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

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

[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))

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Rubi [A]  time = 0.472061, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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]

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 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 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 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]

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*}

Mathematica [C]  time = 1.37601, size = 786, normalized size = 2.51 \[ \frac{\frac{i \text{RootSum}\left [-16 \text{$\#$1}^4 a+\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\& ,\frac{6 i \text{$\#$1}^4 a^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+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}^4 a^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+12 \text{$\#$1}^2 a^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 )-32 i \text{$\#$1}^4 a b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+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 )-4 \text{$\#$1}^6 a b \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 )-64 \text{$\#$1}^2 a b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+i \text{$\#$1}^6 b^2 \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 )-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 )-2 \text{$\#$1}^6 b^2 \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 )+10 \text{$\#$1}^2 b^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$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 )}{-8 \text{$\#$1}^3 a+\text{$\#$1}^7 b-3 \text{$\#$1}^5 b+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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Maple [B]  time = 0.125, size = 1167, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

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)*arctan(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)*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)*arc
tan(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)*arc
tan(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)*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)*arctanh(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)*arctanh(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

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Fricas [B]  time = 10.2006, size = 10311, normalized size = 32.94 \begin{align*} \text{result too large to display} \end{align*}

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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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: TypeError