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

Optimal. Leaf size=313 \[ \frac {\left (-10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} b^{5/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\left (10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} b^{5/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\cos (c+d x) \left (a^2+2 b (2 a+b) \cos ^2(c+d x)-11 a b-2 b^2\right )}{32 a b d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

[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)

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Rubi [A]  time = 0.47, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3215, 1205, 1178, 1166, 205, 208} \[ \frac {\cos (c+d x) \left (a^2+2 b (2 a+b) \cos ^2(c+d x)-11 a b-2 b^2\right )}{32 a b d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}+\frac {\left (-10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} b^{5/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\left (10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} b^{5/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

Antiderivative was successfully verified.

[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 205

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

Rule 208

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

Rule 1166

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 1178

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 1205

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 3215

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*} \int \frac {\sin ^5(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=-\frac {\operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \tan ^{-1}\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 ) \tanh ^{-1}\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*}

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Mathematica [C]  time = 1.40, size = 786, normalized size = 2.51 \[ \frac {\frac {i \text {RootSum}\left [\text {$\#$1}^8 b-4 \text {$\#$1}^6 b-16 \text {$\#$1}^4 a+6 \text {$\#$1}^4 b-4 \text {$\#$1}^2 b+b\& ,\frac {-4 \text {$\#$1}^6 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-2 \text {$\#$1}^6 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-12 \text {$\#$1}^4 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+64 \text {$\#$1}^4 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-10 \text {$\#$1}^4 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-6 i \text {$\#$1}^2 a^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+12 \text {$\#$1}^2 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+32 i \text {$\#$1}^2 a b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-2 i a b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-64 \text {$\#$1}^2 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-5 i \text {$\#$1}^2 b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-i b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+10 \text {$\#$1}^2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 i \text {$\#$1}^6 a b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+i \text {$\#$1}^6 b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+6 i \text {$\#$1}^4 a^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-32 i \text {$\#$1}^4 a b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+5 i \text {$\#$1}^4 b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+4 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )}{\text {$\#$1}^7 b-3 \text {$\#$1}^5 b-8 \text {$\#$1}^3 a+3 \text {$\#$1}^3 b-\text {$\#$1} b}\& \right ]}{a}+\frac {32 \cos (c+d x) \left (a^2+b (2 a+b) \cos (2 (c+d x))-9 a b-b^2\right )}{a (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}-\frac {512 (a-b) \cos (c+d x) (2 a-b \cos (2 (c+d x))+b)}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{128 b d (a-b)^2} \]

Warning: Unable to verify antiderivative.

[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
)

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fricas [B]  time = 1.47, size = 4524, normalized size = 14.45 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the
root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[40,31]Warning,
need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done
assuming [a,b]=[7,-19]-2/d*(-3*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^3+13*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^2*
b-4*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a*b^2-21*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^3+99*((1-cos(c+d*x))/(1+cos
(c+d*x)))^6*a^2*b-24*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a*b^2-63*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^3+225*((1-
cos(c+d*x))/(1+cos(c+d*x)))^5*a^2*b-68*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a*b^2-64*((1-cos(c+d*x))/(1+cos(c+d*x
)))^5*b^3-105*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^3+183*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^2*b-96*((1-cos(c+d
*x))/(1+cos(c+d*x)))^4*a*b^2-384*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*b^3-105*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a
^3-9*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^2*b+452*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a*b^2+64*((1-cos(c+d*x))/(1
+cos(c+d*x)))^3*b^3-63*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^3-87*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^2*b+120*((
1-cos(c+d*x))/(1+cos(c+d*x)))^2*a*b^2-21*(1-cos(c+d*x))/(1+cos(c+d*x))*a^3-37*(1-cos(c+d*x))/(1+cos(c+d*x))*a^
2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a*b^2-3*a^3-3*a^2*b)/(-32*a^3*b+64*a^2*b^2-32*a*b^3)/(((1-cos(c+d*x))/(1+c
os(c+d*x)))^4*a+4*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a+6*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))
/(1+cos(c+d*x)))^2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a+a)^2-2/d/(32*a^3*b-64*a^2*b^2+32*a*b^3)*2/d*(-(2*a+b)/2
*(c+d*x)+(-6*a^5*b+62*a^4*b^2+30*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+6*a^4*a*b-9*a^4*sqrt(a*b)*sqrt(a^2-a*b+sq
rt(a*b)*(a-b))-178*a^3*b^3-132*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-62*a^3*b*a*b+33*a^3*b*sqrt(a*b)*sqrt(a^2-
a*b+sqrt(a*b)*(a-b))+170*a^2*b^4+158*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+178*a^2*b^2*a*b-51*a^2*b^2*sqrt(a*b
)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-48*a*b^5-24*a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-170*a*b^3*a*b+43*a*b^3*sqrt(a*
b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-8*b^5*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+48*b^4*a*b+8*b^4*sqrt(a*b)*sqrt(a^2-a*b+s
qrt(a*b)*(a-b)))*abs(a-b)/(24*a^5*b-96*a^4*b^2+112*a^3*b^3-32*a^2*b^4-8*a*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a+sqr
t(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/2))-(-6*a^5*b+62*a^4*b^2-30*a^4*b*sqrt(a^2-a
*b+sqrt(a*b)*(-a+b))+6*a^4*a*b-9*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-178*a^3*b^3+132*a^3*b^2*sqrt(a^2
-a*b+sqrt(a*b)*(-a+b))-62*a^3*b*a*b+33*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+170*a^2*b^4-158*a^2*b^3*
sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+178*a^2*b^2*a*b-51*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-48*a*b^5+24
*a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-170*a*b^3*a*b+43*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+8*b^5*sq
rt(a^2-a*b+sqrt(a*b)*(-a+b))+48*b^4*a*b+8*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)/(24*a^5*b-96*
a^4*b^2+112*a^3*b^3-32*a^2*b^4-8*a*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a-sqrt(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*
b)))+pi*floor((c+d*x)/pi+1/2)))

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maple [B]  time = 0.39, size = 1167, normalized size = 3.73 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^7*b-1/16/d/(b*cos(d*x+c)^4-2*b*cos(d
*x+c)^2-a+b)^2*b^2/a/(a^2-2*a*b+b^2)*cos(d*x+c)^7-1/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a^2-2*a*b+b^
2)*cos(d*x+c)^5*a+19/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2*b/(a^2-2*a*b+b^2)*cos(d*x+c)^5+3/16/d/(b*cos
(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/a/(a^2-2*a*b+b^2)*cos(d*x+c)^5*b^2+5/16/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a
+b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3*a-7/8/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2*b/(a^2-2*a*b+b^2)*cos(d*x+c
)^3-3/16/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/a/(a^2-2*a*b+b^2)*cos(d*x+c)^3*b^2-3/32/d/(b*cos(d*x+c)^4-2
*b*cos(d*x+c)^2-a+b)^2/b/(a-b)*cos(d*x+c)*a-15/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a-b)*cos(d*x+c)-1
/16/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a-b)/a*b*cos(d*x+c)+1/16/d/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^
(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))+1/32/d/a/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arcta
nh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))*b+3/64/d/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2)*arct
anh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))*a-13/64/d/(a^2-2*a*b+b^2)*b/(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))+1/16/d/a/(a^2-2*a*b+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))*b^2-1/16/d/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(c
os(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-1/32/d/a/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/
(((a*b)^(1/2)-b)*b)^(1/2))*b+3/64/d/(a^2-2*a*b+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))*a-13/64/d/(a^2-2*a*b+b^2)*b/(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))+1/16/d/a/(a^2-2*a*b+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))*b^2

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [B]  time = 20.15, size = 6362, normalized size = 20.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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