3.230 \(\int \frac{\sin ^8(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\)

Optimal. Leaf size=319 \[ -\frac{\left (2 \sqrt{a}-5 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} b^{3/2} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\left (2 \sqrt{a}+5 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} b^{3/2} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}+\frac{\tan ^9(c+d x)}{8 a d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}+\frac{\tan ^3(c+d x)}{32 a b d (a-b)}-\frac{(a+5 b) \tan (c+d x)}{32 a b d (a-b)^2}-\frac{\tan ^5(c+d x) \sec ^2(c+d x)}{32 a b d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )} \]

[Out]

-((2*Sqrt[a] - 5*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(3/4)*(Sqrt[a] - Sqrt[
b])^(5/2)*b^(3/2)*d) + ((2*Sqrt[a] + 5*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^
(3/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) - ((a + 5*b)*Tan[c + d*x])/(32*a*(a - b)^2*b*d) + Tan[c + d*x]^3/(3
2*a*(a - b)*b*d) + Tan[c + d*x]^9/(8*a*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Sec[c + d*x]^
2*Tan[c + d*x]^5)/(32*a*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))

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Rubi [A]  time = 0.530171, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {3217, 1275, 12, 1120, 1279, 1166, 205} \[ -\frac{\left (2 \sqrt{a}-5 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} b^{3/2} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\left (2 \sqrt{a}+5 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} b^{3/2} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}+\frac{\tan ^9(c+d x)}{8 a d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}+\frac{\tan ^3(c+d x)}{32 a b d (a-b)}-\frac{(a+5 b) \tan (c+d x)}{32 a b d (a-b)^2}-\frac{\tan ^5(c+d x) \sec ^2(c+d x)}{32 a b d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

-((2*Sqrt[a] - 5*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(3/4)*(Sqrt[a] - Sqrt[
b])^(5/2)*b^(3/2)*d) + ((2*Sqrt[a] + 5*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^
(3/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) - ((a + 5*b)*Tan[c + d*x])/(32*a*(a - b)^2*b*d) + Tan[c + d*x]^3/(3
2*a*(a - b)*b*d) + Tan[c + d*x]^9/(8*a*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Sec[c + d*x]^
2*Tan[c + d*x]^5)/(32*a*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))

Rule 3217

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]

Rule 1275

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(f*
(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1)*(b*d - 2*a*e - (b*e - 2*c*d)*x^2))/(2*(p + 1)*(b^2 - 4*a*c)), x] - D
ist[f^2/(2*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1)*Simp[(m - 1)*(b*d - 2*a*e) -
(4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[
p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1120

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

Rule 1279

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(e*f
*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 3)), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m -
2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[
{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (Inte
gerQ[p] || IntegerQ[m])

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]

Rubi steps

\begin{align*} \int \frac{\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^8 \left (1+x^2\right )}{\left (a+2 a x^2+(a-b) x^4\right )^3} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int -\frac{2 b x^8}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{16 a b d}\\ &=\frac{\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{x^8}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{8 a d}\\ &=\frac{\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac{\sec ^2(c+d x) \tan ^5(c+d x)}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (10 a+6 a x^2\right )}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{64 a^2 b d}\\ &=\frac{\tan ^3(c+d x)}{32 a (a-b) b d}+\frac{\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac{\sec ^2(c+d x) \tan ^5(c+d x)}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (18 a^2+6 a (a+5 b) x^2\right )}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{192 a^2 (a-b) b d}\\ &=-\frac{(a+5 b) \tan (c+d x)}{32 a (a-b)^2 b d}+\frac{\tan ^3(c+d x)}{32 a (a-b) b d}+\frac{\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac{\sec ^2(c+d x) \tan ^5(c+d x)}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{6 a^2 (a+5 b)-6 a^2 (a-13 b) x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{192 a^2 (a-b)^2 b d}\\ &=-\frac{(a+5 b) \tan (c+d x)}{32 a (a-b)^2 b d}+\frac{\tan ^3(c+d x)}{32 a (a-b) b d}+\frac{\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac{\sec ^2(c+d x) \tan ^5(c+d x)}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac{\left (2 a+3 \sqrt{a} \sqrt{b}-5 b\right ) \operatorname{Subst}\left (\int \frac{1}{a-\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 \sqrt{a} \left (\sqrt{a}+\sqrt{b}\right )^2 b^{3/2} d}-\frac{\left (\left (2 \sqrt{a}-5 \sqrt{b}\right ) \left (\sqrt{a}+\sqrt{b}\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 \sqrt{a} (a-b)^2 b^{3/2} d}\\ &=-\frac{\left (2 \sqrt{a}-5 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} \left (\sqrt{a}-\sqrt{b}\right )^{5/2} b^{3/2} d}+\frac{\left (2 \sqrt{a}+5 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} \left (\sqrt{a}+\sqrt{b}\right )^{5/2} b^{3/2} d}-\frac{(a+5 b) \tan (c+d x)}{32 a (a-b)^2 b d}+\frac{\tan ^3(c+d x)}{32 a (a-b) b d}+\frac{\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac{\sec ^2(c+d x) \tan ^5(c+d x)}{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]  time = 4.01175, size = 331, normalized size = 1.04 \[ \frac{\frac{\left (2 a^{3/2} \sqrt{b}-8 \sqrt{a} b^{3/2}+a b+5 b^2\right ) \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{8 b \sin (2 (c+d x)) ((5 b-2 a) \cos (2 (c+d x))+5 a-14 b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}+\frac{64 a b (a-b) (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}+\frac{\sqrt{b} \left (2 \sqrt{a}-5 \sqrt{b}\right ) \left (\sqrt{a}+\sqrt{b}\right )^2 \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}}{64 b^2 d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

(((2*a^(3/2)*Sqrt[b] + a*b - 8*Sqrt[a]*b^(3/2) + 5*b^2)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqr
t[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + ((2*Sqrt[a] - 5*Sqrt[b])*(Sqrt[a] + Sqrt[b])^2*Sqrt[b]*A
rcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) +
(8*b*(5*a - 14*b + (-2*a + 5*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)]) + (64*a*(a - b)*b*(-6*Sin[2*(c + d*x)] + Sin[4*(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)

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Maple [B]  time = 0.135, size = 1634, normalized size = 5.1 \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)^8/(a-b*sin(d*x+c)^4)^3,x)

[Out]

-1/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a-b)/b*tan(d*x+c)^7*a-19/32/d/(tan(d*x+c)^4*a-ta
n(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a-b)*tan(d*x+c)^7-3/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a
)^2/b/(a^2-2*a*b+b^2)*tan(d*x+c)^5*a^2-15/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a^2-2*a*b
+b^2)*tan(d*x+c)^5*a+9/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*b/(a^2-2*a*b+b^2)*tan(d*x+c)^
5-3/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*a^2/b/(a^2-2*a*b+b^2)*tan(d*x+c)^3-21/32/d/(tan(
d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*a/(a^2-2*a*b+b^2)*tan(d*x+c)^3-1/32/d/(tan(d*x+c)^4*a-tan(d*x+
c)^4*b+2*a*tan(d*x+c)^2+a)^2*a^2/b/(a^2-2*a*b+b^2)*tan(d*x+c)-5/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*
x+c)^2+a)^2*a/(a^2-2*a*b+b^2)*tan(d*x+c)-1/64/d/b/(a^2-2*a*b+b^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))*a^2+7/32/d/(a^2-2*a*b+b^2)*a/(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))-1/32/d/b/(a^2-2*a*b+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))*a^3+11/64/d/(a^2-2*a*b+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))*a^2-1/16/d*b/(
a^2-2*a*b+b^2)*a/(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))-1/64/d/b/(a^2-2*a*b+b^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))*a^2+7/32/d/(a^2-2*a*b+b^2)*a/(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))+1/32/d/b/(a^2-2*a*b+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))*a^3-11/64/d/(a^2-2*a*b+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))*a^2+1/16/d*b/(a^2-2*a*b+b^2)*a/(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))-13/64/d*b/(
a^2-2*a*b+b^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))-5/64
/d*b^2/(a^2-2*a*b+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))-13/64/d*b/(a^2-2*a*b+b^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))+5/64/d*b^2/(a^2-2*a*b+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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \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)^8/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/8*(4*(72*a^2*b^2 - 155*a*b^3 + 26*b^4)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + ((a*b^3 - 4*b^4)*sin(14*d*x + 14
*c) - (32*a^2*b^2 - 58*a*b^3 - b^4)*sin(12*d*x + 12*c) + 3*(48*a^2*b^2 - 73*a*b^3 + 20*b^4)*sin(10*d*x + 10*c)
+ (256*a^3*b - 832*a^2*b^2 + 550*a*b^3 - 175*b^4)*sin(8*d*x + 8*c) + (112*a^2*b^2 - 533*a*b^3 + 220*b^4)*sin(
6*d*x + 6*c) - (32*a^2*b^2 - 158*a*b^3 + 141*b^4)*sin(4*d*x + 4*c) - (17*a*b^3 - 44*b^4)*sin(2*d*x + 2*c))*cos
(16*d*x + 16*c) + 2*(2*(72*a^2*b^2 - 155*a*b^3 + 26*b^4)*sin(12*d*x + 12*c) - 8*(80*a^2*b^2 - 145*a*b^3 + 44*b
^4)*sin(10*d*x + 10*c) - 3*(384*a^3*b - 1312*a^2*b^2 + 873*a*b^3 - 280*b^4)*sin(8*d*x + 8*c) - 16*(32*a^2*b^2
- 151*a*b^3 + 62*b^4)*sin(6*d*x + 6*c) + 2*(72*a^2*b^2 - 355*a*b^3 + 310*b^4)*sin(4*d*x + 4*c) + 24*(3*a*b^3 -
8*b^4)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) - 2*(2*(128*a^3*b - 456*a^2*b^2 + 1233*a*b^3 - 434*b^4)*sin(10*d*
x + 10*c) - (6400*a^3*b - 13888*a^2*b^2 + 8566*a*b^3 - 2485*b^4)*sin(8*d*x + 8*c) - 2*(128*a^3*b + 2744*a^2*b^
2 - 4711*a*b^3 + 1554*b^4)*sin(6*d*x + 6*c) + 4*(400*a^2*b^2 - 918*a*b^3 + 497*b^4)*sin(4*d*x + 4*c) - 2*(72*a
^2*b^2 - 355*a*b^3 + 310*b^4)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) - 2*((2048*a^4 + 18560*a^3*b - 24752*a^2*b^
2 + 13175*a*b^3 - 2800*b^4)*sin(8*d*x + 8*c) + 8*(256*a^3*b + 2400*a^2*b^2 - 2379*a*b^3 + 560*b^4)*sin(6*d*x +
6*c) - 2*(128*a^3*b + 2744*a^2*b^2 - 4711*a*b^3 + 1554*b^4)*sin(4*d*x + 4*c) + 16*(32*a^2*b^2 - 151*a*b^3 + 6
2*b^4)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) - 2*((2048*a^4 + 18560*a^3*b - 24752*a^2*b^2 + 13175*a*b^3 - 2800*
b^4)*sin(6*d*x + 6*c) - (6400*a^3*b - 13888*a^2*b^2 + 8566*a*b^3 - 2485*b^4)*sin(4*d*x + 4*c) + 3*(384*a^3*b -
1312*a^2*b^2 + 873*a*b^3 - 280*b^4)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) - 4*((128*a^3*b - 456*a^2*b^2 + 1233*a
*b^3 - 434*b^4)*sin(4*d*x + 4*c) + 4*(80*a^2*b^2 - 145*a*b^3 + 44*b^4)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) - 8*
((a^2*b^5 - 2*a*b^6 + b^7)*d*cos(16*d*x + 16*c)^2 + 64*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(14*d*x + 14*c)^2 + 16*(
64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^4*b^3 - 736*a^
3*b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^6*b - 57344*a^5*b^2 + 83712*a^4*
b^3 - 67648*a^3*b^4 + 32841*a^2*b^5 - 9170*a*b^6 + 1225*b^7)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^4*b^3 - 736*a^3*
b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 21
0*a*b^6 + 49*b^7)*d*cos(4*d*x + 4*c)^2 + 64*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(2*d*x + 2*c)^2 + (a^2*b^5 - 2*a*b^
6 + b^7)*d*sin(16*d*x + 16*c)^2 + 64*(a^2*b^5 - 2*a*b^6 + b^7)*d*sin(14*d*x + 14*c)^2 + 16*(64*a^4*b^3 - 240*a
^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*sin(12*d*x + 12*c)^2 + 64*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^
5 - 322*a*b^6 + 49*b^7)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^6*b - 57344*a^5*b^2 + 83712*a^4*b^3 - 67648*a^3*b^
4 + 32841*a^2*b^5 - 9170*a*b^6 + 1225*b^7)*d*sin(8*d*x + 8*c)^2 + 64*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^5
- 322*a*b^6 + 49*b^7)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*
d*sin(4*d*x + 4*c)^2 + 64*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64
*(a^2*b^5 - 2*a*b^6 + b^7)*d*sin(2*d*x + 2*c)^2 - 16*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(2*d*x + 2*c) + (a^2*b^5 -
2*a*b^6 + b^7)*d - 2*(8*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(14*d*x + 14*c) + 4*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6
- 7*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(10*d*x + 10*c) - 2*(128*
a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 30
*a*b^6 - 7*b^7)*d*cos(6*d*x + 6*c) + 4*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(4*d*x + 4*c) + 8*(a^2
*b^5 - 2*a*b^6 + b^7)*d*cos(2*d*x + 2*c) - (a^2*b^5 - 2*a*b^6 + b^7)*d)*cos(16*d*x + 16*c) + 16*(4*(8*a^3*b^4
- 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(1
0*d*x + 10*c) - 2*(128*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^
3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(6*d*x + 6*c) + 4*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*co
s(4*d*x + 4*c) + 8*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(2*d*x + 2*c) - (a^2*b^5 - 2*a*b^6 + b^7)*d)*cos(14*d*x + 14
*c) - 8*(8*(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*cos(10*d*x + 10*c) + 2*(1024*a^5*b
^2 - 3712*a^4*b^3 + 5304*a^3*b^4 - 3813*a^2*b^5 + 1442*a*b^6 - 245*b^7)*d*cos(8*d*x + 8*c) + 8*(128*a^4*b^3 -
424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*cos(6*d*x + 6*c) - 4*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5
- 210*a*b^6 + 49*b^7)*d*cos(4*d*x + 4*c) - 8*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(2*d*x + 2*c) +
(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d)*cos(12*d*x + 12*c) + 16*(2*(2048*a^5*b^2 - 6528*a^4*b^3 + 8144
*a^3*b^4 - 5141*a^2*b^5 + 1722*a*b^6 - 245*b^7)*d*cos(8*d*x + 8*c) + 8*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^
5 - 322*a*b^6 + 49*b^7)*d*cos(6*d*x + 6*c) - 4*(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*
d*cos(4*d*x + 4*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(2*d*x + 2*c) + (16*a^3*b^4 - 39*a^2*
b^5 + 30*a*b^6 - 7*b^7)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^5*b^2 - 6528*a^4*b^3 + 8144*a^3*b^4 - 5141*a^2*b^
5 + 1722*a*b^6 - 245*b^7)*d*cos(6*d*x + 6*c) - 4*(1024*a^5*b^2 - 3712*a^4*b^3 + 5304*a^3*b^4 - 3813*a^2*b^5 +
1442*a*b^6 - 245*b^7)*d*cos(4*d*x + 4*c) - 8*(128*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*
cos(2*d*x + 2*c) + (128*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d)*cos(8*d*x + 8*c) - 16*(4*
(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*cos(4*d*x + 4*c) + 8*(16*a^3*b^4 - 39*a^2*b^5
+ 30*a*b^6 - 7*b^7)*d*cos(2*d*x + 2*c) - (16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d)*cos(6*d*x + 6*c) + 8
*(8*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(2*d*x + 2*c) - (8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^
7)*d)*cos(4*d*x + 4*c) - 4*(4*(a^2*b^5 - 2*a*b^6 + b^7)*d*sin(14*d*x + 14*c) + 2*(8*a^3*b^4 - 23*a^2*b^5 + 22*
a*b^6 - 7*b^7)*d*sin(12*d*x + 12*c) - 4*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*sin(10*d*x + 10*c) - (1
28*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*sin(8*d*x + 8*c) - 4*(16*a^3*b^4 - 39*a^2*b^5 +
30*a*b^6 - 7*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*sin(4*d*x + 4*c) + 4*(
a^2*b^5 - 2*a*b^6 + b^7)*d*sin(2*d*x + 2*c))*sin(16*d*x + 16*c) + 32*(2*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7
*b^7)*d*sin(12*d*x + 12*c) - 4*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*sin(10*d*x + 10*c) - (128*a^4*b^
3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*sin(8*d*x + 8*c) - 4*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6
- 7*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*sin(4*d*x + 4*c) + 4*(a^2*b^5 -
2*a*b^6 + b^7)*d*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*
a*b^6 + 49*b^7)*d*sin(10*d*x + 10*c) + (1024*a^5*b^2 - 3712*a^4*b^3 + 5304*a^3*b^4 - 3813*a^2*b^5 + 1442*a*b^6
- 245*b^7)*d*sin(8*d*x + 8*c) + 4*(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*sin(6*d*x
+ 6*c) - 2*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*sin(4*d*x + 4*c) - 4*(8*a^3*b^4 - 2
3*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 32*((2048*a^5*b^2 - 6528*a^4*b^3 + 8144
*a^3*b^4 - 5141*a^2*b^5 + 1722*a*b^6 - 245*b^7)*d*sin(8*d*x + 8*c) + 4*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^
5 - 322*a*b^6 + 49*b^7)*d*sin(6*d*x + 6*c) - 2*(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*
d*sin(4*d*x + 4*c) - 4*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 1
6*(2*(2048*a^5*b^2 - 6528*a^4*b^3 + 8144*a^3*b^4 - 5141*a^2*b^5 + 1722*a*b^6 - 245*b^7)*d*sin(6*d*x + 6*c) - (
1024*a^5*b^2 - 3712*a^4*b^3 + 5304*a^3*b^4 - 3813*a^2*b^5 + 1442*a*b^6 - 245*b^7)*d*sin(4*d*x + 4*c) - 2*(128*
a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^4*
b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*sin(4*d*x + 4*c) + 2*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b
^6 - 7*b^7)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(1/4*(4*(a*b - 4*b^2)*cos(6*d*x + 6*c)^2 + 36*(8*a*
b - 3*b^2)*cos(4*d*x + 4*c)^2 + 4*(a*b - 4*b^2)*cos(2*d*x + 2*c)^2 + 4*(a*b - 4*b^2)*sin(6*d*x + 6*c)^2 + 36*(
8*a*b - 3*b^2)*sin(4*d*x + 4*c)^2 + 2*(8*a^2 - 35*a*b + 48*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(a*b - 4
*b^2)*sin(2*d*x + 2*c)^2 - (18*b^2*cos(4*d*x + 4*c) + (a*b - 4*b^2)*cos(6*d*x + 6*c) + (a*b - 4*b^2)*cos(2*d*x
+ 2*c))*cos(8*d*x + 8*c) - (a*b - 4*b^2 - 2*(8*a^2 - 35*a*b + 48*b^2)*cos(4*d*x + 4*c) - 8*(a*b - 4*b^2)*cos(
2*d*x + 2*c))*cos(6*d*x + 6*c) - 2*(9*b^2 - (8*a^2 - 35*a*b + 48*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (a*
b - 4*b^2)*cos(2*d*x + 2*c) - (18*b^2*sin(4*d*x + 4*c) + (a*b - 4*b^2)*sin(6*d*x + 6*c) + (a*b - 4*b^2)*sin(2*
d*x + 2*c))*sin(8*d*x + 8*c) + 2*((8*a^2 - 35*a*b + 48*b^2)*sin(4*d*x + 4*c) + 4*(a*b - 4*b^2)*sin(2*d*x + 2*c
))*sin(6*d*x + 6*c))/(a^2*b^3 - 2*a*b^4 + b^5 + (a^2*b^3 - 2*a*b^4 + b^5)*cos(8*d*x + 8*c)^2 + 16*(a^2*b^3 - 2
*a*b^4 + b^5)*cos(6*d*x + 6*c)^2 + 4*(64*a^4*b - 176*a^3*b^2 + 169*a^2*b^3 - 66*a*b^4 + 9*b^5)*cos(4*d*x + 4*c
)^2 + 16*(a^2*b^3 - 2*a*b^4 + b^5)*cos(2*d*x + 2*c)^2 + (a^2*b^3 - 2*a*b^4 + b^5)*sin(8*d*x + 8*c)^2 + 16*(a^2
*b^3 - 2*a*b^4 + b^5)*sin(6*d*x + 6*c)^2 + 4*(64*a^4*b - 176*a^3*b^2 + 169*a^2*b^3 - 66*a*b^4 + 9*b^5)*sin(4*d
*x + 4*c)^2 + 16*(8*a^3*b^2 - 19*a^2*b^3 + 14*a*b^4 - 3*b^5)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^2*b^3 -
2*a*b^4 + b^5)*sin(2*d*x + 2*c)^2 + 2*(a^2*b^3 - 2*a*b^4 + b^5 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*cos(6*d*x + 6*c)
- 2*(8*a^3*b^2 - 19*a^2*b^3 + 14*a*b^4 - 3*b^5)*cos(4*d*x + 4*c) - 4*(a^2*b^3 - 2*a*b^4 + b^5)*cos(2*d*x + 2*
c))*cos(8*d*x + 8*c) - 8*(a^2*b^3 - 2*a*b^4 + b^5 - 2*(8*a^3*b^2 - 19*a^2*b^3 + 14*a*b^4 - 3*b^5)*cos(4*d*x +
4*c) - 4*(a^2*b^3 - 2*a*b^4 + b^5)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^3*b^2 - 19*a^2*b^3 + 14*a*b^4 -
3*b^5 - 4*(8*a^3*b^2 - 19*a^2*b^3 + 14*a*b^4 - 3*b^5)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a^2*b^3 - 2*a*b
^4 + b^5)*cos(2*d*x + 2*c) - 4*(2*(a^2*b^3 - 2*a*b^4 + b^5)*sin(6*d*x + 6*c) + (8*a^3*b^2 - 19*a^2*b^3 + 14*a*
b^4 - 3*b^5)*sin(4*d*x + 4*c) + 2*(a^2*b^3 - 2*a*b^4 + b^5)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^3*b^
2 - 19*a^2*b^3 + 14*a*b^4 - 3*b^5)*sin(4*d*x + 4*c) + 2*(a^2*b^3 - 2*a*b^4 + b^5)*sin(2*d*x + 2*c))*sin(6*d*x
+ 6*c)), x) - (2*a*b^3 - 5*b^4 + (a*b^3 - 4*b^4)*cos(14*d*x + 14*c) - (32*a^2*b^2 - 58*a*b^3 - b^4)*cos(12*d*x
+ 12*c) + 3*(48*a^2*b^2 - 73*a*b^3 + 20*b^4)*cos(10*d*x + 10*c) + (256*a^3*b - 832*a^2*b^2 + 550*a*b^3 - 175*
b^4)*cos(8*d*x + 8*c) + (112*a^2*b^2 - 533*a*b^3 + 220*b^4)*cos(6*d*x + 6*c) - (32*a^2*b^2 - 158*a*b^3 + 141*b
^4)*cos(4*d*x + 4*c) - (17*a*b^3 - 44*b^4)*cos(2*d*x + 2*c))*sin(16*d*x + 16*c) + (17*a*b^3 - 44*b^4 - 4*(72*a
^2*b^2 - 155*a*b^3 + 26*b^4)*cos(12*d*x + 12*c) + 16*(80*a^2*b^2 - 145*a*b^3 + 44*b^4)*cos(10*d*x + 10*c) + 6*
(384*a^3*b - 1312*a^2*b^2 + 873*a*b^3 - 280*b^4)*cos(8*d*x + 8*c) + 32*(32*a^2*b^2 - 151*a*b^3 + 62*b^4)*cos(6
*d*x + 6*c) - 4*(72*a^2*b^2 - 355*a*b^3 + 310*b^4)*cos(4*d*x + 4*c) - 48*(3*a*b^3 - 8*b^4)*cos(2*d*x + 2*c))*s
in(14*d*x + 14*c) + (32*a^2*b^2 - 158*a*b^3 + 141*b^4 + 4*(128*a^3*b - 456*a^2*b^2 + 1233*a*b^3 - 434*b^4)*cos
(10*d*x + 10*c) - 2*(6400*a^3*b - 13888*a^2*b^2 + 8566*a*b^3 - 2485*b^4)*cos(8*d*x + 8*c) - 4*(128*a^3*b + 274
4*a^2*b^2 - 4711*a*b^3 + 1554*b^4)*cos(6*d*x + 6*c) + 8*(400*a^2*b^2 - 918*a*b^3 + 497*b^4)*cos(4*d*x + 4*c) -
4*(72*a^2*b^2 - 355*a*b^3 + 310*b^4)*cos(2*d*x + 2*c))*sin(12*d*x + 12*c) - (112*a^2*b^2 - 533*a*b^3 + 220*b^
4 - 2*(2048*a^4 + 18560*a^3*b - 24752*a^2*b^2 + 13175*a*b^3 - 2800*b^4)*cos(8*d*x + 8*c) - 16*(256*a^3*b + 240
0*a^2*b^2 - 2379*a*b^3 + 560*b^4)*cos(6*d*x + 6*c) + 4*(128*a^3*b + 2744*a^2*b^2 - 4711*a*b^3 + 1554*b^4)*cos(
4*d*x + 4*c) - 32*(32*a^2*b^2 - 151*a*b^3 + 62*b^4)*cos(2*d*x + 2*c))*sin(10*d*x + 10*c) - (256*a^3*b - 832*a^
2*b^2 + 550*a*b^3 - 175*b^4 - 2*(2048*a^4 + 18560*a^3*b - 24752*a^2*b^2 + 13175*a*b^3 - 2800*b^4)*cos(6*d*x +
6*c) + 2*(6400*a^3*b - 13888*a^2*b^2 + 8566*a*b^3 - 2485*b^4)*cos(4*d*x + 4*c) - 6*(384*a^3*b - 1312*a^2*b^2 +
873*a*b^3 - 280*b^4)*cos(2*d*x + 2*c))*sin(8*d*x + 8*c) - (144*a^2*b^2 - 219*a*b^3 + 60*b^4 - 4*(128*a^3*b -
456*a^2*b^2 + 1233*a*b^3 - 434*b^4)*cos(4*d*x + 4*c) - 16*(80*a^2*b^2 - 145*a*b^3 + 44*b^4)*cos(2*d*x + 2*c))*
sin(6*d*x + 6*c) + (32*a^2*b^2 - 58*a*b^3 - b^4 - 4*(72*a^2*b^2 - 155*a*b^3 + 26*b^4)*cos(2*d*x + 2*c))*sin(4*
d*x + 4*c) - (a*b^3 - 4*b^4)*sin(2*d*x + 2*c))/((a^2*b^5 - 2*a*b^6 + b^7)*d*cos(16*d*x + 16*c)^2 + 64*(a^2*b^5
- 2*a*b^6 + b^7)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*
cos(12*d*x + 12*c)^2 + 64*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*cos(10*d*x + 10*c)^
2 + 4*(16384*a^6*b - 57344*a^5*b^2 + 83712*a^4*b^3 - 67648*a^3*b^4 + 32841*a^2*b^5 - 9170*a*b^6 + 1225*b^7)*d*
cos(8*d*x + 8*c)^2 + 64*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*cos(6*d*x + 6*c)^2 +
16*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*cos(4*d*x + 4*c)^2 + 64*(a^2*b^5 - 2*a*b^6
+ b^7)*d*cos(2*d*x + 2*c)^2 + (a^2*b^5 - 2*a*b^6 + b^7)*d*sin(16*d*x + 16*c)^2 + 64*(a^2*b^5 - 2*a*b^6 + b^7)*
d*sin(14*d*x + 14*c)^2 + 16*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*sin(12*d*x + 12*c)
^2 + 64*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^6*b
- 57344*a^5*b^2 + 83712*a^4*b^3 - 67648*a^3*b^4 + 32841*a^2*b^5 - 9170*a*b^6 + 1225*b^7)*d*sin(8*d*x + 8*c)^2
+ 64*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^4*b^3 - 2
40*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 -
7*b^7)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^2*b^5 - 2*a*b^6 + b^7)*d*sin(2*d*x + 2*c)^2 - 16*(a^2*b^5 -
2*a*b^6 + b^7)*d*cos(2*d*x + 2*c) + (a^2*b^5 - 2*a*b^6 + b^7)*d - 2*(8*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(14*d*x
+ 14*c) + 4*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 3
0*a*b^6 - 7*b^7)*d*cos(10*d*x + 10*c) - 2*(128*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*cos
(8*d*x + 8*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(6*d*x + 6*c) + 4*(8*a^3*b^4 - 23*a^2*b^5
+ 22*a*b^6 - 7*b^7)*d*cos(4*d*x + 4*c) + 8*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(2*d*x + 2*c) - (a^2*b^5 - 2*a*b^6 +
b^7)*d)*cos(16*d*x + 16*c) + 16*(4*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a
^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(10*d*x + 10*c) - 2*(128*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 16
6*a*b^6 + 35*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(6*d*x + 6*c) + 4*(
8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(4*d*x + 4*c) + 8*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(2*d*x + 2*c)
- (a^2*b^5 - 2*a*b^6 + b^7)*d)*cos(14*d*x + 14*c) - 8*(8*(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6
+ 49*b^7)*d*cos(10*d*x + 10*c) + 2*(1024*a^5*b^2 - 3712*a^4*b^3 + 5304*a^3*b^4 - 3813*a^2*b^5 + 1442*a*b^6 -
245*b^7)*d*cos(8*d*x + 8*c) + 8*(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*cos(6*d*x + 6
*c) - 4*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*cos(4*d*x + 4*c) - 8*(8*a^3*b^4 - 23*a
^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(2*d*x + 2*c) + (8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d)*cos(12*d*x + 12
*c) + 16*(2*(2048*a^5*b^2 - 6528*a^4*b^3 + 8144*a^3*b^4 - 5141*a^2*b^5 + 1722*a*b^6 - 245*b^7)*d*cos(8*d*x + 8
*c) + 8*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*cos(6*d*x + 6*c) - 4*(128*a^4*b^3 - 4
24*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*cos(4*d*x + 4*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*
b^7)*d*cos(2*d*x + 2*c) + (16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^5*
b^2 - 6528*a^4*b^3 + 8144*a^3*b^4 - 5141*a^2*b^5 + 1722*a*b^6 - 245*b^7)*d*cos(6*d*x + 6*c) - 4*(1024*a^5*b^2
- 3712*a^4*b^3 + 5304*a^3*b^4 - 3813*a^2*b^5 + 1442*a*b^6 - 245*b^7)*d*cos(4*d*x + 4*c) - 8*(128*a^4*b^3 - 352
*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*cos(2*d*x + 2*c) + (128*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 1
66*a*b^6 + 35*b^7)*d)*cos(8*d*x + 8*c) - 16*(4*(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*
d*cos(4*d*x + 4*c) + 8*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(2*d*x + 2*c) - (16*a^3*b^4 - 39*a^2*
b^5 + 30*a*b^6 - 7*b^7)*d)*cos(6*d*x + 6*c) + 8*(8*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(2*d*x + 2
*c) - (8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d)*cos(4*d*x + 4*c) - 4*(4*(a^2*b^5 - 2*a*b^6 + b^7)*d*sin(1
4*d*x + 14*c) + 2*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*sin(12*d*x + 12*c) - 4*(16*a^3*b^4 - 39*a^2*b^
5 + 30*a*b^6 - 7*b^7)*d*sin(10*d*x + 10*c) - (128*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*
sin(8*d*x + 8*c) - 4*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^3*b^4 - 23*a^2*b
^5 + 22*a*b^6 - 7*b^7)*d*sin(4*d*x + 4*c) + 4*(a^2*b^5 - 2*a*b^6 + b^7)*d*sin(2*d*x + 2*c))*sin(16*d*x + 16*c)
+ 32*(2*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*sin(12*d*x + 12*c) - 4*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*
b^6 - 7*b^7)*d*sin(10*d*x + 10*c) - (128*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*sin(8*d*x
+ 8*c) - 4*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^3*b^4 - 23*a^2*b^5 + 22*a
*b^6 - 7*b^7)*d*sin(4*d*x + 4*c) + 4*(a^2*b^5 - 2*a*b^6 + b^7)*d*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*
(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*sin(10*d*x + 10*c) + (1024*a^5*b^2 - 3712*a^4
*b^3 + 5304*a^3*b^4 - 3813*a^2*b^5 + 1442*a*b^6 - 245*b^7)*d*sin(8*d*x + 8*c) + 4*(128*a^4*b^3 - 424*a^3*b^4 +
513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*sin(6*d*x + 6*c) - 2*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6
+ 49*b^7)*d*sin(4*d*x + 4*c) - 4*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*sin(2*d*x + 2*c))*sin(12*d*x +
12*c) + 32*((2048*a^5*b^2 - 6528*a^4*b^3 + 8144*a^3*b^4 - 5141*a^2*b^5 + 1722*a*b^6 - 245*b^7)*d*sin(8*d*x + 8
*c) + 4*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*sin(6*d*x + 6*c) - 2*(128*a^4*b^3 - 4
24*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*sin(4*d*x + 4*c) - 4*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*
b^7)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^5*b^2 - 6528*a^4*b^3 + 8144*a^3*b^4 - 5141*a^2*b^5
+ 1722*a*b^6 - 245*b^7)*d*sin(6*d*x + 6*c) - (1024*a^5*b^2 - 3712*a^4*b^3 + 5304*a^3*b^4 - 3813*a^2*b^5 + 144
2*a*b^6 - 245*b^7)*d*sin(4*d*x + 4*c) - 2*(128*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*sin
(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*sin(4*d
*x + 4*c) + 2*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))

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Fricas [B]  time = 14.0834, size = 12051, normalized size = 37.78 \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)^8/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

-1/256*(((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^
2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 +
(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a
^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*
b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a
^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3
*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(35*a^3 - 1491/4*a^2*b + 1875/2*a*b^2 + 625/4*b^3 - 1/4*(140*a^3 - 1491*a^2
*b + 3750*a*b^2 + 625*b^3)*cos(d*x + c)^2 + 1/2*((a^9*b^3 - 18*a^8*b^4 + 75*a^7*b^5 - 140*a^6*b^6 + 135*a^5*b^
7 - 66*a^4*b^8 + 13*a^3*b^9)*d^3*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b
^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^
5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (70*a^5*b - 623*a^4*b^2 + 1161*a^3*b^3 + 99
5*a^2*b^4 + 125*a*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a
^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*
b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a
^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3
*b^6 + 5*a^2*b^7 - a*b^8)*d^2)) + 1/4*(2*(4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129
*a^3*b^6 + 25*a^2*b^7)*d^2*cos(d*x + c)^2 - (4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 -
129*a^3*b^6 + 25*a^2*b^7)*d^2)*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3
- 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*
b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))) - ((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b
^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3
*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4
+ 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3
+ 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 1
20*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 -
5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(35*a^3 - 1491/4*a^2*b + 1875/2*a*b^2 + 625
/4*b^3 - 1/4*(140*a^3 - 1491*a^2*b + 3750*a*b^2 + 625*b^3)*cos(d*x + c)^2 - 1/2*((a^9*b^3 - 18*a^8*b^4 + 75*a^
7*b^5 - 140*a^6*b^6 + 135*a^5*b^7 - 66*a^4*b^8 + 13*a^3*b^9)*d^3*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2
+ 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 21
0*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (70*a^5*b -
623*a^4*b^2 + 1161*a^3*b^3 + 995*a^2*b^4 + 125*a*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^6*b^3 - 5*a^5*b^4
+ 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3
+ 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 1
20*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 -
5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2)) + 1/4*(2*(4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 -
290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2*cos(d*x + c)^2 - (4*a^8*b - 45*a^7*b^2 + 165*a^6*b^
3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2)*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 +
7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*
a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))) + ((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x +
c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c
)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^
5)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^
3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^
7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 4*a^3 - 35*a^2*b
+ 70*a*b^2 + 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(-35*a^3 +
1491/4*a^2*b - 1875/2*a*b^2 - 625/4*b^3 + 1/4*(140*a^3 - 1491*a^2*b + 3750*a*b^2 + 625*b^3)*cos(d*x + c)^2 +
1/2*((a^9*b^3 - 18*a^8*b^4 + 75*a^7*b^5 - 140*a^6*b^6 + 135*a^5*b^7 - 66*a^4*b^8 + 13*a^3*b^9)*d^3*sqrt((1225*
a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^
6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))*cos(d
*x + c)*sin(d*x + c) + (70*a^5*b - 623*a^4*b^2 + 1161*a^3*b^3 + 995*a^2*b^4 + 125*a*b^5)*d*cos(d*x + c)*sin(d*
x + c))*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*
a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*
b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 4*a^3 - 35*a^2*
b + 70*a*b^2 + 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2)) + 1/4*(2*(4
*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2*cos(d*x + c)^2 -
(4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2)*sqrt((1225*a
^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6
+ 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))) - ((a
^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b
^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4
*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a
*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a
^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 +
a^3*b^13)*d^4)) + 4*a^3 - 35*a^2*b + 70*a*b^2 + 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a
^2*b^7 - a*b^8)*d^2))*log(-35*a^3 + 1491/4*a^2*b - 1875/2*a*b^2 - 625/4*b^3 + 1/4*(140*a^3 - 1491*a^2*b + 3750
*a*b^2 + 625*b^3)*cos(d*x + c)^2 - 1/2*((a^9*b^3 - 18*a^8*b^4 + 75*a^7*b^5 - 140*a^6*b^6 + 135*a^5*b^7 - 66*a^
4*b^8 + 13*a^3*b^9)*d^3*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a
^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 -
10*a^4*b^12 + a^3*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) + (70*a^5*b - 623*a^4*b^2 + 1161*a^3*b^3 + 995*a^2*b^4
+ 125*a*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 -
a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45
*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12
+ a^3*b^13)*d^4)) + 4*a^3 - 35*a^2*b + 70*a*b^2 + 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5
*a^2*b^7 - a*b^8)*d^2)) + 1/4*(2*(4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6
+ 25*a^2*b^7)*d^2*cos(d*x + c)^2 - (4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*
b^6 + 25*a^2*b^7)*d^2)*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^
12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 1
0*a^4*b^12 + a^3*b^13)*d^4))) - 8*(2*(2*a*b - 5*b^2)*cos(d*x + c)^7 - 3*(5*a*b - 13*b^2)*cos(d*x + c)^5 + 24*(
a*b - 2*b^2)*cos(d*x + c)^3 - (a^2 + 18*a*b - 19*b^2)*cos(d*x + c))*sin(d*x + c))/((a^2*b^3 - 2*a*b^4 + b^5)*d
*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*c
os(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*
a*b^4 + 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)**8/(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: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError