3.450 \(\int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx\)

Optimal. Leaf size=187 \[ -\frac {a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f (c+d)^2 \sqrt {c^2-d^2}}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}+\frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{2 d f (c+d) (c+d \sin (e+f x))^2}+\frac {a^3 x}{d^3} \]

[Out]

a^3*x/d^3+1/2*(c-d)*cos(f*x+e)*(a^3+a^3*sin(f*x+e))/d/(c+d)/f/(c+d*sin(f*x+e))^2+1/2*a^3*(c-d)*(2*c+5*d)*cos(f
*x+e)/d^2/(c+d)^2/f/(c+d*sin(f*x+e))-a^3*(c-d)*(2*c^2+6*c*d+7*d^2)*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(
1/2))/d^3/(c+d)^2/f/(c^2-d^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.48, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2762, 2968, 3021, 2735, 2660, 618, 204} \[ -\frac {a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f (c+d)^2 \sqrt {c^2-d^2}}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}+\frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{2 d f (c+d) (c+d \sin (e+f x))^2}+\frac {a^3 x}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*x)/d^3 - (a^3*(c - d)*(2*c^2 + 6*c*d + 7*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c +
d)^2*Sqrt[c^2 - d^2]*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(2*d*(c + d)*f*(c + d*Sin[e + f*x])
^2) + (a^3*(c - d)*(2*c + 5*d)*Cos[e + f*x])/(2*d^2*(c + d)^2*f*(c + d*Sin[e + f*x]))

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])

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 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 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 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))^3} \, dx &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {a \int \frac {(a+a \sin (e+f x)) (a (c-5 d)-2 a (c+d) \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx}{2 d (c+d)}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {a \int \frac {a^2 (c-5 d)+\left (a^2 (c-5 d)-2 a^2 (c+d)\right ) \sin (e+f x)-2 a^2 (c+d) \sin ^2(e+f x)}{(c+d \sin (e+f x))^2} \, dx}{2 d (c+d)}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac {a \int \frac {a^2 (c-d) d (c+7 d)+2 a^2 (c-d) (c+d)^2 \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{2 (c-d) d^2 (c+d)^2}\\ &=\frac {a^3 x}{d^3}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 d^3 (c+d)^2}\\ &=\frac {a^3 x}{d^3}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 (c+d)^2 f}\\ &=\frac {a^3 x}{d^3}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\left (2 a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 (c+d)^2 f}\\ &=\frac {a^3 x}{d^3}-\frac {a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 (c+d)^2 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.01, size = 196, normalized size = 1.05 \[ \frac {a^3 (\sin (e+f x)+1)^3 \left (\frac {3 d \left (c^2+c d-2 d^2\right ) \cos (e+f x)}{(c+d)^2 (c+d \sin (e+f x))}-\frac {2 \left (2 c^3+4 c^2 d+c d^2-7 d^3\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{(c+d)^2 \sqrt {c^2-d^2}}-\frac {d (c-d)^2 \cos (e+f x)}{(c+d) (c+d \sin (e+f x))^2}+2 (e+f x)\right )}{2 d^3 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + Sin[e + f*x])^3*(2*(e + f*x) - (2*(2*c^3 + 4*c^2*d + c*d^2 - 7*d^3)*ArcTan[(d + c*Tan[(e + f*x)/2])/
Sqrt[c^2 - d^2]])/((c + d)^2*Sqrt[c^2 - d^2]) - ((c - d)^2*d*Cos[e + f*x])/((c + d)*(c + d*Sin[e + f*x])^2) +
(3*d*(c^2 + c*d - 2*d^2)*Cos[e + f*x])/((c + d)^2*(c + d*Sin[e + f*x]))))/(2*d^3*f*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^6)

________________________________________________________________________________________

fricas [B]  time = 0.54, size = 1064, normalized size = 5.69 \[ \text {result too large to display} \]

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))^3,x, algorithm="fricas")

[Out]

[1/4*(4*(a^3*c^2*d^2 + 2*a^3*c*d^3 + a^3*d^4)*f*x*cos(f*x + e)^2 - 4*(a^3*c^4 + 2*a^3*c^3*d + 2*a^3*c^2*d^2 +
2*a^3*c*d^3 + a^3*d^4)*f*x - (2*a^3*c^4 + 6*a^3*c^3*d + 9*a^3*c^2*d^2 + 6*a^3*c*d^3 + 7*a^3*d^4 - (2*a^3*c^2*d
^2 + 6*a^3*c*d^3 + 7*a^3*d^4)*cos(f*x + e)^2 + 2*(2*a^3*c^3*d + 6*a^3*c^2*d^2 + 7*a^3*c*d^3)*sin(f*x + e))*sqr
t(-(c - d)/(c + d))*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*((c^2 + c*d)*cos(f*
x + e)*sin(f*x + e) + (c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c + d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x +
e) - c^2 - d^2)) - 2*(2*a^3*c^3*d + 4*a^3*c^2*d^2 - 5*a^3*c*d^3 - a^3*d^4)*cos(f*x + e) - 2*(4*(a^3*c^3*d + 2*
a^3*c^2*d^2 + a^3*c*d^3)*f*x + 3*(a^3*c^2*d^2 + a^3*c*d^3 - 2*a^3*d^4)*cos(f*x + e))*sin(f*x + e))/((c^2*d^5 +
2*c*d^6 + d^7)*f*cos(f*x + e)^2 - 2*(c^3*d^4 + 2*c^2*d^5 + c*d^6)*f*sin(f*x + e) - (c^4*d^3 + 2*c^3*d^4 + 2*c
^2*d^5 + 2*c*d^6 + d^7)*f), 1/2*(2*(a^3*c^2*d^2 + 2*a^3*c*d^3 + a^3*d^4)*f*x*cos(f*x + e)^2 - 2*(a^3*c^4 + 2*a
^3*c^3*d + 2*a^3*c^2*d^2 + 2*a^3*c*d^3 + a^3*d^4)*f*x - (2*a^3*c^4 + 6*a^3*c^3*d + 9*a^3*c^2*d^2 + 6*a^3*c*d^3
+ 7*a^3*d^4 - (2*a^3*c^2*d^2 + 6*a^3*c*d^3 + 7*a^3*d^4)*cos(f*x + e)^2 + 2*(2*a^3*c^3*d + 6*a^3*c^2*d^2 + 7*a
^3*c*d^3)*sin(f*x + e))*sqrt((c - d)/(c + d))*arctan(-(c*sin(f*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(
f*x + e))) - (2*a^3*c^3*d + 4*a^3*c^2*d^2 - 5*a^3*c*d^3 - a^3*d^4)*cos(f*x + e) - (4*(a^3*c^3*d + 2*a^3*c^2*d^
2 + a^3*c*d^3)*f*x + 3*(a^3*c^2*d^2 + a^3*c*d^3 - 2*a^3*d^4)*cos(f*x + e))*sin(f*x + e))/((c^2*d^5 + 2*c*d^6 +
d^7)*f*cos(f*x + e)^2 - 2*(c^3*d^4 + 2*c^2*d^5 + c*d^6)*f*sin(f*x + e) - (c^4*d^3 + 2*c^3*d^4 + 2*c^2*d^5 + 2
*c*d^6 + d^7)*f)]

________________________________________________________________________________________

giac [B]  time = 0.41, size = 522, normalized size = 2.79 \[ \frac {\frac {{\left (f x + e\right )} a^{3}}{d^{3}} - \frac {{\left (2 \, a^{3} c^{3} + 4 \, a^{3} c^{2} d + a^{3} c d^{2} - 7 \, a^{3} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (c) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right )} \sqrt {c^{2} - d^{2}}} + \frac {a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 16 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{3} c^{5} + 4 \, a^{3} c^{4} d - 5 \, a^{3} c^{3} d^{2} - a^{3} c^{2} d^{3}}{{\left (c^{4} d^{2} + 2 \, c^{3} d^{3} + c^{2} d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{2}}}{f} \]

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))^3,x, algorithm="giac")

[Out]

((f*x + e)*a^3/d^3 - (2*a^3*c^3 + 4*a^3*c^2*d + a^3*c*d^2 - 7*a^3*d^3)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(c
) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((c^2*d^3 + 2*c*d^4 + d^5)*sqrt(c^2 - d^2)) + (a^3*c
^4*d*tan(1/2*f*x + 1/2*e)^3 + 5*a^3*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 - 4*a^3*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 - 2*
a^3*c*d^4*tan(1/2*f*x + 1/2*e)^3 + 2*a^3*c^5*tan(1/2*f*x + 1/2*e)^2 + 4*a^3*c^4*d*tan(1/2*f*x + 1/2*e)^2 - a^3
*c^3*d^2*tan(1/2*f*x + 1/2*e)^2 + 7*a^3*c^2*d^3*tan(1/2*f*x + 1/2*e)^2 - 10*a^3*c*d^4*tan(1/2*f*x + 1/2*e)^2 -
2*a^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 7*a^3*c^4*d*tan(1/2*f*x + 1/2*e) + 11*a^3*c^3*d^2*tan(1/2*f*x + 1/2*e) - 1
6*a^3*c^2*d^3*tan(1/2*f*x + 1/2*e) - 2*a^3*c*d^4*tan(1/2*f*x + 1/2*e) + 2*a^3*c^5 + 4*a^3*c^4*d - 5*a^3*c^3*d^
2 - a^3*c^2*d^3)/((c^4*d^2 + 2*c^3*d^3 + c^2*d^4)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2)
)/f

________________________________________________________________________________________

maple [B]  time = 0.34, size = 1400, normalized size = 7.49 \[ \text {result too large to display} \]

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))^3,x)

[Out]

a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^2*tan(1/2*f*x+1/2*e)^3+5*a^3/f/(
tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c*tan(1/2*f*x+1/2*e)^3-4*a^3/f*d/(tan(1/2*f
*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3-2*a^3/f*d^2/(tan(1/2*f*x+1/2*e)
^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)/c*tan(1/2*f*x+1/2*e)^3+2*a^3/f/d^2/(tan(1/2*f*x+1/2*e)^2*c+2*
tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^3*tan(1/2*f*x+1/2*e)^2+4*a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2
*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^2*tan(1/2*f*x+1/2*e)^2-a^3/f/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)
*d+c)^2/(c^2+2*c*d+d^2)*c*tan(1/2*f*x+1/2*e)^2+7*a^3/f*d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(
c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^2-10*a^3/f*d^2/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*
d+d^2)/c*tan(1/2*f*x+1/2*e)^2-2*a^3/f*d^3/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)/
c^2*tan(1/2*f*x+1/2*e)^2+7*a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2*c^2/(c^2+2*c*d+d^2)*tan
(1/2*f*x+1/2*e)+11*a^3/f/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2*c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2
*e)-16*a^3/f*d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)-2*a^3/f*
d^2/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)+2*a^3/f/d^2/(tan(
1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^3+4*a^3/f/d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2
*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^2-5*a^3/f/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+
d^2)*c-a^3/f*d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)-2*a^3/f/d^3/(c^2+2*c*d+d^2)
/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*c^3-4*a^3/f/d^2/(c^2+2*c*d+d^2)/(c^2
-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*c^2-a^3/f/d/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/
2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*c+7*a^3/f/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1
/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))+2*a^3/f/d^3*arctan(tan(1/2*f*x+1/2*e))

________________________________________________________________________________________

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

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))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d^2-4*c^2>0)', see `assume?`
for more details)Is 4*d^2-4*c^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 14.27, size = 6246, normalized size = 33.40 \[ \text {result too large to display} \]

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))^3,x)

[Out]

((2*a^3*c^3 - a^3*d^3 - 5*a^3*c*d^2 + 4*a^3*c^2*d)/(d^2*(2*c*d + c^2 + d^2)) + (tan(e/2 + (f*x)/2)^3*(a^3*c^3
- 2*a^3*d^3 - 4*a^3*c*d^2 + 5*a^3*c^2*d))/(c*d*(2*c*d + c^2 + d^2)) + (tan(e/2 + (f*x)/2)*(7*a^3*c^3 - 2*a^3*d
^3 - 16*a^3*c*d^2 + 11*a^3*c^2*d))/(c*d*(2*c*d + c^2 + d^2)) + (tan(e/2 + (f*x)/2)^2*(c^2 + 2*d^2)*(2*a^3*c^3
- a^3*d^3 - 5*a^3*c*d^2 + 4*a^3*c^2*d))/(c^2*d^2*(2*c*d + c^2 + d^2)))/(f*(tan(e/2 + (f*x)/2)^2*(2*c^2 + 4*d^2
) + c^2*tan(e/2 + (f*x)/2)^4 + c^2 + 4*c*d*tan(e/2 + (f*x)/2)^3 + 4*c*d*tan(e/2 + (f*x)/2))) - (2*a^3*atan(-((
(((((8*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d
^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*
c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 - (8*(4*c*d^10 + 2*c^2*d^9 -
6*c^3*d^8 - 2*c^4*d^7 + 2*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)
*(28*c*d^11 + 52*c^2*d^10 + 4*c^3*d^9 - 44*c^4*d^8 - 32*c^5*d^7 - 8*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*
c^3*d^7 + c^4*d^6))*1i)/d^3 + (8*(4*c^2*d^6 + 16*c^3*d^5 + 24*c^4*d^4 + 16*c^5*d^3 + 4*c^6*d^2))/(4*c*d^8 + d^
9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(41*c*d^8 - 46*c^2*d^7 - 99*c^3*d^6 - 36*c^4*d^5
+ 36*c^5*d^4 + 32*c^6*d^3 + 8*c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))/d^3 + ((((8*(4*c*d
^10 + 2*c^2*d^9 - 6*c^3*d^8 - 2*c^4*d^7 + 2*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (((8
*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c
^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^
9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 - (8*tan(e/2 + (f*x)/2)*(28*c*d^11
+ 52*c^2*d^10 + 4*c^3*d^9 - 44*c^4*d^8 - 32*c^5*d^7 - 8*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c
^4*d^6))*1i)/d^3 + (8*(4*c^2*d^6 + 16*c^3*d^5 + 24*c^4*d^4 + 16*c^5*d^3 + 4*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d
^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(41*c*d^8 - 46*c^2*d^7 - 99*c^3*d^6 - 36*c^4*d^5 + 36*c^5*d^
4 + 32*c^6*d^3 + 8*c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))/d^3)/((16*(18*c^4*d - 49*c*d^
4 + 2*c^5 + 29*c^3*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (((((((8*(4*c^2*d^12 + 16*c^3*d^1
1 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f
*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9
+ d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 - (8*(4*c*d^10 + 2*c^2*d^9 - 6*c^3*d^8 - 2*c^4*d^7 + 2*c^5*
d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(28*c*d^11 + 52*c^2*d^10 + 4*c
^3*d^9 - 44*c^4*d^8 - 32*c^5*d^7 - 8*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 + (
8*(4*c^2*d^6 + 16*c^3*d^5 + 24*c^4*d^4 + 16*c^5*d^3 + 4*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4
*d^5) - (8*tan(e/2 + (f*x)/2)*(41*c*d^8 - 46*c^2*d^7 - 99*c^3*d^6 - 36*c^4*d^5 + 36*c^5*d^4 + 32*c^6*d^3 + 8*c
^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 + (((((8*(4*c*d^10 + 2*c^2*d^9 - 6*c^3*d^
8 - 2*c^4*d^7 + 2*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (((8*(4*c^2*d^12 + 16*c^3*d^11
+ 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*
x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 +
d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 - (8*tan(e/2 + (f*x)/2)*(28*c*d^11 + 52*c^2*d^10 + 4*c^3*d^9
- 44*c^4*d^8 - 32*c^5*d^7 - 8*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 + (8*(4*c
^2*d^6 + 16*c^3*d^5 + 24*c^4*d^4 + 16*c^5*d^3 + 4*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5)
- (8*tan(e/2 + (f*x)/2)*(41*c*d^8 - 46*c^2*d^7 - 99*c^3*d^6 - 36*c^4*d^5 + 36*c^5*d^4 + 32*c^6*d^3 + 8*c^7*d^2
))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3 - (16*tan(e/2 + (f*x)/2)*(28*c*d^5 - 32*c^5*d -
8*c^6 + 52*c^2*d^4 + 4*c^3*d^3 - 44*c^4*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))))/(d^3*f) +
(a^3*atan(((a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*((8*(4*a^6*c^2*d^6 + 16*a^6*c^3*d^5 + 24
*a^6*c^4*d^4 + 16*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2
+ (f*x)/2)*(41*a^6*c*d^8 - 46*a^6*c^2*d^7 - 99*a^6*c^3*d^6 - 36*a^6*c^4*d^5 + 36*a^6*c^5*d^4 + 32*a^6*c^6*d^3
+ 8*a^6*c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d
+ c^2 + (7*d^2)/2)*((8*tan(e/2 + (f*x)/2)*(28*a^3*c*d^11 + 52*a^3*c^2*d^10 + 4*a^3*c^3*d^9 - 44*a^3*c^4*d^8 -
32*a^3*c^5*d^7 - 8*a^3*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) - (8*(4*a^3*c*d^10 + 2*a^
3*c^2*d^9 - 6*a^3*c^3*d^8 - 2*a^3*c^4*d^7 + 2*a^3*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5)
+ (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*
d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 +
16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*(3*c*
d + c^2 + (7*d^2)/2))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)))/(5*c*d^7 + d^8 + 10*c^
2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3))*1i)/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)
+ (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*((8*(4*a^6*c^2*d^6 + 16*a^6*c^3*d^5 + 24*a^6*c^4*
d^4 + 16*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/
2)*(41*a^6*c*d^8 - 46*a^6*c^2*d^7 - 99*a^6*c^3*d^6 - 36*a^6*c^4*d^5 + 36*a^6*c^5*d^4 + 32*a^6*c^6*d^3 + 8*a^6*
c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 +
(7*d^2)/2)*((8*(4*a^3*c*d^10 + 2*a^3*c^2*d^9 - 6*a^3*c^3*d^8 - 2*a^3*c^4*d^7 + 2*a^3*c^5*d^6))/(4*c*d^8 + d^9
+ 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(28*a^3*c*d^11 + 52*a^3*c^2*d^10 + 4*a^3*c^3*d^9 -
44*a^3*c^4*d^8 - 32*a^3*c^5*d^7 - 8*a^3*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (a^3*(-
(c + d)^5*(c - d))^(1/2)*((8*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9
+ 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^
11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*(3*c*d + c^2 +
(7*d^2)/2))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 1
0*c^3*d^5 + 5*c^4*d^4 + c^5*d^3))*1i)/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3))/((16*(2
*a^9*c^5 - 49*a^9*c*d^4 + 18*a^9*c^4*d + 29*a^9*c^3*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) +
(16*tan(e/2 + (f*x)/2)*(8*a^9*c^6 - 28*a^9*c*d^5 + 32*a^9*c^5*d - 52*a^9*c^2*d^4 - 4*a^9*c^3*d^3 + 44*a^9*c^4*
d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) - (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d
^2)/2)*((8*(4*a^6*c^2*d^6 + 16*a^6*c^3*d^5 + 24*a^6*c^4*d^4 + 16*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(4*c*d^8 + d^9
+ 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(41*a^6*c*d^8 - 46*a^6*c^2*d^7 - 99*a^6*c^3*d^6 - 3
6*a^6*c^4*d^5 + 36*a^6*c^5*d^4 + 32*a^6*c^6*d^3 + 8*a^6*c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^
4*d^6) + (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*((8*tan(e/2 + (f*x)/2)*(28*a^3*c*d^11 + 52*
a^3*c^2*d^10 + 4*a^3*c^3*d^9 - 44*a^3*c^4*d^8 - 32*a^3*c^5*d^7 - 8*a^3*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 +
4*c^3*d^7 + c^4*d^6) - (8*(4*a^3*c*d^10 + 2*a^3*c^2*d^9 - 6*a^3*c^3*d^8 - 2*a^3*c^4*d^7 + 2*a^3*c^5*d^6))/(4*
c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*c^2*d^12 + 16*c^3*d^11
+ 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*
x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 +
d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*(3*c*d + c^2 + (7*d^2)/2))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5
+ 5*c^4*d^4 + c^5*d^3)))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)))/(5*c*d^7 + d^8 + 10
*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3) + (a^3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*((8*(
4*a^6*c^2*d^6 + 16*a^6*c^3*d^5 + 24*a^6*c^4*d^4 + 16*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7
+ 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(41*a^6*c*d^8 - 46*a^6*c^2*d^7 - 99*a^6*c^3*d^6 - 36*a^6*c^4*d^
5 + 36*a^6*c^5*d^4 + 32*a^6*c^6*d^3 + 8*a^6*c^7*d^2))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (a^
3*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*((8*(4*a^3*c*d^10 + 2*a^3*c^2*d^9 - 6*a^3*c^3*d^8 - 2*a
^3*c^4*d^7 + 2*a^3*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(28*a^3
*c*d^11 + 52*a^3*c^2*d^10 + 4*a^3*c^3*d^9 - 44*a^3*c^4*d^8 - 32*a^3*c^5*d^7 - 8*a^3*c^6*d^6))/(4*c*d^9 + d^10
+ 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (a^3*(-(c + d)^5*(c - d))^(1/2)*((8*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^
10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*
d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^
2*d^8 + 4*c^3*d^7 + c^4*d^6))*(3*c*d + c^2 + (7*d^2)/2))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4
+ c^5*d^3)))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)))/(5*c*d^7 + d^8 + 10*c^2*d^6 + 1
0*c^3*d^5 + 5*c^4*d^4 + c^5*d^3)))*(-(c + d)^5*(c - d))^(1/2)*(3*c*d + c^2 + (7*d^2)/2)*2i)/(f*(5*c*d^7 + d^8
+ 10*c^2*d^6 + 10*c^3*d^5 + 5*c^4*d^4 + c^5*d^3))

________________________________________________________________________________________

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))**3,x)

[Out]

Timed out

________________________________________________________________________________________