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

Optimal. Leaf size=248 \[ \frac{(b c-a d) \left (a^2 b^2 \left (2 c^2-5 d^2\right )+2 a^3 b c d+2 a^4 d^2-8 a b^3 c d+b^4 \left (c^2+6 d^2\right )\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (e+f x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^3 f \left (a^2-b^2\right )^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac{(b c-a d)^2 \left (2 a^2 d+3 a b c-5 b^2 d\right ) \cos (e+f x)}{2 b^2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac{d^3 x}{b^3} \]

[Out]

(d^3*x)/b^3 + ((b*c - a*d)*(2*a^3*b*c*d - 8*a*b^3*c*d + 2*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 6*d^2
))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(5/2)*f) + ((b*c - a*d)^2*(3*a*b*c + 2*a
^2*d - 5*b^2*d)*Cos[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c +
d*Sin[e + f*x]))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)

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Rubi [A]  time = 0.806464, antiderivative size = 248, 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} \[ \frac{(b c-a d) \left (a^2 b^2 \left (2 c^2-5 d^2\right )+2 a^3 b c d+2 a^4 d^2-8 a b^3 c d+b^4 \left (c^2+6 d^2\right )\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (e+f x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^3 f \left (a^2-b^2\right )^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac{(b c-a d)^2 \left (2 a^2 d+3 a b c-5 b^2 d\right ) \cos (e+f x)}{2 b^2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac{d^3 x}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*x)/b^3 + ((b*c - a*d)*(2*a^3*b*c*d - 8*a*b^3*c*d + 2*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 6*d^2
))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(5/2)*f) + ((b*c - a*d)^2*(3*a*b*c + 2*a
^2*d - 5*b^2*d)*Cos[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c +
d*Sin[e + f*x]))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)

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{(c+d \sin (e+f x))^3}{(a+b \sin (e+f x))^3} \, dx &=\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac{\int \frac{5 b^2 c^2 d+a^2 d^3-2 a b c \left (c^2+2 d^2\right )-\left (a^2 c d^2+2 a b d \left (2 c^2+d^2\right )-b^2 \left (c^3+6 c d^2\right )\right ) \sin (e+f x)-2 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)}{(a+b \sin (e+f x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac{(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\int \frac{b \left (a^3 d^3+a^2 b c \left (2 c^2+3 d^2\right )-a b^2 d \left (9 c^2+4 d^2\right )+b^3 c \left (c^2+6 d^2\right )\right )+2 \left (a^2-b^2\right )^2 d^3 \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{d^3 x}{b^3}+\frac{(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac{\left (2 a^5 d^3-5 a^3 b^2 d^3+3 a b^4 d \left (3 c^2+2 d^2\right )-a^2 b^3 c \left (2 c^2+3 d^2\right )-b^5 c \left (c^2+6 d^2\right )\right ) \int \frac{1}{a+b \sin (e+f x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{d^3 x}{b^3}+\frac{(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac{\left (2 a^5 d^3-5 a^3 b^2 d^3+3 a b^4 d \left (3 c^2+2 d^2\right )-a^2 b^3 c \left (2 c^2+3 d^2\right )-b^5 c \left (c^2+6 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{b^3 \left (a^2-b^2\right )^2 f}\\ &=\frac{d^3 x}{b^3}+\frac{(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left (2 \left (2 a^5 d^3-5 a^3 b^2 d^3+3 a b^4 d \left (3 c^2+2 d^2\right )-a^2 b^3 c \left (2 c^2+3 d^2\right )-b^5 c \left (c^2+6 d^2\right )\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{1}{2} (e+f x)\right )\right )}{b^3 \left (a^2-b^2\right )^2 f}\\ &=\frac{d^3 x}{b^3}+\frac{(b c-a d) \left (2 a^2 b^2 c^2+b^4 c^2+2 a^3 b c d-8 a b^3 c d+2 a^4 d^2-5 a^2 b^2 d^2+6 b^4 d^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{a^2-b^2}}\right )}{b^3 \left (a^2-b^2\right )^{5/2} f}+\frac{(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}\\ \end{align*}

Mathematica [B]  time = 2.3726, size = 524, normalized size = 2.11 \[ \frac{\frac{-3 a^2 b^4 c^2 d \sin (2 (e+f x))-3 a^3 b^3 c d^2 \sin (2 (e+f x))-2 b (b c-a d)^2 \left (-4 a^2 b c-2 a^3 d+5 a b^2 d+b^3 c\right ) \cos (e+f x)+3 a^4 b^2 d^3 \sin (2 (e+f x))-16 a^3 b^3 d^3 e \sin (e+f x)-16 a^3 b^3 d^3 f x \sin (e+f x)-6 a^2 b^4 d^3 \sin (2 (e+f x))-2 d^3 \left (b^3-a^2 b\right )^2 (e+f x) \cos (2 (e+f x))-6 a^4 b^2 d^3 e-6 a^4 b^2 d^3 f x+8 a^5 b d^3 e \sin (e+f x)+8 a^5 b d^3 f x \sin (e+f x)+4 a^6 d^3 e+4 a^6 d^3 f x+3 a b^5 c^3 \sin (2 (e+f x))+12 a b^5 c d^2 \sin (2 (e+f x))+8 a b^5 d^3 e \sin (e+f x)+8 a b^5 d^3 f x \sin (e+f x)-6 b^6 c^2 d \sin (2 (e+f x))+2 b^6 d^3 e+2 b^6 d^3 f x}{\left (a^2-b^2\right )^2 (a+b \sin (e+f x))^2}-\frac{4 \left (-a^2 b^3 c \left (2 c^2+3 d^2\right )-5 a^3 b^2 d^3+2 a^5 d^3+3 a b^4 d \left (3 c^2+2 d^2\right )-b^5 c \left (c^2+6 d^2\right )\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (e+f x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}}{4 b^3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(2*a^5*d^3 - 5*a^3*b^2*d^3 + 3*a*b^4*d*(3*c^2 + 2*d^2) - a^2*b^3*c*(2*c^2 + 3*d^2) - b^5*c*(c^2 + 6*d^2))
*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (4*a^6*d^3*e - 6*a^4*b^2*d^3*e + 2*b^6*
d^3*e + 4*a^6*d^3*f*x - 6*a^4*b^2*d^3*f*x + 2*b^6*d^3*f*x - 2*b*(b*c - a*d)^2*(-4*a^2*b*c + b^3*c - 2*a^3*d +
5*a*b^2*d)*Cos[e + f*x] - 2*(-(a^2*b) + b^3)^2*d^3*(e + f*x)*Cos[2*(e + f*x)] + 8*a^5*b*d^3*e*Sin[e + f*x] - 1
6*a^3*b^3*d^3*e*Sin[e + f*x] + 8*a*b^5*d^3*e*Sin[e + f*x] + 8*a^5*b*d^3*f*x*Sin[e + f*x] - 16*a^3*b^3*d^3*f*x*
Sin[e + f*x] + 8*a*b^5*d^3*f*x*Sin[e + f*x] + 3*a*b^5*c^3*Sin[2*(e + f*x)] - 3*a^2*b^4*c^2*d*Sin[2*(e + f*x)]
- 6*b^6*c^2*d*Sin[2*(e + f*x)] - 3*a^3*b^3*c*d^2*Sin[2*(e + f*x)] + 12*a*b^5*c*d^2*Sin[2*(e + f*x)] + 3*a^4*b^
2*d^3*Sin[2*(e + f*x)] - 6*a^2*b^4*d^3*Sin[2*(e + f*x)])/((a^2 - b^2)^2*(a + b*Sin[e + f*x])^2))/(4*b^3*f)

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Maple [B]  time = 0.109, size = 2785, normalized size = 11.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/f*b^3/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*c^3+7/f/b/(tan(1/2*f*x+1/2*e)
^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2*a^4/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*d^3-16/f*b/(tan(1/2*f*x+1/2*e)^2*a
+2*tan(1/2*f*x+1/2*e)*b+a)^2*a^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*d^3+11/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2
*tan(1/2*f*x+1/2*e)*b+a)^2*a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c^3-2/f*b^4/(tan(1/2*f*x+1/2*e)^2*a+2*tan(
1/2*f*x+1/2*e)*b+a)^2/a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c^3+9/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x
+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^2*c*d^2-3/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-
2*a^2*b^2+b^4)*a*c^2*d+2/f*d^3/b^3*arctan(tan(1/2*f*x+1/2*e))+6/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/
2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a*tan(1/2*f*x+1/2*e)^3*c*d^2+9/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e
)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^2*tan(1/2*f*x+1/2*e)^2*c*d^2-15/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2
*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a*tan(1/2*f*x+1/2*e)^2*c^2*d-6/f*b^4/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*
e)*b+a)^2/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^2*c^2*d-15/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)
*b+a)^2*a^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c^2*d+30/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)
*b+a)^2*a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c*d^2+18/f*b^3/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b
+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)^2*c*d^2-12/f*b^3/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a
)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c^2*d+3/f/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1
/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2))*a^2*c*d^2+3/f/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a
^2*b^2+b^4)*a^3*tan(1/2*f*x+1/2*e)^3*c*d^2-6/f/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*
b^2+b^4)*a^3*tan(1/2*f*x+1/2*e)^2*c^2*d-3/f/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2*a^3/(a^4-2*a^2
*b^2+b^4)*tan(1/2*f*x+1/2*e)*c*d^2+5/f/b/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e
)+2*b)/(a^2-b^2)^(1/2))*a^3*d^3-2/f/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*f*x+1/2*e)
+2*b)/(a^2-b^2)^(1/2))*a^5*d^3-6/f*b/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*
b)/(a^2-b^2)^(1/2))*a*d^3+6/f*b^2/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/
(a^2-b^2)^(1/2))*c*d^2+1/f/b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^4*tan(1
/2*f*x+1/2*e)^3*d^3-4/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^2*tan(1/2*
f*x+1/2*e)^3*d^3+5/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a*tan(1/2*f*x
+1/2*e)^3*c^3-2/f*b^4/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/
2*e)^3*c^3+2/f/b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^5*tan(1/2*f*x+1/2
*e)^2*d^3+4/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^2*tan(1/2*f*x+1/2*e)
^2*c^3-10/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a*tan(1/2*f*x+1/2*e)^2
*d^3-2/f*b^5/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)/a^2*tan(1/2*f*x+1/2*e)^2*
c^3+2/f/b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^5*d^3-1/f/(tan(1/2*f*x+1
/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^3*tan(1/2*f*x+1/2*e)^2*d^3-6/f/(tan(1/2*f*x+1/2*e)
^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^3*c^2*d+2/f/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(
1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2))*a^2*c^3+4/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b
+a)^2/(a^4-2*a^2*b^2+b^4)*a^2*c^3+1/f*b^2/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*
e)+2*b)/(a^2-b^2)^(1/2))*c^3+7/f*b^3/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*t
an(1/2*f*x+1/2*e)^2*c^3-9/f*b/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2
-b^2)^(1/2))*a*c^2*d-9/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^2*tan(1/2
*f*x+1/2*e)^3*c^2*d-5/f/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^3*d^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((c+d*sin(f*x+e))^3/(a+b*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.17332, size = 3313, normalized size = 13.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.34669, size = 1197, normalized size = 4.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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