*a*b^2 + b^3)) + (a*tan(c + d*x)*(a + 2*b))/(16*(a^2*
b - 2*a*b^2 + b^3)) + (tan(c + d*x)^7*(15*a*b + 2*a^2 + 3*b^2))/(32*a*(a*b - b^2)) + (tan(c + d*x)^5*(14*a*b +
3*a^2 - 5*b^2))/(16*(a - b)*(a*b - b^2)))/(d*(tan(c + d*x)^8*(a^2 - 2*a*b + b^2) + a^2 - tan(c + d*x)^4*(2*a*
b - 6*a^2) - tan(c + d*x)^6*(4*a*b - 4*a^2) + 4*a^2*tan(c + d*x)^2)) - (atan(((((65536*a^3*b^7 - 163840*a^4*b^
6 + 98304*a^5*b^5 + 32768*a^6*b^4 - 32768*a^7*b^3)/(32768*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)) - (tan(
c + d*x)*((9*b^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^
4 + 16*a^7*b^3 - 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*
b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*(16384*a^3*b^8 - 81920*a^4*b^7 + 163840*a^5*b^6 - 163840*a^6*
b^5 + 81920*a^7*b^4 - 16384*a^8*b^3))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((9*b^3*(a^5*b^9)^(1/2)
- 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 - 86*a*b^2*(a^5*b
^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a
^10*b^6)))^(1/2) + (tan(c + d*x)*(16*a^5 - 116*a^4*b - 101*a*b^4 + 9*b^5 + 331*a^2*b^3 + 149*a^3*b^2))/(256*(a
*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((9*b^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a
^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 - 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2))/(16384
*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*1i - (((65536*a^3*b^7 - 1638
40*a^4*b^6 + 98304*a^5*b^5 + 32768*a^6*b^4 - 32768*a^7*b^3)/(32768*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)
) + (tan(c + d*x)*((9*b^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 1
16*a^6*b^4 + 16*a^7*b^3 - 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10
+ 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*(16384*a^3*b^8 - 81920*a^4*b^7 + 163840*a^5*b^6 - 16
3840*a^6*b^5 + 81920*a^7*b^4 - 16384*a^8*b^3))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((9*b^3*(a^5*b
^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 - 86*a*b
^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^
9*b^7 - a^10*b^6)))^(1/2) - (tan(c + d*x)*(16*a^5 - 116*a^4*b - 101*a*b^4 + 9*b^5 + 331*a^2*b^3 + 149*a^3*b^2)
)/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((9*b^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b
^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 - 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2
))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*1i)/((((65536*a^3*b
^7 - 163840*a^4*b^6 + 98304*a^5*b^5 + 32768*a^6*b^4 - 32768*a^7*b^3)/(32768*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 -
a^5*b^3)) - (tan(c + d*x)*((9*b^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^
5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 - 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*
a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*(16384*a^3*b^8 - 81920*a^4*b^7 + 163840*a^5
*b^6 - 163840*a^6*b^5 + 81920*a^7*b^4 - 16384*a^8*b^3))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((9*b
^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3
- 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b
^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2) + (tan(c + d*x)*(16*a^5 - 116*a^4*b - 101*a*b^4 + 9*b^5 + 331*a^2*b^3 + 149
*a^3*b^2))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((9*b^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) -
15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 - 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*
b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2) - (32*a^4
- 424*a^3*b - 381*a*b^3 + 27*b^4 + 1358*a^2*b^2)/(16384*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)) + (((6553
6*a^3*b^7 - 163840*a^4*b^6 + 98304*a^5*b^5 + 32768*a^6*b^4 - 32768*a^7*b^3)/(32768*(a^2*b^6 - 3*a^3*b^5 + 3*a^
4*b^4 - a^5*b^3)) + (tan(c + d*x)*((9*b^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 +
229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 - 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^
11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*(16384*a^3*b^8 - 81920*a^4*b^7 + 163
840*a^5*b^6 - 163840*a^6*b^5 + 81920*a^7*b^4 - 16384*a^8*b^3))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2))
)*((9*b^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*
a^7*b^3 - 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 1
0*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2) - (tan(c + d*x)*(16*a^5 - 116*a^4*b - 101*a*b^4 + 9*b^5 + 331*a^2*b^
3 + 149*a^3*b^2))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((9*b^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^
(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 - 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*
b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)))*(
(9*b^3*(a^5*b^9)^(1/2) - 80*a^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7
*b^3 - 86*a*b^2*(a^5*b^9)^(1/2) + 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a
^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*2i)/d - (atan(((((65536*a^3*b^7 - 163840*a^4*b^6 + 98304*a^5*b^5 + 3276
8*a^6*b^4 - 32768*a^7*b^3)/(32768*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)) - (tan(c + d*x)*((80*a^3*(a^5*b
^9)^(1/2) - 9*b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 + 86*a*b^
2*(a^5*b^9)^(1/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9
*b^7 - a^10*b^6)))^(1/2)*(16384*a^3*b^8 - 81920*a^4*b^7 + 163840*a^5*b^6 - 163840*a^6*b^5 + 81920*a^7*b^4 - 16
384*a^8*b^3))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((80*a^3*(a^5*b^9)^(1/2) - 9*b^3*(a^5*b^9)^(1/2
) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 + 86*a*b^2*(a^5*b^9)^(1/2) - 301*a^2*b*(a
^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2) + (tan(
c + d*x)*(16*a^5 - 116*a^4*b - 101*a*b^4 + 9*b^5 + 331*a^2*b^3 + 149*a^3*b^2))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3
*b^3 - a^4*b^2)))*((80*a^3*(a^5*b^9)^(1/2) - 9*b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 1
16*a^6*b^4 + 16*a^7*b^3 + 86*a*b^2*(a^5*b^9)^(1/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10
+ 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*1i - (((65536*a^3*b^7 - 163840*a^4*b^6 + 98304*a^5*b
^5 + 32768*a^6*b^4 - 32768*a^7*b^3)/(32768*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)) + (tan(c + d*x)*((80*a
^3*(a^5*b^9)^(1/2) - 9*b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3
+ 86*a*b^2*(a^5*b^9)^(1/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^
8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*(16384*a^3*b^8 - 81920*a^4*b^7 + 163840*a^5*b^6 - 163840*a^6*b^5 + 81920*a^7
*b^4 - 16384*a^8*b^3))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((80*a^3*(a^5*b^9)^(1/2) - 9*b^3*(a^5*
b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 + 86*a*b^2*(a^5*b^9)^(1/2) - 301
*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2
) - (tan(c + d*x)*(16*a^5 - 116*a^4*b - 101*a*b^4 + 9*b^5 + 331*a^2*b^3 + 149*a^3*b^2))/(256*(a*b^5 - 3*a^2*b^
4 + 3*a^3*b^3 - a^4*b^2)))*((80*a^3*(a^5*b^9)^(1/2) - 9*b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^
5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 + 86*a*b^2*(a^5*b^9)^(1/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*
a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*1i)/((((65536*a^3*b^7 - 163840*a^4*b^6 + 98
304*a^5*b^5 + 32768*a^6*b^4 - 32768*a^7*b^3)/(32768*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)) - (tan(c + d*
x)*((80*a^3*(a^5*b^9)^(1/2) - 9*b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16
*a^7*b^3 + 86*a*b^2*(a^5*b^9)^(1/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 -
10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*(16384*a^3*b^8 - 81920*a^4*b^7 + 163840*a^5*b^6 - 163840*a^6*b^5 +
81920*a^7*b^4 - 16384*a^8*b^3))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((80*a^3*(a^5*b^9)^(1/2) - 9*
b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 + 86*a*b^2*(a^5*b^9)^(1
/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^
6)))^(1/2) + (tan(c + d*x)*(16*a^5 - 116*a^4*b - 101*a*b^4 + 9*b^5 + 331*a^2*b^3 + 149*a^3*b^2))/(256*(a*b^5 -
3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((80*a^3*(a^5*b^9)^(1/2) - 9*b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6
+ 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 + 86*a*b^2*(a^5*b^9)^(1/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*
b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2) - (32*a^4 - 424*a^3*b - 381*a*b^3
+ 27*b^4 + 1358*a^2*b^2)/(16384*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)) + (((65536*a^3*b^7 - 163840*a^4*b
^6 + 98304*a^5*b^5 + 32768*a^6*b^4 - 32768*a^7*b^3)/(32768*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)) + (tan
(c + d*x)*((80*a^3*(a^5*b^9)^(1/2) - 9*b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b
^4 + 16*a^7*b^3 + 86*a*b^2*(a^5*b^9)^(1/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7
*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)*(16384*a^3*b^8 - 81920*a^4*b^7 + 163840*a^5*b^6 - 163840*a^6
*b^5 + 81920*a^7*b^4 - 16384*a^8*b^3))/(256*(a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((80*a^3*(a^5*b^9)^(1/
2) - 9*b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 + 86*a*b^2*(a^5*
b^9)^(1/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 -
a^10*b^6)))^(1/2) - (tan(c + d*x)*(16*a^5 - 116*a^4*b - 101*a*b^4 + 9*b^5 + 331*a^2*b^3 + 149*a^3*b^2))/(256*(
a*b^5 - 3*a^2*b^4 + 3*a^3*b^3 - a^4*b^2)))*((80*a^3*(a^5*b^9)^(1/2) - 9*b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*
a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 + 86*a*b^2*(a^5*b^9)^(1/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(1638
4*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^10*b^6)))^(1/2)))*((80*a^3*(a^5*b^9)^(1/2)
- 9*b^3*(a^5*b^9)^(1/2) - 15*a^3*b^7 + 30*a^4*b^6 + 229*a^5*b^5 - 116*a^6*b^4 + 16*a^7*b^3 + 86*a*b^2*(a^5*b^9
)^(1/2) - 301*a^2*b*(a^5*b^9)^(1/2))/(16384*(a^5*b^11 - 5*a^6*b^10 + 10*a^7*b^9 - 10*a^8*b^8 + 5*a^9*b^7 - a^1
0*b^6)))^(1/2)*2i)/d