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

Optimal. Leaf size=208 \[ \frac {b \left (-a^2 d^2+2 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \cos (e+f x)}{d^2 f \left (c^2-d^2\right )}-\frac {b^2 x (2 b c-3 a d)}{d^3}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) (c+d \sin (e+f x))}+\frac {2 (b c-a d)^2 \left (a c d+2 b c^2-3 b d^2\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f \left (c^2-d^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.50, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2792, 3023, 2735, 2660, 618, 204} \[ \frac {b \left (-a^2 d^2+2 a b c d+b^2 \left (-\left (2 c^2-d^2\right )\right )\right ) \cos (e+f x)}{d^2 f \left (c^2-d^2\right )}-\frac {b^2 x (2 b c-3 a d)}{d^3}+\frac {2 (b c-a d)^2 \left (a c d+2 b c^2-3 b d^2\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f \left (c^2-d^2\right )^{3/2}}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) (c+d \sin (e+f x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*(2*b*c - 3*a*d)*x)/d^3) + (2*(b*c - a*d)^2*(2*b*c^2 + a*c*d - 3*b*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/
Sqrt[c^2 - d^2]])/(d^3*(c^2 - d^2)^(3/2)*f) + (b*(2*a*b*c*d - a^2*d^2 - b^2*(2*c^2 - d^2))*Cos[e + f*x])/(d^2*
(c^2 - d^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(d*(c^2 - 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 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 3023

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

Rubi steps

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

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Mathematica [A]  time = 1.10, size = 152, normalized size = 0.73 \[ \frac {-b^2 (e+f x) (2 b c-3 a d)+\frac {2 (b c-a d)^2 \left (a c d+2 b c^2-3 b d^2\right ) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{3/2}}+\frac {d (a d-b c)^3 \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}+b^3 (-d) \cos (e+f x)}{d^3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.55, size = 1062, normalized size = 5.11 \[ \left [-\frac {2 \, {\left (2 \, b^{3} c^{6} - 3 \, a b^{2} c^{5} d - 4 \, b^{3} c^{4} d^{2} + 6 \, a b^{2} c^{3} d^{3} + 2 \, b^{3} c^{2} d^{4} - 3 \, a b^{2} c d^{5}\right )} f x + {\left (2 \, b^{3} c^{5} - 3 \, a b^{2} c^{4} d - 3 \, b^{3} c^{3} d^{2} - 3 \, a^{2} b c d^{4} + {\left (a^{3} + 6 \, a b^{2}\right )} c^{2} d^{3} + {\left (2 \, b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} - 3 \, b^{3} c^{2} d^{3} - 3 \, a^{2} b d^{5} + {\left (a^{3} + 6 \, a b^{2}\right )} c d^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) + 2 \, {\left (2 \, b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + a^{3} d^{6} + 3 \, {\left (a^{2} b - b^{3}\right )} c^{3} d^{3} - {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d^{4} - {\left (3 \, a^{2} b - b^{3}\right )} c d^{5}\right )} \cos \left (f x + e\right ) + 2 \, {\left ({\left (2 \, b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} - 4 \, b^{3} c^{3} d^{3} + 6 \, a b^{2} c^{2} d^{4} + 2 \, b^{3} c d^{5} - 3 \, a b^{2} d^{6}\right )} f x + {\left (b^{3} c^{4} d^{2} - 2 \, b^{3} c^{2} d^{4} + b^{3} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{4} d^{4} - 2 \, c^{2} d^{6} + d^{8}\right )} f \sin \left (f x + e\right ) + {\left (c^{5} d^{3} - 2 \, c^{3} d^{5} + c d^{7}\right )} f\right )}}, -\frac {{\left (2 \, b^{3} c^{6} - 3 \, a b^{2} c^{5} d - 4 \, b^{3} c^{4} d^{2} + 6 \, a b^{2} c^{3} d^{3} + 2 \, b^{3} c^{2} d^{4} - 3 \, a b^{2} c d^{5}\right )} f x + {\left (2 \, b^{3} c^{5} - 3 \, a b^{2} c^{4} d - 3 \, b^{3} c^{3} d^{2} - 3 \, a^{2} b c d^{4} + {\left (a^{3} + 6 \, a b^{2}\right )} c^{2} d^{3} + {\left (2 \, b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} - 3 \, b^{3} c^{2} d^{3} - 3 \, a^{2} b d^{5} + {\left (a^{3} + 6 \, a b^{2}\right )} c d^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) + {\left (2 \, b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + a^{3} d^{6} + 3 \, {\left (a^{2} b - b^{3}\right )} c^{3} d^{3} - {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d^{4} - {\left (3 \, a^{2} b - b^{3}\right )} c d^{5}\right )} \cos \left (f x + e\right ) + {\left ({\left (2 \, b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} - 4 \, b^{3} c^{3} d^{3} + 6 \, a b^{2} c^{2} d^{4} + 2 \, b^{3} c d^{5} - 3 \, a b^{2} d^{6}\right )} f x + {\left (b^{3} c^{4} d^{2} - 2 \, b^{3} c^{2} d^{4} + b^{3} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{{\left (c^{4} d^{4} - 2 \, c^{2} d^{6} + d^{8}\right )} f \sin \left (f x + e\right ) + {\left (c^{5} d^{3} - 2 \, c^{3} d^{5} + c d^{7}\right )} f}\right ] \]

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))^2,x, algorithm="fricas")

[Out]

[-1/2*(2*(2*b^3*c^6 - 3*a*b^2*c^5*d - 4*b^3*c^4*d^2 + 6*a*b^2*c^3*d^3 + 2*b^3*c^2*d^4 - 3*a*b^2*c*d^5)*f*x + (
2*b^3*c^5 - 3*a*b^2*c^4*d - 3*b^3*c^3*d^2 - 3*a^2*b*c*d^4 + (a^3 + 6*a*b^2)*c^2*d^3 + (2*b^3*c^4*d - 3*a*b^2*c
^3*d^2 - 3*b^3*c^2*d^3 - 3*a^2*b*d^5 + (a^3 + 6*a*b^2)*c*d^4)*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^5*d - 3*a*b^2*c^4*d^2 + a^3*d^6 +
3*(a^2*b - b^3)*c^3*d^3 - (a^3 - 3*a*b^2)*c^2*d^4 - (3*a^2*b - b^3)*c*d^5)*cos(f*x + e) + 2*((2*b^3*c^5*d - 3
*a*b^2*c^4*d^2 - 4*b^3*c^3*d^3 + 6*a*b^2*c^2*d^4 + 2*b^3*c*d^5 - 3*a*b^2*d^6)*f*x + (b^3*c^4*d^2 - 2*b^3*c^2*d
^4 + b^3*d^6)*cos(f*x + e))*sin(f*x + e))/((c^4*d^4 - 2*c^2*d^6 + d^8)*f*sin(f*x + e) + (c^5*d^3 - 2*c^3*d^5 +
c*d^7)*f), -((2*b^3*c^6 - 3*a*b^2*c^5*d - 4*b^3*c^4*d^2 + 6*a*b^2*c^3*d^3 + 2*b^3*c^2*d^4 - 3*a*b^2*c*d^5)*f*
x + (2*b^3*c^5 - 3*a*b^2*c^4*d - 3*b^3*c^3*d^2 - 3*a^2*b*c*d^4 + (a^3 + 6*a*b^2)*c^2*d^3 + (2*b^3*c^4*d - 3*a*
b^2*c^3*d^2 - 3*b^3*c^2*d^3 - 3*a^2*b*d^5 + (a^3 + 6*a*b^2)*c*d^4)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*si
n(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (2*b^3*c^5*d - 3*a*b^2*c^4*d^2 + a^3*d^6 + 3*(a^2*b - b^3)*c
^3*d^3 - (a^3 - 3*a*b^2)*c^2*d^4 - (3*a^2*b - b^3)*c*d^5)*cos(f*x + e) + ((2*b^3*c^5*d - 3*a*b^2*c^4*d^2 - 4*b
^3*c^3*d^3 + 6*a*b^2*c^2*d^4 + 2*b^3*c*d^5 - 3*a*b^2*d^6)*f*x + (b^3*c^4*d^2 - 2*b^3*c^2*d^4 + b^3*d^6)*cos(f*
x + e))*sin(f*x + e))/((c^4*d^4 - 2*c^2*d^6 + d^8)*f*sin(f*x + e) + (c^5*d^3 - 2*c^3*d^5 + c*d^7)*f)]

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giac [B]  time = 0.21, size = 585, normalized size = 2.81 \[ \frac {\frac {2 \, {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 3 \, b^{3} c^{2} d^{2} + a^{3} c d^{3} + 6 \, a b^{2} c d^{3} - 3 \, a^{2} b d^{4}\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} - d^{5}\right )} \sqrt {c^{2} - d^{2}}} - \frac {2 \, {\left (b^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{2} b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, b^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a b^{2} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, a^{2} b c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a^{3} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, b^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{2} b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, b^{3} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - b^{3} c^{2} d^{2} - a^{3} c d^{3}\right )}}{{\left (c^{3} d^{2} - c d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, 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 )}} - \frac {{\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} {\left (f x + e\right )}}{d^{3}}}{f} \]

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))^2,x, algorithm="giac")

[Out]

(2*(2*b^3*c^4 - 3*a*b^2*c^3*d - 3*b^3*c^2*d^2 + a^3*c*d^3 + 6*a*b^2*c*d^3 - 3*a^2*b*d^4)*(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 - d^5)*sqrt(c^2 - d^2))
- 2*(b^3*c^3*d*tan(1/2*f*x + 1/2*e)^3 - 3*a*b^2*c^2*d^2*tan(1/2*f*x + 1/2*e)^3 + 3*a^2*b*c*d^3*tan(1/2*f*x + 1
/2*e)^3 - a^3*d^4*tan(1/2*f*x + 1/2*e)^3 + 2*b^3*c^4*tan(1/2*f*x + 1/2*e)^2 - 3*a*b^2*c^3*d*tan(1/2*f*x + 1/2*
e)^2 + 3*a^2*b*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 - b^3*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 - a^3*c*d^3*tan(1/2*f*x + 1
/2*e)^2 + 3*b^3*c^3*d*tan(1/2*f*x + 1/2*e) - 3*a*b^2*c^2*d^2*tan(1/2*f*x + 1/2*e) + 3*a^2*b*c*d^3*tan(1/2*f*x
+ 1/2*e) - 2*b^3*c*d^3*tan(1/2*f*x + 1/2*e) - a^3*d^4*tan(1/2*f*x + 1/2*e) + 2*b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2
*b*c^2*d^2 - b^3*c^2*d^2 - a^3*c*d^3)/((c^3*d^2 - c*d^4)*(c*tan(1/2*f*x + 1/2*e)^4 + 2*d*tan(1/2*f*x + 1/2*e)^
3 + 2*c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)) - (2*b^3*c - 3*a*b^2*d)*(f*x + e)/d^3)/f

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maple [B]  time = 0.29, size = 842, normalized size = 4.05 \[ \frac {2 d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{3}}{f \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right ) c}-\frac {6 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2} b}{f \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right )}+\frac {6 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a \,b^{2}}{f \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right )}-\frac {2 c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b^{3}}{f d \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right )}+\frac {2 d \,a^{3}}{f \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right )}-\frac {6 a^{2} b c}{f \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right )}+\frac {6 a \,b^{2} c^{2}}{f d \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right )}-\frac {2 c^{3} b^{3}}{f \,d^{2} \left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d +c \right ) \left (c^{2}-d^{2}\right )}+\frac {2 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) a^{3} c}{f \left (c^{2}-d^{2}\right )^{\frac {3}{2}}}-\frac {6 d \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) a^{2} b}{f \left (c^{2}-d^{2}\right )^{\frac {3}{2}}}-\frac {6 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) a \,b^{2} c^{3}}{f \,d^{2} \left (c^{2}-d^{2}\right )^{\frac {3}{2}}}+\frac {12 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) a \,b^{2} c}{f \left (c^{2}-d^{2}\right )^{\frac {3}{2}}}+\frac {4 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) b^{3} c^{4}}{f \,d^{3} \left (c^{2}-d^{2}\right )^{\frac {3}{2}}}-\frac {6 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) b^{3} c^{2}}{f d \left (c^{2}-d^{2}\right )^{\frac {3}{2}}}-\frac {2 b^{3}}{f \,d^{2} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}+\frac {6 b^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a}{f \,d^{2}}-\frac {4 b^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c}{f \,d^{3}} \]

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))^2,x)

[Out]

2/f*d^2/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)/(c^2-d^2)/c*tan(1/2*f*x+1/2*e)*a^3-6/f*d/(tan(1/2*f*
x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)/(c^2-d^2)*tan(1/2*f*x+1/2*e)*a^2*b+6/f/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/
2*f*x+1/2*e)*d+c)/(c^2-d^2)*c*tan(1/2*f*x+1/2*e)*a*b^2-2/f/d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)
/(c^2-d^2)*c^2*tan(1/2*f*x+1/2*e)*b^3+2/f*d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)/(c^2-d^2)*a^3-6/
f/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)/(c^2-d^2)*a^2*b*c+6/f/d/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*
f*x+1/2*e)*d+c)/(c^2-d^2)*a*b^2*c^2-2/f/d^2/(tan(1/2*f*x+1/2*e)^2*c+2*tan(1/2*f*x+1/2*e)*d+c)/(c^2-d^2)*c^3*b^
3+2/f/(c^2-d^2)^(3/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*a^3*c-6/f*d/(c^2-d^2)^(3/2)*arc
tan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*a^2*b-6/f/d^2/(c^2-d^2)^(3/2)*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^3+12/f/(c^2-d^2)^(3/2)*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+4/f/d^3/(c^2-d^2)^(3/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*b^3*c^4-6/f/
d/(c^2-d^2)^(3/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*b^3*c^2-2/f*b^3/d^2/(1+tan(1/2*f*x+
1/2*e)^2)+6/f*b^2/d^2*arctan(tan(1/2*f*x+1/2*e))*a-4/f*b^3/d^3*arctan(tan(1/2*f*x+1/2*e))*c

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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+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^2,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?

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mupad [B]  time = 17.64, size = 8953, normalized size = 43.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*(a^3*d^3 - 2*b^3*c^3 + b^3*c*d^2 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(d^2*(c^2 - d^2)) + (2*tan(e/2 + (f*x)/
2)^2*(a^3*d^3 - 2*b^3*c^3 + b^3*c*d^2 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(d^2*(c^2 - d^2)) + (2*tan(e/2 + (f*x)
/2)*(a^3*d^3 - 3*b^3*c^3 + 2*b^3*c*d^2 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(c*d*(c^2 - d^2)) + (2*tan(e/2 + (f*x
)/2)^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(c*d*(c^2 - d^2)))/(f*(c + 2*d*tan(e/2 + (f*x)/2)
+ 2*c*tan(e/2 + (f*x)/2)^2 + c*tan(e/2 + (f*x)/2)^4 + 2*d*tan(e/2 + (f*x)/2)^3)) + (2*b^2*atan(((b^2*(3*a*d -
2*b*c)*((32*(4*b^6*c^4*d^6 - 8*b^6*c^6*d^4 + 4*b^6*c^8*d^2 - 12*a*b^5*c^3*d^7 + 24*a*b^5*c^5*d^5 - 12*a*b^5*c^
7*d^3 + 9*a^2*b^4*c^2*d^8 - 18*a^2*b^4*c^4*d^6 + 9*a^2*b^4*c^6*d^4))/(d^9 - 2*c^2*d^7 + c^4*d^5) - (32*tan(e/2
+ (f*x)/2)*(a^6*c^3*d^8 - 8*b^6*c^3*d^8 + 29*b^6*c^5*d^6 - 28*b^6*c^7*d^4 + 8*b^6*c^9*d^2 + 24*a*b^5*c^2*d^9
- 96*a*b^5*c^4*d^7 + 90*a*b^5*c^6*d^5 - 24*a*b^5*c^8*d^3 - 18*a^2*b^4*c*d^10 + 9*a^4*b^2*c*d^10 - 6*a^5*b*c^2*
d^9 + 99*a^2*b^4*c^3*d^8 - 84*a^2*b^4*c^5*d^6 + 18*a^2*b^4*c^7*d^4 - 36*a^3*b^3*c^2*d^9 + 12*a^3*b^3*c^4*d^7 +
4*a^3*b^3*c^6*d^5 + 12*a^4*b^2*c^3*d^8 - 6*a^4*b^2*c^5*d^6))/(d^10 - 2*c^2*d^8 + c^4*d^6) + (b^2*(3*a*d - 2*b
*c)*((32*tan(e/2 + (f*x)/2)*(2*a^3*c^2*d^11 - 2*a^3*c^4*d^9 - 6*b^3*c^3*d^10 + 10*b^3*c^5*d^8 - 4*b^3*c^7*d^6
+ 12*a*b^2*c^2*d^11 - 18*a*b^2*c^4*d^9 + 6*a*b^2*c^6*d^7 + 6*a^2*b*c^3*d^10 - 6*a^2*b*c*d^12))/(d^10 - 2*c^2*d
^8 + c^4*d^6) - (32*(a^3*c^5*d^7 - a^3*c^3*d^9 + 2*b^3*c^2*d^10 - 3*b^3*c^4*d^8 + b^3*c^6*d^6 + 3*a*b^2*c^3*d^
9 + 3*a^2*b*c^2*d^10 - 3*a^2*b*c^4*d^8 - 3*a*b^2*c*d^11))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (b^2*((32*(c^2*d^12 -
2*c^4*d^10 + c^6*d^8))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^14 - 8*c^3*d^12 + 7*c^5*d^1
0 - 2*c^7*d^8))/(d^10 - 2*c^2*d^8 + c^4*d^6))*(3*a*d - 2*b*c)*1i)/d^3)*1i)/d^3))/d^3 + (b^2*(3*a*d - 2*b*c)*((
32*(4*b^6*c^4*d^6 - 8*b^6*c^6*d^4 + 4*b^6*c^8*d^2 - 12*a*b^5*c^3*d^7 + 24*a*b^5*c^5*d^5 - 12*a*b^5*c^7*d^3 + 9
*a^2*b^4*c^2*d^8 - 18*a^2*b^4*c^4*d^6 + 9*a^2*b^4*c^6*d^4))/(d^9 - 2*c^2*d^7 + c^4*d^5) - (32*tan(e/2 + (f*x)/
2)*(a^6*c^3*d^8 - 8*b^6*c^3*d^8 + 29*b^6*c^5*d^6 - 28*b^6*c^7*d^4 + 8*b^6*c^9*d^2 + 24*a*b^5*c^2*d^9 - 96*a*b^
5*c^4*d^7 + 90*a*b^5*c^6*d^5 - 24*a*b^5*c^8*d^3 - 18*a^2*b^4*c*d^10 + 9*a^4*b^2*c*d^10 - 6*a^5*b*c^2*d^9 + 99*
a^2*b^4*c^3*d^8 - 84*a^2*b^4*c^5*d^6 + 18*a^2*b^4*c^7*d^4 - 36*a^3*b^3*c^2*d^9 + 12*a^3*b^3*c^4*d^7 + 4*a^3*b^
3*c^6*d^5 + 12*a^4*b^2*c^3*d^8 - 6*a^4*b^2*c^5*d^6))/(d^10 - 2*c^2*d^8 + c^4*d^6) + (b^2*(3*a*d - 2*b*c)*((32*
(a^3*c^5*d^7 - a^3*c^3*d^9 + 2*b^3*c^2*d^10 - 3*b^3*c^4*d^8 + b^3*c^6*d^6 + 3*a*b^2*c^3*d^9 + 3*a^2*b*c^2*d^10
- 3*a^2*b*c^4*d^8 - 3*a*b^2*c*d^11))/(d^9 - 2*c^2*d^7 + c^4*d^5) - (32*tan(e/2 + (f*x)/2)*(2*a^3*c^2*d^11 - 2
*a^3*c^4*d^9 - 6*b^3*c^3*d^10 + 10*b^3*c^5*d^8 - 4*b^3*c^7*d^6 + 12*a*b^2*c^2*d^11 - 18*a*b^2*c^4*d^9 + 6*a*b^
2*c^6*d^7 + 6*a^2*b*c^3*d^10 - 6*a^2*b*c*d^12))/(d^10 - 2*c^2*d^8 + c^4*d^6) + (b^2*((32*(c^2*d^12 - 2*c^4*d^1
0 + c^6*d^8))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^14 - 8*c^3*d^12 + 7*c^5*d^10 - 2*c^7
*d^8))/(d^10 - 2*c^2*d^8 + c^4*d^6))*(3*a*d - 2*b*c)*1i)/d^3)*1i)/d^3))/d^3)/((64*(6*b^9*c^6*d^2 - 4*b^9*c^8 -
39*a*b^8*c^5*d^3 + 4*a^3*b^6*c^7*d - 27*a^5*b^4*c*d^7 + 105*a^2*b^7*c^4*d^4 - 57*a^2*b^7*c^6*d^2 - 144*a^3*b^
6*c^3*d^5 + 55*a^3*b^6*c^5*d^3 + 99*a^4*b^5*c^2*d^6 + 3*a^4*b^5*c^4*d^4 - 12*a^4*b^5*c^6*d^2 - 39*a^5*b^4*c^3*
d^5 + 9*a^5*b^4*c^5*d^3 + 18*a^6*b^3*c^2*d^6 + 2*a^6*b^3*c^4*d^4 - 3*a^7*b^2*c^3*d^5 + 24*a*b^8*c^7*d))/(d^9 -
2*c^2*d^7 + c^4*d^5) + (64*tan(e/2 + (f*x)/2)*(40*b^9*c^7*d^2 - 24*b^9*c^5*d^4 - 16*b^9*c^9 + 120*a*b^8*c^4*d
^5 - 192*a*b^8*c^6*d^3 - 54*a^4*b^5*c*d^8 - 222*a^2*b^7*c^3*d^6 + 330*a^2*b^7*c^5*d^4 - 108*a^2*b^7*c^7*d^2 +
180*a^3*b^6*c^2*d^7 - 226*a^3*b^6*c^4*d^5 + 46*a^3*b^6*c^6*d^3 + 30*a^4*b^5*c^3*d^6 + 24*a^4*b^5*c^5*d^4 + 18*
a^5*b^4*c^2*d^7 - 18*a^5*b^4*c^4*d^5 + 72*a*b^8*c^8*d))/(d^10 - 2*c^2*d^8 + c^4*d^6) + (b^2*(3*a*d - 2*b*c)*((
32*(4*b^6*c^4*d^6 - 8*b^6*c^6*d^4 + 4*b^6*c^8*d^2 - 12*a*b^5*c^3*d^7 + 24*a*b^5*c^5*d^5 - 12*a*b^5*c^7*d^3 + 9
*a^2*b^4*c^2*d^8 - 18*a^2*b^4*c^4*d^6 + 9*a^2*b^4*c^6*d^4))/(d^9 - 2*c^2*d^7 + c^4*d^5) - (32*tan(e/2 + (f*x)/
2)*(a^6*c^3*d^8 - 8*b^6*c^3*d^8 + 29*b^6*c^5*d^6 - 28*b^6*c^7*d^4 + 8*b^6*c^9*d^2 + 24*a*b^5*c^2*d^9 - 96*a*b^
5*c^4*d^7 + 90*a*b^5*c^6*d^5 - 24*a*b^5*c^8*d^3 - 18*a^2*b^4*c*d^10 + 9*a^4*b^2*c*d^10 - 6*a^5*b*c^2*d^9 + 99*
a^2*b^4*c^3*d^8 - 84*a^2*b^4*c^5*d^6 + 18*a^2*b^4*c^7*d^4 - 36*a^3*b^3*c^2*d^9 + 12*a^3*b^3*c^4*d^7 + 4*a^3*b^
3*c^6*d^5 + 12*a^4*b^2*c^3*d^8 - 6*a^4*b^2*c^5*d^6))/(d^10 - 2*c^2*d^8 + c^4*d^6) + (b^2*(3*a*d - 2*b*c)*((32*
tan(e/2 + (f*x)/2)*(2*a^3*c^2*d^11 - 2*a^3*c^4*d^9 - 6*b^3*c^3*d^10 + 10*b^3*c^5*d^8 - 4*b^3*c^7*d^6 + 12*a*b^
2*c^2*d^11 - 18*a*b^2*c^4*d^9 + 6*a*b^2*c^6*d^7 + 6*a^2*b*c^3*d^10 - 6*a^2*b*c*d^12))/(d^10 - 2*c^2*d^8 + c^4*
d^6) - (32*(a^3*c^5*d^7 - a^3*c^3*d^9 + 2*b^3*c^2*d^10 - 3*b^3*c^4*d^8 + b^3*c^6*d^6 + 3*a*b^2*c^3*d^9 + 3*a^2
*b*c^2*d^10 - 3*a^2*b*c^4*d^8 - 3*a*b^2*c*d^11))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (b^2*((32*(c^2*d^12 - 2*c^4*d^1
0 + c^6*d^8))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^14 - 8*c^3*d^12 + 7*c^5*d^10 - 2*c^7
*d^8))/(d^10 - 2*c^2*d^8 + c^4*d^6))*(3*a*d - 2*b*c)*1i)/d^3)*1i)/d^3)*1i)/d^3 - (b^2*(3*a*d - 2*b*c)*((32*(4*
b^6*c^4*d^6 - 8*b^6*c^6*d^4 + 4*b^6*c^8*d^2 - 12*a*b^5*c^3*d^7 + 24*a*b^5*c^5*d^5 - 12*a*b^5*c^7*d^3 + 9*a^2*b
^4*c^2*d^8 - 18*a^2*b^4*c^4*d^6 + 9*a^2*b^4*c^6*d^4))/(d^9 - 2*c^2*d^7 + c^4*d^5) - (32*tan(e/2 + (f*x)/2)*(a^
6*c^3*d^8 - 8*b^6*c^3*d^8 + 29*b^6*c^5*d^6 - 28*b^6*c^7*d^4 + 8*b^6*c^9*d^2 + 24*a*b^5*c^2*d^9 - 96*a*b^5*c^4*
d^7 + 90*a*b^5*c^6*d^5 - 24*a*b^5*c^8*d^3 - 18*a^2*b^4*c*d^10 + 9*a^4*b^2*c*d^10 - 6*a^5*b*c^2*d^9 + 99*a^2*b^
4*c^3*d^8 - 84*a^2*b^4*c^5*d^6 + 18*a^2*b^4*c^7*d^4 - 36*a^3*b^3*c^2*d^9 + 12*a^3*b^3*c^4*d^7 + 4*a^3*b^3*c^6*
d^5 + 12*a^4*b^2*c^3*d^8 - 6*a^4*b^2*c^5*d^6))/(d^10 - 2*c^2*d^8 + c^4*d^6) + (b^2*(3*a*d - 2*b*c)*((32*(a^3*c
^5*d^7 - a^3*c^3*d^9 + 2*b^3*c^2*d^10 - 3*b^3*c^4*d^8 + b^3*c^6*d^6 + 3*a*b^2*c^3*d^9 + 3*a^2*b*c^2*d^10 - 3*a
^2*b*c^4*d^8 - 3*a*b^2*c*d^11))/(d^9 - 2*c^2*d^7 + c^4*d^5) - (32*tan(e/2 + (f*x)/2)*(2*a^3*c^2*d^11 - 2*a^3*c
^4*d^9 - 6*b^3*c^3*d^10 + 10*b^3*c^5*d^8 - 4*b^3*c^7*d^6 + 12*a*b^2*c^2*d^11 - 18*a*b^2*c^4*d^9 + 6*a*b^2*c^6*
d^7 + 6*a^2*b*c^3*d^10 - 6*a^2*b*c*d^12))/(d^10 - 2*c^2*d^8 + c^4*d^6) + (b^2*((32*(c^2*d^12 - 2*c^4*d^10 + c^
6*d^8))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^14 - 8*c^3*d^12 + 7*c^5*d^10 - 2*c^7*d^8))
/(d^10 - 2*c^2*d^8 + c^4*d^6))*(3*a*d - 2*b*c)*1i)/d^3)*1i)/d^3)*1i)/d^3))*(3*a*d - 2*b*c))/(d^3*f) + (atan(((
(a*d - b*c)^2*(-(c + d)^3*(c - d)^3)^(1/2)*((32*(4*b^6*c^4*d^6 - 8*b^6*c^6*d^4 + 4*b^6*c^8*d^2 - 12*a*b^5*c^3*
d^7 + 24*a*b^5*c^5*d^5 - 12*a*b^5*c^7*d^3 + 9*a^2*b^4*c^2*d^8 - 18*a^2*b^4*c^4*d^6 + 9*a^2*b^4*c^6*d^4))/(d^9
- 2*c^2*d^7 + c^4*d^5) - (32*tan(e/2 + (f*x)/2)*(a^6*c^3*d^8 - 8*b^6*c^3*d^8 + 29*b^6*c^5*d^6 - 28*b^6*c^7*d^4
+ 8*b^6*c^9*d^2 + 24*a*b^5*c^2*d^9 - 96*a*b^5*c^4*d^7 + 90*a*b^5*c^6*d^5 - 24*a*b^5*c^8*d^3 - 18*a^2*b^4*c*d^
10 + 9*a^4*b^2*c*d^10 - 6*a^5*b*c^2*d^9 + 99*a^2*b^4*c^3*d^8 - 84*a^2*b^4*c^5*d^6 + 18*a^2*b^4*c^7*d^4 - 36*a^
3*b^3*c^2*d^9 + 12*a^3*b^3*c^4*d^7 + 4*a^3*b^3*c^6*d^5 + 12*a^4*b^2*c^3*d^8 - 6*a^4*b^2*c^5*d^6))/(d^10 - 2*c^
2*d^8 + c^4*d^6) + ((a*d - b*c)^2*(-(c + d)^3*(c - d)^3)^(1/2)*((32*tan(e/2 + (f*x)/2)*(2*a^3*c^2*d^11 - 2*a^3
*c^4*d^9 - 6*b^3*c^3*d^10 + 10*b^3*c^5*d^8 - 4*b^3*c^7*d^6 + 12*a*b^2*c^2*d^11 - 18*a*b^2*c^4*d^9 + 6*a*b^2*c^
6*d^7 + 6*a^2*b*c^3*d^10 - 6*a^2*b*c*d^12))/(d^10 - 2*c^2*d^8 + c^4*d^6) - (32*(a^3*c^5*d^7 - a^3*c^3*d^9 + 2*
b^3*c^2*d^10 - 3*b^3*c^4*d^8 + b^3*c^6*d^6 + 3*a*b^2*c^3*d^9 + 3*a^2*b*c^2*d^10 - 3*a^2*b*c^4*d^8 - 3*a*b^2*c*
d^11))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (((32*(c^2*d^12 - 2*c^4*d^10 + c^6*d^8))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (3
2*tan(e/2 + (f*x)/2)*(3*c*d^14 - 8*c^3*d^12 + 7*c^5*d^10 - 2*c^7*d^8))/(d^10 - 2*c^2*d^8 + c^4*d^6))*(a*d - b*
c)^2*(-(c + d)^3*(c - d)^3)^(1/2)*(2*b*c^2 - 3*b*d^2 + a*c*d))/(d^9 - 3*c^2*d^7 + 3*c^4*d^5 - c^6*d^3))*(2*b*c
^2 - 3*b*d^2 + a*c*d))/(d^9 - 3*c^2*d^7 + 3*c^4*d^5 - c^6*d^3))*(2*b*c^2 - 3*b*d^2 + a*c*d)*1i)/(d^9 - 3*c^2*d
^7 + 3*c^4*d^5 - c^6*d^3) + ((a*d - b*c)^2*(-(c + d)^3*(c - d)^3)^(1/2)*((32*(4*b^6*c^4*d^6 - 8*b^6*c^6*d^4 +
4*b^6*c^8*d^2 - 12*a*b^5*c^3*d^7 + 24*a*b^5*c^5*d^5 - 12*a*b^5*c^7*d^3 + 9*a^2*b^4*c^2*d^8 - 18*a^2*b^4*c^4*d^
6 + 9*a^2*b^4*c^6*d^4))/(d^9 - 2*c^2*d^7 + c^4*d^5) - (32*tan(e/2 + (f*x)/2)*(a^6*c^3*d^8 - 8*b^6*c^3*d^8 + 29
*b^6*c^5*d^6 - 28*b^6*c^7*d^4 + 8*b^6*c^9*d^2 + 24*a*b^5*c^2*d^9 - 96*a*b^5*c^4*d^7 + 90*a*b^5*c^6*d^5 - 24*a*
b^5*c^8*d^3 - 18*a^2*b^4*c*d^10 + 9*a^4*b^2*c*d^10 - 6*a^5*b*c^2*d^9 + 99*a^2*b^4*c^3*d^8 - 84*a^2*b^4*c^5*d^6
+ 18*a^2*b^4*c^7*d^4 - 36*a^3*b^3*c^2*d^9 + 12*a^3*b^3*c^4*d^7 + 4*a^3*b^3*c^6*d^5 + 12*a^4*b^2*c^3*d^8 - 6*a
^4*b^2*c^5*d^6))/(d^10 - 2*c^2*d^8 + c^4*d^6) + ((a*d - b*c)^2*(-(c + d)^3*(c - d)^3)^(1/2)*((32*(a^3*c^5*d^7
- a^3*c^3*d^9 + 2*b^3*c^2*d^10 - 3*b^3*c^4*d^8 + b^3*c^6*d^6 + 3*a*b^2*c^3*d^9 + 3*a^2*b*c^2*d^10 - 3*a^2*b*c^
4*d^8 - 3*a*b^2*c*d^11))/(d^9 - 2*c^2*d^7 + c^4*d^5) - (32*tan(e/2 + (f*x)/2)*(2*a^3*c^2*d^11 - 2*a^3*c^4*d^9
- 6*b^3*c^3*d^10 + 10*b^3*c^5*d^8 - 4*b^3*c^7*d^6 + 12*a*b^2*c^2*d^11 - 18*a*b^2*c^4*d^9 + 6*a*b^2*c^6*d^7 + 6
*a^2*b*c^3*d^10 - 6*a^2*b*c*d^12))/(d^10 - 2*c^2*d^8 + c^4*d^6) + (((32*(c^2*d^12 - 2*c^4*d^10 + c^6*d^8))/(d^
9 - 2*c^2*d^7 + c^4*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^14 - 8*c^3*d^12 + 7*c^5*d^10 - 2*c^7*d^8))/(d^10 - 2*
c^2*d^8 + c^4*d^6))*(a*d - b*c)^2*(-(c + d)^3*(c - d)^3)^(1/2)*(2*b*c^2 - 3*b*d^2 + a*c*d))/(d^9 - 3*c^2*d^7 +
3*c^4*d^5 - c^6*d^3))*(2*b*c^2 - 3*b*d^2 + a*c*d))/(d^9 - 3*c^2*d^7 + 3*c^4*d^5 - c^6*d^3))*(2*b*c^2 - 3*b*d^
2 + a*c*d)*1i)/(d^9 - 3*c^2*d^7 + 3*c^4*d^5 - c^6*d^3))/((64*(6*b^9*c^6*d^2 - 4*b^9*c^8 - 39*a*b^8*c^5*d^3 + 4
*a^3*b^6*c^7*d - 27*a^5*b^4*c*d^7 + 105*a^2*b^7*c^4*d^4 - 57*a^2*b^7*c^6*d^2 - 144*a^3*b^6*c^3*d^5 + 55*a^3*b^
6*c^5*d^3 + 99*a^4*b^5*c^2*d^6 + 3*a^4*b^5*c^4*d^4 - 12*a^4*b^5*c^6*d^2 - 39*a^5*b^4*c^3*d^5 + 9*a^5*b^4*c^5*d
^3 + 18*a^6*b^3*c^2*d^6 + 2*a^6*b^3*c^4*d^4 - 3*a^7*b^2*c^3*d^5 + 24*a*b^8*c^7*d))/(d^9 - 2*c^2*d^7 + c^4*d^5)
+ (64*tan(e/2 + (f*x)/2)*(40*b^9*c^7*d^2 - 24*b^9*c^5*d^4 - 16*b^9*c^9 + 120*a*b^8*c^4*d^5 - 192*a*b^8*c^6*d^
3 - 54*a^4*b^5*c*d^8 - 222*a^2*b^7*c^3*d^6 + 330*a^2*b^7*c^5*d^4 - 108*a^2*b^7*c^7*d^2 + 180*a^3*b^6*c^2*d^7 -
226*a^3*b^6*c^4*d^5 + 46*a^3*b^6*c^6*d^3 + 30*a^4*b^5*c^3*d^6 + 24*a^4*b^5*c^5*d^4 + 18*a^5*b^4*c^2*d^7 - 18*
a^5*b^4*c^4*d^5 + 72*a*b^8*c^8*d))/(d^10 - 2*c^2*d^8 + c^4*d^6) + ((a*d - b*c)^2*(-(c + d)^3*(c - d)^3)^(1/2)*
((32*(4*b^6*c^4*d^6 - 8*b^6*c^6*d^4 + 4*b^6*c^8*d^2 - 12*a*b^5*c^3*d^7 + 24*a*b^5*c^5*d^5 - 12*a*b^5*c^7*d^3 +
9*a^2*b^4*c^2*d^8 - 18*a^2*b^4*c^4*d^6 + 9*a^2*b^4*c^6*d^4))/(d^9 - 2*c^2*d^7 + c^4*d^5) - (32*tan(e/2 + (f*x
)/2)*(a^6*c^3*d^8 - 8*b^6*c^3*d^8 + 29*b^6*c^5*d^6 - 28*b^6*c^7*d^4 + 8*b^6*c^9*d^2 + 24*a*b^5*c^2*d^9 - 96*a*
b^5*c^4*d^7 + 90*a*b^5*c^6*d^5 - 24*a*b^5*c^8*d^3 - 18*a^2*b^4*c*d^10 + 9*a^4*b^2*c*d^10 - 6*a^5*b*c^2*d^9 + 9
9*a^2*b^4*c^3*d^8 - 84*a^2*b^4*c^5*d^6 + 18*a^2*b^4*c^7*d^4 - 36*a^3*b^3*c^2*d^9 + 12*a^3*b^3*c^4*d^7 + 4*a^3*
b^3*c^6*d^5 + 12*a^4*b^2*c^3*d^8 - 6*a^4*b^2*c^5*d^6))/(d^10 - 2*c^2*d^8 + c^4*d^6) + ((a*d - b*c)^2*(-(c + d)
^3*(c - d)^3)^(1/2)*((32*tan(e/2 + (f*x)/2)*(2*a^3*c^2*d^11 - 2*a^3*c^4*d^9 - 6*b^3*c^3*d^10 + 10*b^3*c^5*d^8
- 4*b^3*c^7*d^6 + 12*a*b^2*c^2*d^11 - 18*a*b^2*c^4*d^9 + 6*a*b^2*c^6*d^7 + 6*a^2*b*c^3*d^10 - 6*a^2*b*c*d^12))
/(d^10 - 2*c^2*d^8 + c^4*d^6) - (32*(a^3*c^5*d^7 - a^3*c^3*d^9 + 2*b^3*c^2*d^10 - 3*b^3*c^4*d^8 + b^3*c^6*d^6
+ 3*a*b^2*c^3*d^9 + 3*a^2*b*c^2*d^10 - 3*a^2*b*c^4*d^8 - 3*a*b^2*c*d^11))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (((32*
(c^2*d^12 - 2*c^4*d^10 + c^6*d^8))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^14 - 8*c^3*d^12
+ 7*c^5*d^10 - 2*c^7*d^8))/(d^10 - 2*c^2*d^8 + c^4*d^6))*(a*d - b*c)^2*(-(c + d)^3*(c - d)^3)^(1/2)*(2*b*c^2
- 3*b*d^2 + a*c*d))/(d^9 - 3*c^2*d^7 + 3*c^4*d^5 - c^6*d^3))*(2*b*c^2 - 3*b*d^2 + a*c*d))/(d^9 - 3*c^2*d^7 + 3
*c^4*d^5 - c^6*d^3))*(2*b*c^2 - 3*b*d^2 + a*c*d))/(d^9 - 3*c^2*d^7 + 3*c^4*d^5 - c^6*d^3) - ((a*d - b*c)^2*(-(
c + d)^3*(c - d)^3)^(1/2)*((32*(4*b^6*c^4*d^6 - 8*b^6*c^6*d^4 + 4*b^6*c^8*d^2 - 12*a*b^5*c^3*d^7 + 24*a*b^5*c^
5*d^5 - 12*a*b^5*c^7*d^3 + 9*a^2*b^4*c^2*d^8 - 18*a^2*b^4*c^4*d^6 + 9*a^2*b^4*c^6*d^4))/(d^9 - 2*c^2*d^7 + c^4
*d^5) - (32*tan(e/2 + (f*x)/2)*(a^6*c^3*d^8 - 8*b^6*c^3*d^8 + 29*b^6*c^5*d^6 - 28*b^6*c^7*d^4 + 8*b^6*c^9*d^2
+ 24*a*b^5*c^2*d^9 - 96*a*b^5*c^4*d^7 + 90*a*b^5*c^6*d^5 - 24*a*b^5*c^8*d^3 - 18*a^2*b^4*c*d^10 + 9*a^4*b^2*c*
d^10 - 6*a^5*b*c^2*d^9 + 99*a^2*b^4*c^3*d^8 - 84*a^2*b^4*c^5*d^6 + 18*a^2*b^4*c^7*d^4 - 36*a^3*b^3*c^2*d^9 + 1
2*a^3*b^3*c^4*d^7 + 4*a^3*b^3*c^6*d^5 + 12*a^4*b^2*c^3*d^8 - 6*a^4*b^2*c^5*d^6))/(d^10 - 2*c^2*d^8 + c^4*d^6)
+ ((a*d - b*c)^2*(-(c + d)^3*(c - d)^3)^(1/2)*((32*(a^3*c^5*d^7 - a^3*c^3*d^9 + 2*b^3*c^2*d^10 - 3*b^3*c^4*d^8
+ b^3*c^6*d^6 + 3*a*b^2*c^3*d^9 + 3*a^2*b*c^2*d^10 - 3*a^2*b*c^4*d^8 - 3*a*b^2*c*d^11))/(d^9 - 2*c^2*d^7 + c^
4*d^5) - (32*tan(e/2 + (f*x)/2)*(2*a^3*c^2*d^11 - 2*a^3*c^4*d^9 - 6*b^3*c^3*d^10 + 10*b^3*c^5*d^8 - 4*b^3*c^7*
d^6 + 12*a*b^2*c^2*d^11 - 18*a*b^2*c^4*d^9 + 6*a*b^2*c^6*d^7 + 6*a^2*b*c^3*d^10 - 6*a^2*b*c*d^12))/(d^10 - 2*c
^2*d^8 + c^4*d^6) + (((32*(c^2*d^12 - 2*c^4*d^10 + c^6*d^8))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (32*tan(e/2 + (f*x)
/2)*(3*c*d^14 - 8*c^3*d^12 + 7*c^5*d^10 - 2*c^7*d^8))/(d^10 - 2*c^2*d^8 + c^4*d^6))*(a*d - b*c)^2*(-(c + d)^3*
(c - d)^3)^(1/2)*(2*b*c^2 - 3*b*d^2 + a*c*d))/(d^9 - 3*c^2*d^7 + 3*c^4*d^5 - c^6*d^3))*(2*b*c^2 - 3*b*d^2 + a*
c*d))/(d^9 - 3*c^2*d^7 + 3*c^4*d^5 - c^6*d^3))*(2*b*c^2 - 3*b*d^2 + a*c*d))/(d^9 - 3*c^2*d^7 + 3*c^4*d^5 - c^6
*d^3)))*(a*d - b*c)^2*(-(c + d)^3*(c - d)^3)^(1/2)*(2*b*c^2 - 3*b*d^2 + a*c*d)*2i)/(f*(d^9 - 3*c^2*d^7 + 3*c^4
*d^5 - c^6*d^3))

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

[Out]

Timed out

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