3.501 \(\int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=280 \[ \frac {4 a^3 \left (4 c^2+5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d^3 f (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 a^3 (c-d) (4 c+5 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d^3 f (c+d) \sqrt {c+d \sin (e+f x)}}+\frac {8 a^3 (c-d) (c+2 d) \cos (e+f x)}{3 d^2 f (c+d)^2 \sqrt {c+d \sin (e+f x)}}+\frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}} \]

[Out]

2/3*(c-d)*cos(f*x+e)*(a^3+a^3*sin(f*x+e))/d/(c+d)/f/(c+d*sin(f*x+e))^(3/2)+8/3*a^3*(c-d)*(c+2*d)*cos(f*x+e)/d^
2/(c+d)^2/f/(c+d*sin(f*x+e))^(1/2)-4/3*a^3*(4*c^2+5*c*d-3*d^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1
/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/d^3/(c+d)^2
/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+4/3*a^3*(c-d)*(4*c+5*d)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi
+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d^3/(c+d
)/f/(c+d*sin(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.58, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2762, 2968, 3021, 2752, 2663, 2661, 2655, 2653} \[ \frac {4 a^3 \left (4 c^2+5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d^3 f (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 a^3 (c-d) (c+2 d) \cos (e+f x)}{3 d^2 f (c+d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {4 a^3 (c-d) (4 c+5 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d^3 f (c+d) \sqrt {c+d \sin (e+f x)}}+\frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2),x]

[Out]

(2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) + (8*a^3*(c - d)*
(c + 2*d)*Cos[e + f*x])/(3*d^2*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) + (4*a^3*(4*c^2 + 5*c*d - 3*d^2)*Elliptic
E[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c
+ d)]) - (4*a^3*(c - d)*(4*c + 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c
+ d)])/(3*d^3*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2762

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c
+ a*d)), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*
Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
&& (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3021

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

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx &=\frac {2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {(2 a) \int \frac {(a+a \sin (e+f x)) (a (c-4 d)-a (2 c+d) \sin (e+f x))}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 d (c+d)}\\ &=\frac {2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {(2 a) \int \frac {a^2 (c-4 d)+\left (a^2 (c-4 d)-a^2 (2 c+d)\right ) \sin (e+f x)-a^2 (2 c+d) \sin ^2(e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 d (c+d)}\\ &=\frac {2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^{3/2}}+\frac {8 a^3 (c-d) (c+2 d) \cos (e+f x)}{3 d^2 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(4 a) \int \frac {\frac {1}{2} a^2 (c-d) d (c+5 d)+\frac {1}{2} a^2 (c-d) \left (4 c^2+5 c d-3 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 (c-d) d^2 (c+d)^2}\\ &=\frac {2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^{3/2}}+\frac {8 a^3 (c-d) (c+2 d) \cos (e+f x)}{3 d^2 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (2 a^3 (c-d) (4 c+5 d)\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d^3 (c+d)}+\frac {\left (2 a^3 \left (4 c^2+5 c d-3 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 d^3 (c+d)^2}\\ &=\frac {2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^{3/2}}+\frac {8 a^3 (c-d) (c+2 d) \cos (e+f x)}{3 d^2 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}+\frac {\left (2 a^3 \left (4 c^2+5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{3 d^3 (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (2 a^3 (c-d) (4 c+5 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{3 d^3 (c+d) \sqrt {c+d \sin (e+f x)}}\\ &=\frac {2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^{3/2}}+\frac {8 a^3 (c-d) (c+2 d) \cos (e+f x)}{3 d^2 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^3 \left (4 c^2+5 c d-3 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^3 (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 a^3 (c-d) (4 c+5 d) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^3 (c+d) f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.55, size = 232, normalized size = 0.83 \[ -\frac {2 a^3 (\sin (e+f x)+1)^3 \left (d (d-c) \cos (e+f x) \left (4 c^2+d (5 c+9 d) \sin (e+f x)+9 c d+d^2\right )+2 (c+d) \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{3/2} \left (\left (4 c^2+5 c d-3 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )+d^2 (c+5 d) F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )\right )}{3 d^3 f (c+d)^2 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^6 (c+d \sin (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2),x]

[Out]

(-2*a^3*(1 + Sin[e + f*x])^3*(2*(c + d)*(d^2*(c + 5*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*c^
2 + 5*c*d - 3*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/
4, (2*d)/(c + d)]))*((c + d*Sin[e + f*x])/(c + d))^(3/2) + d*(-c + d)*Cos[e + f*x]*(4*c^2 + 9*c*d + d^2 + d*(5
*c + 9*d)*Sin[e + f*x])))/(3*d^3*(c + d)^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c + d*Sin[e + f*x])^(3/2
))

________________________________________________________________________________________

fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{3 \, c d^{2} \cos \left (f x + e\right )^{2} - c^{3} - 3 \, c d^{2} + {\left (d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{2} d - d^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

integral((3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c)/(
3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(5/2), x)

________________________________________________________________________________________

maple [B]  time = 5.41, size = 1257, normalized size = 4.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*a^3*(-2/d^4/(cos(f*x+e)^2*sin(f*x+e)*d+c*cos(f*x+e)^2)^(1/2)*(-d/(c-d)
*sin(f*x+e)-d/(c-d))^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(EllipticE
((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*c^2-EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),
((c-d)/(c+d))^(1/2))*d^2+2*EllipticF((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*c^2-6*EllipticF
((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*c*d+4*EllipticF((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2
),((c-d)/(c+d))^(1/2))*d^2)+3/d^3*(c^2-2*c*d+d^2)*(2*d*cos(f*x+e)^2/(c^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2
)^(1/2)+2*c/(c^2-d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d
/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(
1/2))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(
c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/
(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+1/d^3*(-c^3+3*c^2*d-3*c*d^2+d^3)
*(2/3/(c^2-d^2)/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))
^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*
EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+8/3*c*d/(c^2-d^2)^2*(c/d-1)*((c+d*sin(f*x+e))/(c
-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1
/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))
^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(5/2), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(5/2),x)

[Out]

int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(5/2), x)

________________________________________________________________________________________

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((a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________