3.479 \(\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx\)

Optimal. Leaf size=378 \[ -\frac{d^3 \left (20 c^2+30 c d+13 d^2\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{a^3 f (c-d)^5 (c+d)^2 \sqrt{c^2-d^2}}-\frac{d \left (142 c^2 d^2-30 c^3 d+4 c^4+525 c d^3+304 d^4\right ) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{d \left (-30 c^2 d+4 c^3+146 c d^2+195 d^3\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{\left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{15 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2} \]

[Out]

-((d^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^5*(c + d)^2*S
qrt[c^2 - d^2]*f)) - (d*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3)*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c
+ d*Sin[e + f*x])^2) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2) - ((2*c - 11*d
)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - ((2*c^2 - 15*c*d + 76*d^2)*
Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - (d*(4*c^4 - 30*c^3*d + 142*c^
2*d^2 + 525*c*d^3 + 304*d^4)*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*Sin[e + f*x]))

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Rubi [A]  time = 0.96211, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {2766, 2978, 2754, 12, 2660, 618, 204} \[ -\frac{d^3 \left (20 c^2+30 c d+13 d^2\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{a^3 f (c-d)^5 (c+d)^2 \sqrt{c^2-d^2}}-\frac{d \left (142 c^2 d^2-30 c^3 d+4 c^4+525 c d^3+304 d^4\right ) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{d \left (-30 c^2 d+4 c^3+146 c d^2+195 d^3\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{\left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{15 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]

[Out]

-((d^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^5*(c + d)^2*S
qrt[c^2 - d^2]*f)) - (d*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3)*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c
+ d*Sin[e + f*x])^2) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2) - ((2*c - 11*d
)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - ((2*c^2 - 15*c*d + 76*d^2)*
Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - (d*(4*c^4 - 30*c^3*d + 142*c^
2*d^2 + 525*c*d^3 + 304*d^4)*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*Sin[e + f*x]))

Rule 2766

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

Rule 2978

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

Rule 2754

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*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 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{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{\int \frac{-a (2 c-7 d)-4 a d \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx}{5 a^2 (c-d)}\\ &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac{\int \frac{a^2 \left (2 c^2-9 c d+43 d^2\right )+3 a^2 (2 c-11 d) d \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx}{15 a^4 (c-d)^2}\\ &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{\int \frac{-3 a^3 (2 c-65 d) d^2-2 a^3 d \left (2 c^2-15 c d+76 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{15 a^6 (c-d)^3}\\ &=-\frac{d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}+\frac{\int \frac{2 a^3 d^2 \left (2 c^2-165 c d-152 d^2\right )+a^3 d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{30 a^6 (c-d)^4 (c+d)}\\ &=-\frac{d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))}-\frac{\int \frac{15 a^3 d^3 \left (20 c^2+30 c d+13 d^2\right )}{c+d \sin (e+f x)} \, dx}{30 a^6 (c-d)^5 (c+d)^2}\\ &=-\frac{d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))}-\frac{\left (d^3 \left (20 c^2+30 c d+13 d^2\right )\right ) \int \frac{1}{c+d \sin (e+f x)} \, dx}{2 a^3 (c-d)^5 (c+d)^2}\\ &=-\frac{d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))}-\frac{\left (d^3 \left (20 c^2+30 c d+13 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 )}{a^3 (c-d)^5 (c+d)^2 f}\\ &=-\frac{d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))}+\frac{\left (2 d^3 \left (20 c^2+30 c d+13 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 )}{a^3 (c-d)^5 (c+d)^2 f}\\ &=-\frac{d^3 \left (20 c^2+30 c d+13 d^2\right ) \tan ^{-1}\left (\frac{d+c \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{a^3 (c-d)^5 (c+d)^2 \sqrt{c^2-d^2} f}-\frac{d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{\left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac{d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))}\\ \end{align*}

Mathematica [B]  time = 6.26632, size = 914, normalized size = 2.42 \[ \frac{\left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\frac{320 \sin \left (\frac{1}{2} (e+f x)\right ) c^6-32 \sin \left (\frac{5}{2} (e+f x)\right ) c^6+32 d \cos \left (\frac{7}{2} (e+f x)\right ) c^5-1520 d \sin \left (\frac{1}{2} (e+f x)\right ) c^5+80 d \sin \left (\frac{5}{2} (e+f x)\right ) c^5-200 d^2 \cos \left (\frac{7}{2} (e+f x)\right ) c^4+4568 d^2 \sin \left (\frac{1}{2} (e+f x)\right ) c^4+800 d^2 \sin \left (\frac{3}{2} (e+f x)\right ) c^4-32 d^2 \sin \left (\frac{5}{2} (e+f x)\right ) c^4+8 d^2 \sin \left (\frac{9}{2} (e+f x)\right ) c^4-1260 d^3 \cos \left (\frac{5}{2} (e+f x)\right ) c^3+836 d^3 \cos \left (\frac{7}{2} (e+f x)\right ) c^3+27340 d^3 \sin \left (\frac{1}{2} (e+f x)\right ) c^3+7500 d^3 \sin \left (\frac{3}{2} (e+f x)\right ) c^3-6820 d^3 \sin \left (\frac{5}{2} (e+f x)\right ) c^3-60 d^3 \sin \left (\frac{9}{2} (e+f x)\right ) c^3-2640 d^4 \cos \left (\frac{5}{2} (e+f x)\right ) c^2+4480 d^4 \cos \left (\frac{7}{2} (e+f x)\right ) c^2+40904 d^4 \sin \left (\frac{1}{2} (e+f x)\right ) c^2+13280 d^4 \sin \left (\frac{3}{2} (e+f x)\right ) c^2-18080 d^4 \sin \left (\frac{5}{2} (e+f x)\right ) c^2-60 d^4 \sin \left (\frac{7}{2} (e+f x)\right ) c^2+284 d^4 \sin \left (\frac{9}{2} (e+f x)\right ) c^2-2250 d^5 \cos \left (\frac{5}{2} (e+f x)\right ) c+5747 d^5 \cos \left (\frac{7}{2} (e+f x)\right ) c-135 d^5 \cos \left (\frac{9}{2} (e+f x)\right ) c+26020 d^5 \sin \left (\frac{1}{2} (e+f x)\right ) c+9690 d^5 \sin \left (\frac{3}{2} (e+f x)\right ) c-15670 d^5 \sin \left (\frac{5}{2} (e+f x)\right ) c+135 d^5 \sin \left (\frac{7}{2} (e+f x)\right ) c+915 d^5 \sin \left (\frac{9}{2} (e+f x)\right ) c+10 d \left (-40 c^5+340 d c^4+1934 d^2 c^3+3040 d^3 c^2+1994 d^4 c+481 d^5\right ) \cos \left (\frac{1}{2} (e+f x)\right )-2 \left (80 c^6-424 d c^5+1200 d^2 c^4+9698 d^3 c^3+17640 d^4 c^2+12371 d^5 c+2905 d^6\right ) \cos \left (\frac{3}{2} (e+f x)\right )-870 d^6 \cos \left (\frac{5}{2} (e+f x)\right )+2200 d^6 \cos \left (\frac{7}{2} (e+f x)\right )-90 d^6 \cos \left (\frac{9}{2} (e+f x)\right )+6318 d^6 \sin \left (\frac{1}{2} (e+f x)\right )+2750 d^6 \sin \left (\frac{3}{2} (e+f x)\right )-4266 d^6 \sin \left (\frac{5}{2} (e+f x)\right )+60 d^6 \sin \left (\frac{7}{2} (e+f x)\right )+518 d^6 \sin \left (\frac{9}{2} (e+f x)\right )}{(c+d \sin (e+f x))^2}-\frac{480 d^3 \left (20 c^2+30 d c+13 d^2\right ) \tan ^{-1}\left (\frac{d+c \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{\sqrt{c^2-d^2}}\right )}{480 a^3 (c-d)^5 (c+d)^2 f (\sin (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((-480*d^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/S
qrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/Sqrt[c^2 - d^2] + (10*d*(-40*c^5 + 340*c^4*d + 1934*c
^3*d^2 + 3040*c^2*d^3 + 1994*c*d^4 + 481*d^5)*Cos[(e + f*x)/2] - 2*(80*c^6 - 424*c^5*d + 1200*c^4*d^2 + 9698*c
^3*d^3 + 17640*c^2*d^4 + 12371*c*d^5 + 2905*d^6)*Cos[(3*(e + f*x))/2] - 1260*c^3*d^3*Cos[(5*(e + f*x))/2] - 26
40*c^2*d^4*Cos[(5*(e + f*x))/2] - 2250*c*d^5*Cos[(5*(e + f*x))/2] - 870*d^6*Cos[(5*(e + f*x))/2] + 32*c^5*d*Co
s[(7*(e + f*x))/2] - 200*c^4*d^2*Cos[(7*(e + f*x))/2] + 836*c^3*d^3*Cos[(7*(e + f*x))/2] + 4480*c^2*d^4*Cos[(7
*(e + f*x))/2] + 5747*c*d^5*Cos[(7*(e + f*x))/2] + 2200*d^6*Cos[(7*(e + f*x))/2] - 135*c*d^5*Cos[(9*(e + f*x))
/2] - 90*d^6*Cos[(9*(e + f*x))/2] + 320*c^6*Sin[(e + f*x)/2] - 1520*c^5*d*Sin[(e + f*x)/2] + 4568*c^4*d^2*Sin[
(e + f*x)/2] + 27340*c^3*d^3*Sin[(e + f*x)/2] + 40904*c^2*d^4*Sin[(e + f*x)/2] + 26020*c*d^5*Sin[(e + f*x)/2]
+ 6318*d^6*Sin[(e + f*x)/2] + 800*c^4*d^2*Sin[(3*(e + f*x))/2] + 7500*c^3*d^3*Sin[(3*(e + f*x))/2] + 13280*c^2
*d^4*Sin[(3*(e + f*x))/2] + 9690*c*d^5*Sin[(3*(e + f*x))/2] + 2750*d^6*Sin[(3*(e + f*x))/2] - 32*c^6*Sin[(5*(e
+ f*x))/2] + 80*c^5*d*Sin[(5*(e + f*x))/2] - 32*c^4*d^2*Sin[(5*(e + f*x))/2] - 6820*c^3*d^3*Sin[(5*(e + f*x))
/2] - 18080*c^2*d^4*Sin[(5*(e + f*x))/2] - 15670*c*d^5*Sin[(5*(e + f*x))/2] - 4266*d^6*Sin[(5*(e + f*x))/2] -
60*c^2*d^4*Sin[(7*(e + f*x))/2] + 135*c*d^5*Sin[(7*(e + f*x))/2] + 60*d^6*Sin[(7*(e + f*x))/2] + 8*c^4*d^2*Sin
[(9*(e + f*x))/2] - 60*c^3*d^3*Sin[(9*(e + f*x))/2] + 284*c^2*d^4*Sin[(9*(e + f*x))/2] + 915*c*d^5*Sin[(9*(e +
f*x))/2] + 518*d^6*Sin[(9*(e + f*x))/2])/(c + d*Sin[e + f*x])^2))/(480*a^3*(c - d)^5*(c + d)^2*f*(1 + Sin[e +
f*x])^3)

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Maple [B]  time = 0.141, size = 1462, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x)

[Out]

-20/f/a^3/(c-d)^5/(tan(1/2*f*x+1/2*e)+1)*d^2+4/f/a^3/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)^2*c-10/f/a^3/(c-d)^4/(tan(
1/2*f*x+1/2*e)+1)^2*d-16/3/f/a^3/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)^3*c+28/3/f/a^3/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)^
3*d+2/f/a^3*d^7/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+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/f/a^3*d^7/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+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
)^3-10/f/a^3*d^4/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+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)^2+2/f/a^3*d^8/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+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)^2-29/f/a^3*d^5/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+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)-6/f/a^3*d^5/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c-13/f/a^3*
d^5/(c-d)^5/(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))-6/f/a^3*d
^6/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+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)^3-19/f/a^3*d
^6/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+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)^2-18/f/a^3*d
^6/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+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)-10/f/a^3*d^4
/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^2)*c^2-12/f/a^3*d^7/(c-d)^5/(c*tan(1
/2*f*x+1/2*e)^2+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)^2-6/f/a^3*d^5/(c-d)^5/(c*tan(
1/2*f*x+1/2*e)^2+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)^2-20/f/a^3*d^3/(c-d)^5/(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^2-30/f/a^3*d^4/(c-d)^5/(
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-11/f/a^3*d^5/(c-d)^5
/(c*tan(1/2*f*x+1/2*e)^2+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)^3-2/f/a^3/(c-d)^5/(t
an(1/2*f*x+1/2*e)+1)*c^2+1/f/a^3*d^6/(c-d)^5/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^2+2*c*d+d^
2)+10/f/a^3/(c-d)^5/(tan(1/2*f*x+1/2*e)+1)*c*d+4/f/a^3/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^4-8/5/f/a^3/(c-d)^3/(tan
(1/2*f*x+1/2*e)+1)^5

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.51949, size = 11534, normalized size = 30.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[-1/60*(12*c^8 - 24*c^7*d - 24*c^6*d^2 + 72*c^5*d^3 - 72*c^3*d^5 + 24*c^2*d^6 + 24*c*d^7 - 12*d^8 + 2*(4*c^6*d
^2 - 30*c^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 - 525*c*d^7 - 304*d^8)*cos(f*x + e)^5 - 2*(8*c^7*d -
52*c^6*d^2 + 216*c^5*d^3 + 1086*c^4*d^4 + 984*c^3*d^5 - 621*c^2*d^6 - 1208*c*d^7 - 413*d^8)*cos(f*x + e)^4 -
2*(4*c^8 - 6*c^7*d - 20*c^6*d^2 + 768*c^5*d^3 + 2676*c^4*d^4 + 2307*c^3*d^5 - 1573*c^2*d^6 - 3069*c*d^7 - 1087
*d^8)*cos(f*x + e)^3 + 4*(4*c^8 - 20*c^7*d + 19*c^6*d^2 + 330*c^5*d^3 + 699*c^4*d^4 + 345*c^3*d^5 - 526*c^2*d^
6 - 655*c*d^7 - 196*d^8)*cos(f*x + e)^2 - 15*(80*c^4*d^3 + 280*c^3*d^4 + 372*c^2*d^5 + 224*c*d^6 + 52*d^7 + (2
0*c^2*d^5 + 30*c*d^6 + 13*d^7)*cos(f*x + e)^5 + (40*c^3*d^4 + 120*c^2*d^5 + 116*c*d^6 + 39*d^7)*cos(f*x + e)^4
- (20*c^4*d^3 + 110*c^3*d^4 + 193*c^2*d^5 + 142*c*d^6 + 39*d^7)*cos(f*x + e)^3 - (60*c^4*d^3 + 290*c^3*d^4 +
479*c^2*d^5 + 340*c*d^6 + 91*d^7)*cos(f*x + e)^2 + 2*(20*c^4*d^3 + 70*c^3*d^4 + 93*c^2*d^5 + 56*c*d^6 + 13*d^7
)*cos(f*x + e) + (80*c^4*d^3 + 280*c^3*d^4 + 372*c^2*d^5 + 224*c*d^6 + 52*d^7 + (20*c^2*d^5 + 30*c*d^6 + 13*d^
7)*cos(f*x + e)^4 - 2*(20*c^3*d^4 + 50*c^2*d^5 + 43*c*d^6 + 13*d^7)*cos(f*x + e)^3 - (20*c^4*d^3 + 150*c^3*d^4
+ 293*c^2*d^5 + 228*c*d^6 + 65*d^7)*cos(f*x + e)^2 + 2*(20*c^4*d^3 + 70*c^3*d^4 + 93*c^2*d^5 + 56*c*d^6 + 13*
d^7)*cos(f*x + e))*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2
- d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*
x + e) - c^2 - d^2)) + 12*(3*c^8 - 11*c^7*d + 9*c^6*d^2 + 213*c^5*d^3 + 475*c^4*d^4 + 237*c^3*d^5 - 359*c^2*d^
6 - 439*c*d^7 - 128*d^8)*cos(f*x + e) - 2*(6*c^8 - 12*c^7*d - 12*c^6*d^2 + 36*c^5*d^3 - 36*c^3*d^5 + 12*c^2*d^
6 + 12*c*d^7 - 6*d^8 + (4*c^6*d^2 - 30*c^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 - 525*c*d^7 - 304*d^8
)*cos(f*x + e)^4 + (8*c^7*d - 48*c^6*d^2 + 186*c^5*d^3 + 1224*c^4*d^4 + 1539*c^3*d^5 - 459*c^2*d^6 - 1733*c*d^
7 - 717*d^8)*cos(f*x + e)^3 - 2*(2*c^8 - 7*c^7*d + 14*c^6*d^2 + 291*c^5*d^3 + 726*c^4*d^4 + 384*c^3*d^5 - 557*
c^2*d^6 - 668*c*d^7 - 185*d^8)*cos(f*x + e)^2 - 6*(2*c^8 - 9*c^7*d + 11*c^6*d^2 + 207*c^5*d^3 + 475*c^4*d^4 +
243*c^3*d^5 - 361*c^2*d^6 - 441*c*d^7 - 127*d^8)*cos(f*x + e))*sin(f*x + e))/((a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8
*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^5 + (2*
a^3*c^10*d - 3*a^3*c^9*d^2 - 9*a^3*c^8*d^3 + 16*a^3*c^7*d^4 + 12*a^3*c^6*d^5 - 30*a^3*c^5*d^6 - 2*a^3*c^4*d^7
+ 24*a^3*c^3*d^8 - 6*a^3*c^2*d^9 - 7*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^4 - (a^3*c^11 + a^3*c^10*d - 9*a^
3*c^9*d^2 - a^3*c^8*d^3 + 26*a^3*c^7*d^4 - 6*a^3*c^6*d^5 - 34*a^3*c^5*d^6 + 14*a^3*c^4*d^7 + 21*a^3*c^3*d^8 -
11*a^3*c^2*d^9 - 5*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^3 - (3*a^3*c^11 + a^3*c^10*d - 23*a^3*c^9*d^2 + 3*a
^3*c^8*d^3 + 62*a^3*c^7*d^4 - 22*a^3*c^6*d^5 - 78*a^3*c^5*d^6 + 38*a^3*c^4*d^7 + 47*a^3*c^3*d^8 - 27*a^3*c^2*d
^9 - 11*a^3*c*d^10 + 7*a^3*d^11)*f*cos(f*x + e)^2 + 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 +
10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^1
0 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*
a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f + ((a
^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^
3*d^11)*f*cos(f*x + e)^4 - 2*(a^3*c^10*d - 2*a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^7*d^4 + 2*a^3*c^6*d^5 - 12*
a^3*c^5*d^6 + 2*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c^2*d^9 - 2*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^3 - (a^3
*c^11 + 3*a^3*c^10*d - 13*a^3*c^9*d^2 - 7*a^3*c^8*d^3 + 42*a^3*c^7*d^4 - 2*a^3*c^6*d^5 - 58*a^3*c^5*d^6 + 18*a
^3*c^4*d^7 + 37*a^3*c^3*d^8 - 17*a^3*c^2*d^9 - 9*a^3*c*d^10 + 5*a^3*d^11)*f*cos(f*x + e)^2 + 2*(a^3*c^11 - a^3
*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 +
5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d
^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3
*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f)*sin(f*x + e)), -1/30*(6*c^8 - 12*c^7*d - 12*c^6*d^2 + 36*c^5*d^3 - 36*c^3
*d^5 + 12*c^2*d^6 + 12*c*d^7 - 6*d^8 + (4*c^6*d^2 - 30*c^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 - 525
*c*d^7 - 304*d^8)*cos(f*x + e)^5 - (8*c^7*d - 52*c^6*d^2 + 216*c^5*d^3 + 1086*c^4*d^4 + 984*c^3*d^5 - 621*c^2*
d^6 - 1208*c*d^7 - 413*d^8)*cos(f*x + e)^4 - (4*c^8 - 6*c^7*d - 20*c^6*d^2 + 768*c^5*d^3 + 2676*c^4*d^4 + 2307
*c^3*d^5 - 1573*c^2*d^6 - 3069*c*d^7 - 1087*d^8)*cos(f*x + e)^3 + 2*(4*c^8 - 20*c^7*d + 19*c^6*d^2 + 330*c^5*d
^3 + 699*c^4*d^4 + 345*c^3*d^5 - 526*c^2*d^6 - 655*c*d^7 - 196*d^8)*cos(f*x + e)^2 - 15*(80*c^4*d^3 + 280*c^3*
d^4 + 372*c^2*d^5 + 224*c*d^6 + 52*d^7 + (20*c^2*d^5 + 30*c*d^6 + 13*d^7)*cos(f*x + e)^5 + (40*c^3*d^4 + 120*c
^2*d^5 + 116*c*d^6 + 39*d^7)*cos(f*x + e)^4 - (20*c^4*d^3 + 110*c^3*d^4 + 193*c^2*d^5 + 142*c*d^6 + 39*d^7)*co
s(f*x + e)^3 - (60*c^4*d^3 + 290*c^3*d^4 + 479*c^2*d^5 + 340*c*d^6 + 91*d^7)*cos(f*x + e)^2 + 2*(20*c^4*d^3 +
70*c^3*d^4 + 93*c^2*d^5 + 56*c*d^6 + 13*d^7)*cos(f*x + e) + (80*c^4*d^3 + 280*c^3*d^4 + 372*c^2*d^5 + 224*c*d^
6 + 52*d^7 + (20*c^2*d^5 + 30*c*d^6 + 13*d^7)*cos(f*x + e)^4 - 2*(20*c^3*d^4 + 50*c^2*d^5 + 43*c*d^6 + 13*d^7)
*cos(f*x + e)^3 - (20*c^4*d^3 + 150*c^3*d^4 + 293*c^2*d^5 + 228*c*d^6 + 65*d^7)*cos(f*x + e)^2 + 2*(20*c^4*d^3
+ 70*c^3*d^4 + 93*c^2*d^5 + 56*c*d^6 + 13*d^7)*cos(f*x + e))*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x
+ e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + 6*(3*c^8 - 11*c^7*d + 9*c^6*d^2 + 213*c^5*d^3 + 475*c^4*d^4 + 237
*c^3*d^5 - 359*c^2*d^6 - 439*c*d^7 - 128*d^8)*cos(f*x + e) - (6*c^8 - 12*c^7*d - 12*c^6*d^2 + 36*c^5*d^3 - 36*
c^3*d^5 + 12*c^2*d^6 + 12*c*d^7 - 6*d^8 + (4*c^6*d^2 - 30*c^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 -
525*c*d^7 - 304*d^8)*cos(f*x + e)^4 + (8*c^7*d - 48*c^6*d^2 + 186*c^5*d^3 + 1224*c^4*d^4 + 1539*c^3*d^5 - 459*
c^2*d^6 - 1733*c*d^7 - 717*d^8)*cos(f*x + e)^3 - 2*(2*c^8 - 7*c^7*d + 14*c^6*d^2 + 291*c^5*d^3 + 726*c^4*d^4 +
384*c^3*d^5 - 557*c^2*d^6 - 668*c*d^7 - 185*d^8)*cos(f*x + e)^2 - 6*(2*c^8 - 9*c^7*d + 11*c^6*d^2 + 207*c^5*d
^3 + 475*c^4*d^4 + 243*c^3*d^5 - 361*c^2*d^6 - 441*c*d^7 - 127*d^8)*cos(f*x + e))*sin(f*x + e))/((a^3*c^9*d^2
- 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f*c
os(f*x + e)^5 + (2*a^3*c^10*d - 3*a^3*c^9*d^2 - 9*a^3*c^8*d^3 + 16*a^3*c^7*d^4 + 12*a^3*c^6*d^5 - 30*a^3*c^5*d
^6 - 2*a^3*c^4*d^7 + 24*a^3*c^3*d^8 - 6*a^3*c^2*d^9 - 7*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^4 - (a^3*c^11
+ a^3*c^10*d - 9*a^3*c^9*d^2 - a^3*c^8*d^3 + 26*a^3*c^7*d^4 - 6*a^3*c^6*d^5 - 34*a^3*c^5*d^6 + 14*a^3*c^4*d^7
+ 21*a^3*c^3*d^8 - 11*a^3*c^2*d^9 - 5*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^3 - (3*a^3*c^11 + a^3*c^10*d - 2
3*a^3*c^9*d^2 + 3*a^3*c^8*d^3 + 62*a^3*c^7*d^4 - 22*a^3*c^6*d^5 - 78*a^3*c^5*d^6 + 38*a^3*c^4*d^7 + 47*a^3*c^3
*d^8 - 27*a^3*c^2*d^9 - 11*a^3*c*d^10 + 7*a^3*d^11)*f*cos(f*x + e)^2 + 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^
2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*
c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 1
0*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10
+ a^3*d^11)*f + ((a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8
- 3*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^4 - 2*(a^3*c^10*d - 2*a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^7*d^4 +
2*a^3*c^6*d^5 - 12*a^3*c^5*d^6 + 2*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c^2*d^9 - 2*a^3*c*d^10 + a^3*d^11)*f*co
s(f*x + e)^3 - (a^3*c^11 + 3*a^3*c^10*d - 13*a^3*c^9*d^2 - 7*a^3*c^8*d^3 + 42*a^3*c^7*d^4 - 2*a^3*c^6*d^5 - 58
*a^3*c^5*d^6 + 18*a^3*c^4*d^7 + 37*a^3*c^3*d^8 - 17*a^3*c^2*d^9 - 9*a^3*c*d^10 + 5*a^3*d^11)*f*cos(f*x + e)^2
+ 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6
+ 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c
^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*
a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f)*sin(f*x + e))]

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

[Out]

Timed out

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Giac [B]  time = 1.56968, size = 1072, normalized size = 2.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

-1/15*(15*(20*c^2*d^3 + 30*c*d^4 + 13*d^5)*(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)))/((a^3*c^7 - 3*a^3*c^6*d + a^3*c^5*d^2 + 5*a^3*c^4*d^3 - 5*a^3*c^3*d^4 - a^3*c^2*
d^5 + 3*a^3*c*d^6 - a^3*d^7)*sqrt(c^2 - d^2)) + 15*(11*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 + 6*c^2*d^6*tan(1/2*f*x
+ 1/2*e)^3 - 2*c*d^7*tan(1/2*f*x + 1/2*e)^3 + 10*c^4*d^4*tan(1/2*f*x + 1/2*e)^2 + 6*c^3*d^5*tan(1/2*f*x + 1/2*
e)^2 + 19*c^2*d^6*tan(1/2*f*x + 1/2*e)^2 + 12*c*d^7*tan(1/2*f*x + 1/2*e)^2 - 2*d^8*tan(1/2*f*x + 1/2*e)^2 + 29
*c^3*d^5*tan(1/2*f*x + 1/2*e) + 18*c^2*d^6*tan(1/2*f*x + 1/2*e) - 2*c*d^7*tan(1/2*f*x + 1/2*e) + 10*c^4*d^4 +
6*c^3*d^5 - c^2*d^6)/((a^3*c^9 - 3*a^3*c^8*d + a^3*c^7*d^2 + 5*a^3*c^6*d^3 - 5*a^3*c^5*d^4 - a^3*c^4*d^5 + 3*a
^3*c^3*d^6 - a^3*c^2*d^7)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2) + 2*(15*c^2*tan(1/2*f*x
+ 1/2*e)^4 - 75*c*d*tan(1/2*f*x + 1/2*e)^4 + 150*d^2*tan(1/2*f*x + 1/2*e)^4 + 30*c^2*tan(1/2*f*x + 1/2*e)^3 -
195*c*d*tan(1/2*f*x + 1/2*e)^3 + 525*d^2*tan(1/2*f*x + 1/2*e)^3 + 40*c^2*tan(1/2*f*x + 1/2*e)^2 - 245*c*d*tan
(1/2*f*x + 1/2*e)^2 + 745*d^2*tan(1/2*f*x + 1/2*e)^2 + 20*c^2*tan(1/2*f*x + 1/2*e) - 145*c*d*tan(1/2*f*x + 1/2
*e) + 485*d^2*tan(1/2*f*x + 1/2*e) + 7*c^2 - 44*c*d + 127*d^2)/((a^3*c^5 - 5*a^3*c^4*d + 10*a^3*c^3*d^2 - 10*a
^3*c^2*d^3 + 5*a^3*c*d^4 - a^3*d^5)*(tan(1/2*f*x + 1/2*e) + 1)^5))/f