3.193 \(\int \frac{\sin ^5(x)}{(a+b \sin (x))^3} \, dx\)

Optimal. Leaf size=243 \[ \frac{x \left (12 a^2+b^2\right )}{2 b^5}+\frac{3 a \left (-7 a^2 b^2+4 a^4+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac{a^3 \left (-29 a^2 b^2+12 a^4+20 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2}}+\frac{a^2 \left (4 a^2-7 b^2\right ) \sin ^2(x) \cos (x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac{a^2 \sin ^3(x) \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{\left (-10 a^2 b^2+6 a^4+b^4\right ) \sin (x) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2} \]

[Out]

((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^
5*(a^2 - b^2)^(5/2)) + (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[x])/(2*b^4*(a^2 - b^2)^2) - ((6*a^4 - 10*a^2*b^2 +
b^4)*Cos[x]*Sin[x])/(2*b^3*(a^2 - b^2)^2) + (a^2*Cos[x]*Sin[x]^3)/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) + (a^2*(
4*a^2 - 7*b^2)*Cos[x]*Sin[x]^2)/(2*b^2*(a^2 - b^2)^2*(a + b*Sin[x]))

________________________________________________________________________________________

Rubi [A]  time = 0.661085, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {2792, 3047, 3049, 3023, 2735, 2660, 618, 204} \[ \frac{x \left (12 a^2+b^2\right )}{2 b^5}+\frac{3 a \left (-7 a^2 b^2+4 a^4+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac{a^3 \left (-29 a^2 b^2+12 a^4+20 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2}}+\frac{a^2 \left (4 a^2-7 b^2\right ) \sin ^2(x) \cos (x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac{a^2 \sin ^3(x) \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{\left (-10 a^2 b^2+6 a^4+b^4\right ) \sin (x) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^5/(a + b*Sin[x])^3,x]

[Out]

((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^
5*(a^2 - b^2)^(5/2)) + (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[x])/(2*b^4*(a^2 - b^2)^2) - ((6*a^4 - 10*a^2*b^2 +
b^4)*Cos[x]*Sin[x])/(2*b^3*(a^2 - b^2)^2) + (a^2*Cos[x]*Sin[x]^3)/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) + (a^2*(
4*a^2 - 7*b^2)*Cos[x]*Sin[x]^2)/(2*b^2*(a^2 - b^2)^2*(a + b*Sin[x]))

Rule 2792

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(
d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e +
f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*
d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*
n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] &
& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e +
f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sin ^5(x)}{(a+b \sin (x))^3} \, dx &=\frac{a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{\int \frac{\sin ^2(x) \left (3 a^2-2 a b \sin (x)-2 \left (2 a^2-b^2\right ) \sin ^2(x)\right )}{(a+b \sin (x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac{a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac{a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac{\int \frac{\sin (x) \left (-2 a^2 \left (4 a^2-7 b^2\right )+a b \left (a^2-4 b^2\right ) \sin (x)+2 \left (6 a^4-10 a^2 b^2+b^4\right ) \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac{a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac{a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac{\int \frac{2 a \left (6 a^4-10 a^2 b^2+b^4\right )-2 b \left (2 a^4-4 a^2 b^2-b^4\right ) \sin (x)-6 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin ^2(x)}{a+b \sin (x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac{\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac{a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac{a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac{\int \frac{2 a b \left (6 a^4-10 a^2 b^2+b^4\right )+2 \left (a^2-b^2\right )^2 \left (12 a^2+b^2\right ) \sin (x)}{a+b \sin (x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (12 a^2+b^2\right ) x}{2 b^5}+\frac{3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac{\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac{a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac{a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac{\left (a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right )\right ) \int \frac{1}{a+b \sin (x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2}\\ &=\frac{\left (12 a^2+b^2\right ) x}{2 b^5}+\frac{3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac{\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac{a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac{a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac{\left (a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{b^5 \left (a^2-b^2\right )^2}\\ &=\frac{\left (12 a^2+b^2\right ) x}{2 b^5}+\frac{3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac{\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac{a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac{a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac{\left (2 a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{x}{2}\right )\right )}{b^5 \left (a^2-b^2\right )^2}\\ &=\frac{\left (12 a^2+b^2\right ) x}{2 b^5}-\frac{a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2}}+\frac{3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac{\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac{a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac{a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}\\ \end{align*}

Mathematica [A]  time = 0.9175, size = 164, normalized size = 0.67 \[ \frac{2 x \left (12 a^2+b^2\right )-\frac{4 a^3 \left (-29 a^2 b^2+12 a^4+20 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac{2 a^4 b \left (7 a^2-10 b^2\right ) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}-\frac{2 a^5 b \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+12 a b \cos (x)-b^2 \sin (2 x)}{4 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(12*a^2 + b^2)*x - (4*a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b
^2)^(5/2) + 12*a*b*Cos[x] - (2*a^5*b*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (2*a^4*b*(7*a^2 - 10*b^2)*Co
s[x])/((a - b)^2*(a + b)^2*(a + b*Sin[x])) - b^2*Sin[2*x])/(4*b^5)

________________________________________________________________________________________

Maple [B]  time = 0.066, size = 712, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^5/(a+b*sin(x))^3,x)

[Out]

1/b^3/(tan(1/2*x)^2+1)^2*tan(1/2*x)^3+6/b^4/(tan(1/2*x)^2+1)^2*tan(1/2*x)^2*a-1/b^3/(tan(1/2*x)^2+1)^2*tan(1/2
*x)+6/b^4/(tan(1/2*x)^2+1)^2*a+12/b^5*arctan(tan(1/2*x))*a^2+5*a^6/b^3/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^
4-2*a^2*b^2+b^4)*tan(1/2*x)^3-8*a^4/b/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^3+6*a
^7/b^4/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2+3*a^5/b^2/(tan(1/2*x)^2*a+2*tan(1/
2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2-18*a^3/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*ta
n(1/2*x)^2+19*a^6/b^3/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)-28*a^4/b/(tan(1/2*x)^
2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)+6*a^7/b^4/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^
2*b^2+b^4)-9*a^5/b^2/(tan(1/2*x)^2*a+2*tan(1/2*x)*b+a)^2/(a^4-2*a^2*b^2+b^4)-12*a^7/b^5/(a^4-2*a^2*b^2+b^4)/(a
^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*x)+2*b)/(a^2-b^2)^(1/2))+29*a^5/b^3/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*
arctan(1/2*(2*a*tan(1/2*x)+2*b)/(a^2-b^2)^(1/2))-20*a^3/b/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*
tan(1/2*x)+2*b)/(a^2-b^2)^(1/2))+1/2*x/b^3

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.445, size = 2384, normalized size = 9.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*(12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*x*cos(x)^2 + 8*(a^7*b^3 - 3*a^5*b^5 + 3*a^3
*b^7 - a*b^9)*cos(x)^3 + (12*a^9 - 17*a^7*b^2 - 9*a^5*b^4 + 20*a^3*b^6 - (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6
)*cos(x)^2 + 2*(12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*sin(x))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(x)^2 - 2*
a*b*sin(x) - a^2 - b^2 - 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 -
b^2)) - 2*(12*a^10 - 23*a^8*b^2 - 2*a^6*b^4 + 24*a^4*b^6 - 10*a^2*b^8 - b^10)*x - 2*(12*a^9*b - 29*a^7*b^3 +
15*a^5*b^5 + 6*a^3*b^7 - 4*a*b^9)*cos(x) - 2*((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cos(x)^3 + 2*(12*a^9*b
- 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*x + (18*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8 + b^10)*
cos(x))*sin(x))/(a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^13 - (a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*cos(x)^2
+ 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*sin(x)), -1/2*((12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b
^8 - b^10)*x*cos(x)^2 + 4*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cos(x)^3 - (12*a^9 - 17*a^7*b^2 - 9*a^5*b^
4 + 20*a^3*b^6 - (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*cos(x)^2 + 2*(12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*sin(
x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (12*a^10 - 23*a^8*b^2 - 2*a^6*b^4 + 24*
a^4*b^6 - 10*a^2*b^8 - b^10)*x - (12*a^9*b - 29*a^7*b^3 + 15*a^5*b^5 + 6*a^3*b^7 - 4*a*b^9)*cos(x) - ((a^6*b^4
- 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cos(x)^3 + 2*(12*a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*x + (18
*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8 + b^10)*cos(x))*sin(x))/(a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^
13 - (a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*cos(x)^2 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*sin(x)
)]

________________________________________________________________________________________

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(x)**5/(a+b*sin(x))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.70847, size = 697, normalized size = 2.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(12*a^7 - 29*a^5*b^2 + 20*a^3*b^4)*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^
2)))/((a^4*b^5 - 2*a^2*b^7 + b^9)*sqrt(a^2 - b^2)) + (6*a^6*b*tan(1/2*x)^7 - 10*a^4*b^3*tan(1/2*x)^7 + a^2*b^5
*tan(1/2*x)^7 + 12*a^7*tan(1/2*x)^6 - 5*a^5*b^2*tan(1/2*x)^6 - 20*a^3*b^4*tan(1/2*x)^6 + 4*a*b^6*tan(1/2*x)^6
+ 54*a^6*b*tan(1/2*x)^5 - 90*a^4*b^3*tan(1/2*x)^5 + 17*a^2*b^5*tan(1/2*x)^5 + 4*b^7*tan(1/2*x)^5 + 36*a^7*tan(
1/2*x)^4 - 15*a^5*b^2*tan(1/2*x)^4 - 66*a^3*b^4*tan(1/2*x)^4 + 24*a*b^6*tan(1/2*x)^4 + 90*a^6*b*tan(1/2*x)^3 -
162*a^4*b^3*tan(1/2*x)^3 + 55*a^2*b^5*tan(1/2*x)^3 - 4*b^7*tan(1/2*x)^3 + 36*a^7*tan(1/2*x)^2 - 31*a^5*b^2*ta
n(1/2*x)^2 - 40*a^3*b^4*tan(1/2*x)^2 + 20*a*b^6*tan(1/2*x)^2 + 42*a^6*b*tan(1/2*x) - 74*a^4*b^3*tan(1/2*x) + 2
3*a^2*b^5*tan(1/2*x) + 12*a^7 - 21*a^5*b^2 + 6*a^3*b^4)/((a^4*b^4 - 2*a^2*b^6 + b^8)*(a*tan(1/2*x)^4 + 2*b*tan
(1/2*x)^3 + 2*a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)^2) + 1/2*(12*a^2 + b^2)*x/b^5