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

Optimal. Leaf size=255 \[ -\frac{\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (-5 c^3 d^2+2 c^5+6 c d^4\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{d^3 f \left (c^2-d^2\right )^{5/2}}+\frac{(b c-a d)^2 \left (3 a c d+2 b c^2-5 b d^2\right ) \cos (e+f x)}{2 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}+\frac{b^3 x}{d^3} \]

[Out]

(b^3*x)/d^3 - ((9*a^2*b*c*d^4 - a^3*d^3*(2*c^2 + d^2) - 3*a*b^2*d^3*(c^2 + 2*d^2) + b^3*(2*c^5 - 5*c^3*d^2 + 6
*c*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c^2 - d^2)^(5/2)*f) + ((b*c - a*d)^2*Cos[e +
f*x]*(a + b*Sin[e + f*x]))/(2*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((b*c - a*d)^2*(2*b*c^2 + 3*a*c*d - 5*
b*d^2)*Cos[e + f*x])/(2*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))

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Rubi [A]  time = 0.631105, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2792, 3021, 2735, 2660, 618, 204} \[ -\frac{\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (-5 c^3 d^2+2 c^5+6 c d^4\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{d^3 f \left (c^2-d^2\right )^{5/2}}+\frac{(b c-a d)^2 \left (3 a c d+2 b c^2-5 b d^2\right ) \cos (e+f x)}{2 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}+\frac{b^3 x}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*x)/d^3 - ((9*a^2*b*c*d^4 - a^3*d^3*(2*c^2 + d^2) - 3*a*b^2*d^3*(c^2 + 2*d^2) + b^3*(2*c^5 - 5*c^3*d^2 + 6
*c*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c^2 - d^2)^(5/2)*f) + ((b*c - a*d)^2*Cos[e +
f*x]*(a + b*Sin[e + f*x]))/(2*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((b*c - a*d)^2*(2*b*c^2 + 3*a*c*d - 5*
b*d^2)*Cos[e + f*x])/(2*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*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 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]

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{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx &=\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}-\frac{\int \frac{b^3 c^2-2 a^3 c d-4 a b^2 c d+5 a^2 b d^2-\left (4 a^2 b c d+2 b^3 c d-a^3 d^2+a b^2 \left (c^2-6 d^2\right )\right ) \sin (e+f x)-2 b^3 \left (c^2-d^2\right ) \sin ^2(e+f x)}{(c+d \sin (e+f x))^2} \, dx}{2 d \left (c^2-d^2\right )}\\ &=\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac{\int \frac{-d \left (9 a^2 b c d^2-a^3 d \left (2 c^2+d^2\right )-3 a b^2 d \left (c^2+2 d^2\right )-b^3 \left (c^3-4 c d^2\right )\right )+2 b^3 \left (c^2-d^2\right )^2 \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{2 d^2 \left (c^2-d^2\right )^2}\\ &=\frac{b^3 x}{d^3}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac{\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right ) \int \frac{1}{c+d \sin (e+f x)} \, dx}{2 d^3 \left (c^2-d^2\right )^2}\\ &=\frac{b^3 x}{d^3}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac{\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\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 \left (c^2-d^2\right )^2 f}\\ &=\frac{b^3 x}{d^3}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac{\left (2 \left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\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 \left (c^2-d^2\right )^2 f}\\ &=\frac{b^3 x}{d^3}-\frac{\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right ) \tan ^{-1}\left (\frac{d+c \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{d^3 \left (c^2-d^2\right )^{5/2} f}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac{(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}\\ \end{align*}

Mathematica [B]  time = 2.32157, size = 521, normalized size = 2.04 \[ \frac{\frac{-3 a^2 b c^2 d^4 \sin (2 (e+f x))-6 a^2 b d^6 \sin (2 (e+f x))+3 a^3 c d^5 \sin (2 (e+f x))-3 a b^2 c^3 d^3 \sin (2 (e+f x))+12 a b^2 c d^5 \sin (2 (e+f x))-2 d (b c-a d)^2 \left (-4 a c^2 d+a d^3-2 b c^3+5 b c d^2\right ) \cos (e+f x)+3 b^3 c^4 d^2 \sin (2 (e+f x))-16 b^3 c^3 d^3 e \sin (e+f x)-16 b^3 c^3 d^3 f x \sin (e+f x)-6 b^3 c^2 d^4 \sin (2 (e+f x))-2 b^3 \left (d^3-c^2 d\right )^2 (e+f x) \cos (2 (e+f x))-6 b^3 c^4 d^2 e-6 b^3 c^4 d^2 f x+8 b^3 c^5 d e \sin (e+f x)+8 b^3 c^5 d f x \sin (e+f x)+4 b^3 c^6 e+4 b^3 c^6 f x+8 b^3 c d^5 e \sin (e+f x)+8 b^3 c d^5 f x \sin (e+f x)+2 b^3 d^6 e+2 b^3 d^6 f x}{\left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2}-\frac{4 \left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (-5 c^3 d^2+2 c^5+6 c d^4\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{\left (c^2-d^2\right )^{5/2}}}{4 d^3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(9*a^2*b*c*d^4 - a^3*d^3*(2*c^2 + d^2) - 3*a*b^2*d^3*(c^2 + 2*d^2) + b^3*(2*c^5 - 5*c^3*d^2 + 6*c*d^4))*A
rcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(5/2) + (4*b^3*c^6*e - 6*b^3*c^4*d^2*e + 2*b^3*d^
6*e + 4*b^3*c^6*f*x - 6*b^3*c^4*d^2*f*x + 2*b^3*d^6*f*x - 2*d*(b*c - a*d)^2*(-2*b*c^3 - 4*a*c^2*d + 5*b*c*d^2
+ a*d^3)*Cos[e + f*x] - 2*b^3*(-(c^2*d) + d^3)^2*(e + f*x)*Cos[2*(e + f*x)] + 8*b^3*c^5*d*e*Sin[e + f*x] - 16*
b^3*c^3*d^3*e*Sin[e + f*x] + 8*b^3*c*d^5*e*Sin[e + f*x] + 8*b^3*c^5*d*f*x*Sin[e + f*x] - 16*b^3*c^3*d^3*f*x*Si
n[e + f*x] + 8*b^3*c*d^5*f*x*Sin[e + f*x] + 3*b^3*c^4*d^2*Sin[2*(e + f*x)] - 3*a*b^2*c^3*d^3*Sin[2*(e + f*x)]
- 3*a^2*b*c^2*d^4*Sin[2*(e + f*x)] - 6*b^3*c^2*d^4*Sin[2*(e + f*x)] + 3*a^3*c*d^5*Sin[2*(e + f*x)] + 12*a*b^2*
c*d^5*Sin[2*(e + f*x)] - 6*a^2*b*d^6*Sin[2*(e + f*x)])/((c^2 - d^2)^2*(c + d*Sin[e + f*x])^2))/(4*d^3*f)

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Maple [B]  time = 0.108, size = 2785, normalized size = 10.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/f*d^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*a^3+3/f/(c^4-2*c^2*d^2+d^4)/(
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))*a*b^2*c^2-5/f/(c*tan(1/2*f*x+1/2*e)^2+
2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*b^3*c^3+2/f*b^3/d^3*arctan(tan(1/2*f*x+1/2*e))+2/f/(c^4-2*c^2*
d^2+d^4)/(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))*a^3*c^2-6/f/(c*tan(1/2*f*x+1
/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*a^2*b*c^3+4/f*d/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+
1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*a^3*c^2+2/f/d^2/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c
^2*d^2+d^4)*b^3*c^5+7/f*d^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*tan(1/2*f*
x+1/2*e)^2*a^3+1/f*d^2/(c^4-2*c^2*d^2+d^4)/(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))*a^3-6/f*d^4/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)/c*tan(1/2*f*x+1/2*e)
^2*a^2*b+9/f*d/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c^2*tan(1/2*f*x+1/2*e)^
2*a*b^2-15/f*d/(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^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/2*e)*
a^2*b+30/f*d^2/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*c/(c^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/2*e)*a*
b^2-9/f*d/(c^4-2*c^2*d^2+d^4)/(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))*a^2*b*c
-9/f*d/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c^2*tan(1/2*f*x+1/2*e)^3*a^2*b+
6/f*d^2/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c*tan(1/2*f*x+1/2*e)^3*a*b^2-1
5/f*d^2/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c*tan(1/2*f*x+1/2*e)^2*a^2*b-2
/f*d^4/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/c/(c^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/2*e)*a^3+5/f/d/
(c^4-2*c^2*d^2+d^4)/(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))*b^3*c^3-6/f*d/(c^
4-2*c^2*d^2+d^4)/(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))*b^3*c+3/f/(c*tan(1/2
*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c^3*tan(1/2*f*x+1/2*e)^3*a*b^2-6/f/(c*tan(1/2*f*
x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c^3*tan(1/2*f*x+1/2*e)^2*a^2*b-3/f/(c*tan(1/2*f*x+1
/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*c^3/(c^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/2*e)*a*b^2+7/f/d/(c*tan(1/2*f*x+1/2*
e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*c^4/(c^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/2*e)*b^3-16/f*d/(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^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/2*e)*b^3-3/f*d^2/(c*tan(1/2*f*x+1/2*e)^2+2*
tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*a^2*b*c+9/f*d/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^
2/(c^4-2*c^2*d^2+d^4)*a*b^2*c^2+5/f*d^2/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4
)*c*tan(1/2*f*x+1/2*e)^3*a^3-2/f*d^4/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)/c
*tan(1/2*f*x+1/2*e)^3*a^3+1/f/d/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c^4*ta
n(1/2*f*x+1/2*e)^3*b^3-4/f*d/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c^2*tan(1
/2*f*x+1/2*e)^3*b^3+4/f*d/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c^2*tan(1/2*
f*x+1/2*e)^2*a^3-2/f*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^4-2*c^2*d^2+d^4)/c^2*tan(1/2*f
*x+1/2*e)^2*a^3+2/f/d^2/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c^5*tan(1/2*f*
x+1/2*e)^2*b^3-10/f*d^2/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c*tan(1/2*f*x+
1/2*e)^2*b^3+11/f*d^2/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*c/(c^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/
2*e)*a^3+18/f*d^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/2*e)^2
*a*b^2-12/f*d^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/2*e)*a^2
*b-2/f/d^3/(c^4-2*c^2*d^2+d^4)/(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))*b^3*c^
5+6/f*d^2/(c^4-2*c^2*d^2+d^4)/(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))*a*b^2-1
/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2/(c^4-2*c^2*d^2+d^4)*c^3*tan(1/2*f*x+1/2*e)^2*b^3

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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((a+b*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 = 2.51203, size = 3468, normalized size = 13.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(b^3*c^6*d^2 - 3*b^3*c^4*d^4 + 3*b^3*c^2*d^6 - b^3*d^8)*f*x*cos(f*x + e)^2 - 4*(b^3*c^8 - 2*b^3*c^6*d^
2 + 2*b^3*c^2*d^6 - b^3*d^8)*f*x - (2*b^3*c^7 - 3*b^3*c^5*d^2 - (2*a^3 + 3*a*b^2)*c^4*d^3 + (9*a^2*b + b^3)*c^
3*d^4 - 3*(a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)*d^7 - (2*b^3*c^5*d^2 - 5*b^3*c
^3*d^4 - (2*a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)*d^7)*cos(f*x + e)^2 + 2*(2*b^
3*c^6*d - 5*b^3*c^4*d^3 - (2*a^3 + 3*a*b^2)*c^3*d^4 + 3*(3*a^2*b + 2*b^3)*c^2*d^5 - (a^3 + 6*a*b^2)*c*d^6)*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)) -
2*(2*b^3*c^7*d + 3*a^2*b*c*d^7 + a^3*d^8 - (6*a^2*b + 7*b^3)*c^5*d^3 + (4*a^3 + 9*a*b^2)*c^4*d^4 + (3*a^2*b +
5*b^3)*c^3*d^5 - (5*a^3 + 9*a*b^2)*c^2*d^6)*cos(f*x + e) - 2*(4*(b^3*c^7*d - 3*b^3*c^5*d^3 + 3*b^3*c^3*d^5 -
b^3*c*d^7)*f*x + 3*(b^3*c^6*d^2 - a*b^2*c^5*d^3 + 2*a^2*b*d^8 - (a^2*b + 3*b^3)*c^4*d^4 + (a^3 + 5*a*b^2)*c^3*
d^5 - (a^2*b - 2*b^3)*c^2*d^6 - (a^3 + 4*a*b^2)*c*d^7)*cos(f*x + e))*sin(f*x + e))/((c^6*d^5 - 3*c^4*d^7 + 3*c
^2*d^9 - d^11)*f*cos(f*x + e)^2 - 2*(c^7*d^4 - 3*c^5*d^6 + 3*c^3*d^8 - c*d^10)*f*sin(f*x + e) - (c^8*d^3 - 2*c
^6*d^5 + 2*c^2*d^9 - d^11)*f), 1/2*(2*(b^3*c^6*d^2 - 3*b^3*c^4*d^4 + 3*b^3*c^2*d^6 - b^3*d^8)*f*x*cos(f*x + e)
^2 - 2*(b^3*c^8 - 2*b^3*c^6*d^2 + 2*b^3*c^2*d^6 - b^3*d^8)*f*x - (2*b^3*c^7 - 3*b^3*c^5*d^2 - (2*a^3 + 3*a*b^2
)*c^4*d^3 + (9*a^2*b + b^3)*c^3*d^4 - 3*(a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)*
d^7 - (2*b^3*c^5*d^2 - 5*b^3*c^3*d^4 - (2*a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)
*d^7)*cos(f*x + e)^2 + 2*(2*b^3*c^6*d - 5*b^3*c^4*d^3 - (2*a^3 + 3*a*b^2)*c^3*d^4 + 3*(3*a^2*b + 2*b^3)*c^2*d^
5 - (a^3 + 6*a*b^2)*c*d^6)*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))) - (2*b^3*c^7*d + 3*a^2*b*c*d^7 + a^3*d^8 - (6*a^2*b + 7*b^3)*c^5*d^3 + (4*a^3 + 9*a*b^2)*c^4*d^4 + (3*
a^2*b + 5*b^3)*c^3*d^5 - (5*a^3 + 9*a*b^2)*c^2*d^6)*cos(f*x + e) - (4*(b^3*c^7*d - 3*b^3*c^5*d^3 + 3*b^3*c^3*d
^5 - b^3*c*d^7)*f*x + 3*(b^3*c^6*d^2 - a*b^2*c^5*d^3 + 2*a^2*b*d^8 - (a^2*b + 3*b^3)*c^4*d^4 + (a^3 + 5*a*b^2)
*c^3*d^5 - (a^2*b - 2*b^3)*c^2*d^6 - (a^3 + 4*a*b^2)*c*d^7)*cos(f*x + e))*sin(f*x + e))/((c^6*d^5 - 3*c^4*d^7
+ 3*c^2*d^9 - d^11)*f*cos(f*x + e)^2 - 2*(c^7*d^4 - 3*c^5*d^6 + 3*c^3*d^8 - c*d^10)*f*sin(f*x + e) - (c^8*d^3
- 2*c^6*d^5 + 2*c^2*d^9 - d^11)*f)]

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

[Out]

Timed out

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Giac [B]  time = 1.45529, size = 1200, normalized size = 4.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((f*x + e)*b^3/d^3 - (2*b^3*c^5 - 5*b^3*c^3*d^2 - 2*a^3*c^2*d^3 - 3*a*b^2*c^2*d^3 + 9*a^2*b*c*d^4 + 6*b^3*c*d^
4 - a^3*d^5 - 6*a*b^2*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)))/((c^4*d^3 - 2*c^2*d^5 + d^7)*sqrt(c^2 - d^2)) + (b^3*c^6*d*tan(1/2*f*x + 1/2*e)^3 + 3*a*b^2*c^5*
d^2*tan(1/2*f*x + 1/2*e)^3 - 9*a^2*b*c^4*d^3*tan(1/2*f*x + 1/2*e)^3 - 4*b^3*c^4*d^3*tan(1/2*f*x + 1/2*e)^3 + 5
*a^3*c^3*d^4*tan(1/2*f*x + 1/2*e)^3 + 6*a*b^2*c^3*d^4*tan(1/2*f*x + 1/2*e)^3 - 2*a^3*c*d^6*tan(1/2*f*x + 1/2*e
)^3 + 2*b^3*c^7*tan(1/2*f*x + 1/2*e)^2 - 6*a^2*b*c^5*d^2*tan(1/2*f*x + 1/2*e)^2 - b^3*c^5*d^2*tan(1/2*f*x + 1/
2*e)^2 + 4*a^3*c^4*d^3*tan(1/2*f*x + 1/2*e)^2 + 9*a*b^2*c^4*d^3*tan(1/2*f*x + 1/2*e)^2 - 15*a^2*b*c^3*d^4*tan(
1/2*f*x + 1/2*e)^2 - 10*b^3*c^3*d^4*tan(1/2*f*x + 1/2*e)^2 + 7*a^3*c^2*d^5*tan(1/2*f*x + 1/2*e)^2 + 18*a*b^2*c
^2*d^5*tan(1/2*f*x + 1/2*e)^2 - 6*a^2*b*c*d^6*tan(1/2*f*x + 1/2*e)^2 - 2*a^3*d^7*tan(1/2*f*x + 1/2*e)^2 + 7*b^
3*c^6*d*tan(1/2*f*x + 1/2*e) - 3*a*b^2*c^5*d^2*tan(1/2*f*x + 1/2*e) - 15*a^2*b*c^4*d^3*tan(1/2*f*x + 1/2*e) -
16*b^3*c^4*d^3*tan(1/2*f*x + 1/2*e) + 11*a^3*c^3*d^4*tan(1/2*f*x + 1/2*e) + 30*a*b^2*c^3*d^4*tan(1/2*f*x + 1/2
*e) - 12*a^2*b*c^2*d^5*tan(1/2*f*x + 1/2*e) - 2*a^3*c*d^6*tan(1/2*f*x + 1/2*e) + 2*b^3*c^7 - 6*a^2*b*c^5*d^2 -
5*b^3*c^5*d^2 + 4*a^3*c^4*d^3 + 9*a*b^2*c^4*d^3 - 3*a^2*b*c^3*d^4 - a^3*c^2*d^5)/((c^6*d^2 - 2*c^4*d^4 + c^2*
d^6)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2))/f