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

Optimal. Leaf size=156 \[ -\frac {b x \left (-6 a^2 d^2+6 a b c d-\left (b^2 \left (2 c^2+d^2\right )\right )\right )}{2 d^3}+\frac {b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}-\frac {2 (b c-a d)^3 \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f \sqrt {c^2-d^2}} \]

[Out]

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

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Rubi [A]  time = 0.38, antiderivative size = 156, 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 = {2793, 3023, 2735, 2660, 618, 204} \[ -\frac {b x \left (-6 a^2 d^2+6 a b c d+b^2 \left (-\left (2 c^2+d^2\right )\right )\right )}{2 d^3}+\frac {b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}-\frac {2 (b c-a d)^3 \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f \sqrt {c^2-d^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2793

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d
*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a*d
*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n -
2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] ||
(EqQ[a, 0] && NeQ[c, 0])))

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

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Mathematica [A]  time = 0.37, size = 137, normalized size = 0.88 \[ \frac {2 b (e+f x) \left (6 a^2 d^2-6 a b c d+b^2 \left (2 c^2+d^2\right )\right )+4 b^2 d (b c-3 a d) \cos (e+f x)-\frac {8 (b c-a d)^3 \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{\sqrt {c^2-d^2}}+b^3 \left (-d^2\right ) \sin (2 (e+f x))}{4 d^3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 578, normalized size = 3.71 \[ \left [\frac {{\left (2 \, b^{3} c^{4} - 6 \, a b^{2} c^{3} d + 6 \, a b^{2} c d^{3} + {\left (6 \, a^{2} b - b^{3}\right )} c^{2} d^{2} - {\left (6 \, a^{2} b + b^{3}\right )} d^{4}\right )} f x - {\left (b^{3} c^{2} d^{2} - b^{3} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-c^{2} + d^{2}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) + 2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} \cos \left (f x + e\right )}{2 \, {\left (c^{2} d^{3} - d^{5}\right )} f}, \frac {{\left (2 \, b^{3} c^{4} - 6 \, a b^{2} c^{3} d + 6 \, a b^{2} c d^{3} + {\left (6 \, a^{2} b - b^{3}\right )} c^{2} d^{2} - {\left (6 \, a^{2} b + b^{3}\right )} d^{4}\right )} f x - {\left (b^{3} c^{2} d^{2} - b^{3} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) + 2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} \cos \left (f x + e\right )}{2 \, {\left (c^{2} d^{3} - d^{5}\right )} f}\right ] \]

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)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.60, size = 252, normalized size = 1.62 \[ \frac {\frac {{\left (2 \, b^{3} c^{2} - 6 \, a b^{2} c d + 6 \, a^{2} b d^{2} + b^{3} d^{2}\right )} {\left (f x + e\right )}}{d^{3}} - \frac {4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (c) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{\sqrt {c^{2} - d^{2}} d^{3}} + \frac {2 \, {\left (b^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, b^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, a b^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, b^{3} c - 6 \, a b^{2} d\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} d^{2}}}{2 \, f} \]

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)),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.24, size = 506, normalized size = 3.24 \[ \frac {2 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) a^{3}}{f \sqrt {c^{2}-d^{2}}}-\frac {6 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) a^{2} b c}{f d \sqrt {c^{2}-d^{2}}}+\frac {6 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) a \,b^{2} c^{2}}{f \,d^{2} \sqrt {c^{2}-d^{2}}}-\frac {2 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right ) b^{3} c^{3}}{f \,d^{3} \sqrt {c^{2}-d^{2}}}+\frac {b^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f d \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {6 b^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a}{f d \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {2 b^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c}{f \,d^{2} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {b^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f d \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {6 b^{2} a}{f d \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {2 b^{3} c}{f \,d^{2} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {6 b \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2}}{f d}-\frac {6 b^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a c}{f \,d^{2}}+\frac {2 b^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c^{2}}{f \,d^{3}}+\frac {b^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f d} \]

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)),x)

[Out]

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

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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)),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 = 14.56, size = 5902, normalized size = 37.83 \[ \text {result too large to display} \]

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)),x)

[Out]

((2*(b^3*c - 3*a*b^2*d))/d^2 + (b^3*tan(e/2 + (f*x)/2)^3)/d + (2*tan(e/2 + (f*x)/2)^2*(b^3*c - 3*a*b^2*d))/d^2
- (b^3*tan(e/2 + (f*x)/2))/d)/(f*(2*tan(e/2 + (f*x)/2)^2 + tan(e/2 + (f*x)/2)^4 + 1)) + (atan((((b^3*c^2*1i +
(b*d^2*(6*a^2 + b^2)*1i)/2 - a*b^2*c*d*3i)*((8*(b^6*c^2*d^6 + 4*b^6*c^4*d^4 + 4*b^6*c^6*d^2 - 12*a*b^5*c^3*d^
5 - 24*a*b^5*c^5*d^3 + 12*a^2*b^4*c^2*d^6 + 60*a^2*b^4*c^4*d^4 - 72*a^3*b^3*c^3*d^5 + 36*a^4*b^2*c^2*d^6))/d^5
+ (8*tan(e/2 + (f*x)/2)*(2*b^6*c*d^8 - 4*a^6*c*d^8 + 7*b^6*c^3*d^6 + 4*b^6*c^5*d^4 - 8*b^6*c^7*d^2 - 24*a*b^5
*c^2*d^7 - 36*a*b^5*c^4*d^5 + 48*a*b^5*c^6*d^3 + 24*a^2*b^4*c*d^8 + 72*a^4*b^2*c*d^8 + 24*a^5*b*c^2*d^7 + 108*
a^2*b^4*c^3*d^6 - 120*a^2*b^4*c^5*d^4 - 144*a^3*b^3*c^2*d^7 + 152*a^3*b^3*c^4*d^5 - 96*a^4*b^2*c^3*d^6))/d^6 +
((b^3*c^2*1i + (b*d^2*(6*a^2 + b^2)*1i)/2 - a*b^2*c*d*3i)*((8*tan(e/2 + (f*x)/2)*(8*a^3*c*d^9 - 8*b^3*c^4*d^6
+ 24*a*b^2*c^3*d^7 - 24*a^2*b*c^2*d^8))/d^6 - (8*(2*b^3*c*d^8 - 4*a^3*c^2*d^7 + 2*b^3*c^3*d^6 - 12*a*b^2*c^2*
d^7 + 12*a^2*b*c*d^8))/d^5 + ((32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^10 - 8*c^3*d^8))/d^6)*(b^3*c^2*1i +
(b*d^2*(6*a^2 + b^2)*1i)/2 - a*b^2*c*d*3i))/d^3))/d^3)*1i)/d^3 + ((b^3*c^2*1i + (b*d^2*(6*a^2 + b^2)*1i)/2 - a
*b^2*c*d*3i)*((8*(b^6*c^2*d^6 + 4*b^6*c^4*d^4 + 4*b^6*c^6*d^2 - 12*a*b^5*c^3*d^5 - 24*a*b^5*c^5*d^3 + 12*a^2*b
^4*c^2*d^6 + 60*a^2*b^4*c^4*d^4 - 72*a^3*b^3*c^3*d^5 + 36*a^4*b^2*c^2*d^6))/d^5 + (8*tan(e/2 + (f*x)/2)*(2*b^6
*c*d^8 - 4*a^6*c*d^8 + 7*b^6*c^3*d^6 + 4*b^6*c^5*d^4 - 8*b^6*c^7*d^2 - 24*a*b^5*c^2*d^7 - 36*a*b^5*c^4*d^5 + 4
8*a*b^5*c^6*d^3 + 24*a^2*b^4*c*d^8 + 72*a^4*b^2*c*d^8 + 24*a^5*b*c^2*d^7 + 108*a^2*b^4*c^3*d^6 - 120*a^2*b^4*c
^5*d^4 - 144*a^3*b^3*c^2*d^7 + 152*a^3*b^3*c^4*d^5 - 96*a^4*b^2*c^3*d^6))/d^6 + ((b^3*c^2*1i + (b*d^2*(6*a^2 +
b^2)*1i)/2 - a*b^2*c*d*3i)*((8*(2*b^3*c*d^8 - 4*a^3*c^2*d^7 + 2*b^3*c^3*d^6 - 12*a*b^2*c^2*d^7 + 12*a^2*b*c*d
^8))/d^5 - (8*tan(e/2 + (f*x)/2)*(8*a^3*c*d^9 - 8*b^3*c^4*d^6 + 24*a*b^2*c^3*d^7 - 24*a^2*b*c^2*d^8))/d^6 + ((
32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^10 - 8*c^3*d^8))/d^6)*(b^3*c^2*1i + (b*d^2*(6*a^2 + b^2)*1i)/2 - a*
b^2*c*d*3i))/d^3))/d^3)*1i)/d^3)/((16*(2*b^9*c^7 + b^9*c^5*d^2 - 3*a*b^8*c^4*d^3 + 4*a^3*b^6*c^6*d - 2*a^6*b^3
*c*d^6 + 3*a^2*b^7*c^3*d^4 + 30*a^2*b^7*c^5*d^2 - a^3*b^6*c^2*d^5 - 36*a^3*b^6*c^4*d^3 + 18*a^4*b^5*c^3*d^4 -
24*a^4*b^5*c^5*d^2 + 60*a^5*b^4*c^4*d^3 - 76*a^6*b^3*c^3*d^4 + 48*a^7*b^2*c^2*d^5 - 12*a*b^8*c^6*d - 12*a^8*b*
c*d^6))/d^5 - ((b^3*c^2*1i + (b*d^2*(6*a^2 + b^2)*1i)/2 - a*b^2*c*d*3i)*((8*(b^6*c^2*d^6 + 4*b^6*c^4*d^4 + 4*b
^6*c^6*d^2 - 12*a*b^5*c^3*d^5 - 24*a*b^5*c^5*d^3 + 12*a^2*b^4*c^2*d^6 + 60*a^2*b^4*c^4*d^4 - 72*a^3*b^3*c^3*d^
5 + 36*a^4*b^2*c^2*d^6))/d^5 + (8*tan(e/2 + (f*x)/2)*(2*b^6*c*d^8 - 4*a^6*c*d^8 + 7*b^6*c^3*d^6 + 4*b^6*c^5*d^
4 - 8*b^6*c^7*d^2 - 24*a*b^5*c^2*d^7 - 36*a*b^5*c^4*d^5 + 48*a*b^5*c^6*d^3 + 24*a^2*b^4*c*d^8 + 72*a^4*b^2*c*d
^8 + 24*a^5*b*c^2*d^7 + 108*a^2*b^4*c^3*d^6 - 120*a^2*b^4*c^5*d^4 - 144*a^3*b^3*c^2*d^7 + 152*a^3*b^3*c^4*d^5
- 96*a^4*b^2*c^3*d^6))/d^6 + ((b^3*c^2*1i + (b*d^2*(6*a^2 + b^2)*1i)/2 - a*b^2*c*d*3i)*((8*tan(e/2 + (f*x)/2)*
(8*a^3*c*d^9 - 8*b^3*c^4*d^6 + 24*a*b^2*c^3*d^7 - 24*a^2*b*c^2*d^8))/d^6 - (8*(2*b^3*c*d^8 - 4*a^3*c^2*d^7 + 2
*b^3*c^3*d^6 - 12*a*b^2*c^2*d^7 + 12*a^2*b*c*d^8))/d^5 + ((32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^10 - 8*c
^3*d^8))/d^6)*(b^3*c^2*1i + (b*d^2*(6*a^2 + b^2)*1i)/2 - a*b^2*c*d*3i))/d^3))/d^3))/d^3 + ((b^3*c^2*1i + (b*d^
2*(6*a^2 + b^2)*1i)/2 - a*b^2*c*d*3i)*((8*(b^6*c^2*d^6 + 4*b^6*c^4*d^4 + 4*b^6*c^6*d^2 - 12*a*b^5*c^3*d^5 - 24
*a*b^5*c^5*d^3 + 12*a^2*b^4*c^2*d^6 + 60*a^2*b^4*c^4*d^4 - 72*a^3*b^3*c^3*d^5 + 36*a^4*b^2*c^2*d^6))/d^5 + (8*
tan(e/2 + (f*x)/2)*(2*b^6*c*d^8 - 4*a^6*c*d^8 + 7*b^6*c^3*d^6 + 4*b^6*c^5*d^4 - 8*b^6*c^7*d^2 - 24*a*b^5*c^2*d
^7 - 36*a*b^5*c^4*d^5 + 48*a*b^5*c^6*d^3 + 24*a^2*b^4*c*d^8 + 72*a^4*b^2*c*d^8 + 24*a^5*b*c^2*d^7 + 108*a^2*b^
4*c^3*d^6 - 120*a^2*b^4*c^5*d^4 - 144*a^3*b^3*c^2*d^7 + 152*a^3*b^3*c^4*d^5 - 96*a^4*b^2*c^3*d^6))/d^6 + ((b^3
*c^2*1i + (b*d^2*(6*a^2 + b^2)*1i)/2 - a*b^2*c*d*3i)*((8*(2*b^3*c*d^8 - 4*a^3*c^2*d^7 + 2*b^3*c^3*d^6 - 12*a*b
^2*c^2*d^7 + 12*a^2*b*c*d^8))/d^5 - (8*tan(e/2 + (f*x)/2)*(8*a^3*c*d^9 - 8*b^3*c^4*d^6 + 24*a*b^2*c^3*d^7 - 24
*a^2*b*c^2*d^8))/d^6 + ((32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^10 - 8*c^3*d^8))/d^6)*(b^3*c^2*1i + (b*d^2
*(6*a^2 + b^2)*1i)/2 - a*b^2*c*d*3i))/d^3))/d^3))/d^3 + (16*tan(e/2 + (f*x)/2)*(8*b^9*c^8 + 2*b^9*c^4*d^4 + 8*
b^9*c^6*d^2 - 6*a*b^8*c^3*d^5 - 48*a*b^8*c^5*d^3 - 2*a^3*b^6*c*d^7 - 24*a^5*b^4*c*d^7 - 72*a^7*b^2*c*d^7 + 6*a
^2*b^7*c^2*d^6 + 120*a^2*b^7*c^4*d^4 + 288*a^2*b^7*c^6*d^2 - 152*a^3*b^6*c^3*d^5 - 656*a^3*b^6*c^5*d^3 + 96*a^
4*b^5*c^2*d^6 + 912*a^4*b^5*c^4*d^4 - 768*a^5*b^4*c^3*d^5 + 360*a^6*b^3*c^2*d^6 - 72*a*b^8*c^7*d))/d^6))*(b^3*
c^2*1i + (b*d^2*(6*a^2 + b^2)*1i)/2 - a*b^2*c*d*3i)*2i)/(d^3*f) + (atan((((-(c + d)*(c - d))^(1/2)*(a*d - b*c)
^3*((8*(b^6*c^2*d^6 + 4*b^6*c^4*d^4 + 4*b^6*c^6*d^2 - 12*a*b^5*c^3*d^5 - 24*a*b^5*c^5*d^3 + 12*a^2*b^4*c^2*d^6
+ 60*a^2*b^4*c^4*d^4 - 72*a^3*b^3*c^3*d^5 + 36*a^4*b^2*c^2*d^6))/d^5 + (8*tan(e/2 + (f*x)/2)*(2*b^6*c*d^8 - 4
*a^6*c*d^8 + 7*b^6*c^3*d^6 + 4*b^6*c^5*d^4 - 8*b^6*c^7*d^2 - 24*a*b^5*c^2*d^7 - 36*a*b^5*c^4*d^5 + 48*a*b^5*c^
6*d^3 + 24*a^2*b^4*c*d^8 + 72*a^4*b^2*c*d^8 + 24*a^5*b*c^2*d^7 + 108*a^2*b^4*c^3*d^6 - 120*a^2*b^4*c^5*d^4 - 1
44*a^3*b^3*c^2*d^7 + 152*a^3*b^3*c^4*d^5 - 96*a^4*b^2*c^3*d^6))/d^6 + ((-(c + d)*(c - d))^(1/2)*(a*d - b*c)^3*
((8*tan(e/2 + (f*x)/2)*(8*a^3*c*d^9 - 8*b^3*c^4*d^6 + 24*a*b^2*c^3*d^7 - 24*a^2*b*c^2*d^8))/d^6 - (8*(2*b^3*c*
d^8 - 4*a^3*c^2*d^7 + 2*b^3*c^3*d^6 - 12*a*b^2*c^2*d^7 + 12*a^2*b*c*d^8))/d^5 + ((32*c^2*d^3 + (8*tan(e/2 + (f
*x)/2)*(12*c*d^10 - 8*c^3*d^8))/d^6)*(-(c + d)*(c - d))^(1/2)*(a*d - b*c)^3)/(d^5 - c^2*d^3)))/(d^5 - c^2*d^3)
)*1i)/(d^5 - c^2*d^3) + ((-(c + d)*(c - d))^(1/2)*(a*d - b*c)^3*((8*(b^6*c^2*d^6 + 4*b^6*c^4*d^4 + 4*b^6*c^6*d
^2 - 12*a*b^5*c^3*d^5 - 24*a*b^5*c^5*d^3 + 12*a^2*b^4*c^2*d^6 + 60*a^2*b^4*c^4*d^4 - 72*a^3*b^3*c^3*d^5 + 36*a
^4*b^2*c^2*d^6))/d^5 + (8*tan(e/2 + (f*x)/2)*(2*b^6*c*d^8 - 4*a^6*c*d^8 + 7*b^6*c^3*d^6 + 4*b^6*c^5*d^4 - 8*b^
6*c^7*d^2 - 24*a*b^5*c^2*d^7 - 36*a*b^5*c^4*d^5 + 48*a*b^5*c^6*d^3 + 24*a^2*b^4*c*d^8 + 72*a^4*b^2*c*d^8 + 24*
a^5*b*c^2*d^7 + 108*a^2*b^4*c^3*d^6 - 120*a^2*b^4*c^5*d^4 - 144*a^3*b^3*c^2*d^7 + 152*a^3*b^3*c^4*d^5 - 96*a^4
*b^2*c^3*d^6))/d^6 + ((-(c + d)*(c - d))^(1/2)*(a*d - b*c)^3*((8*(2*b^3*c*d^8 - 4*a^3*c^2*d^7 + 2*b^3*c^3*d^6
- 12*a*b^2*c^2*d^7 + 12*a^2*b*c*d^8))/d^5 - (8*tan(e/2 + (f*x)/2)*(8*a^3*c*d^9 - 8*b^3*c^4*d^6 + 24*a*b^2*c^3*
d^7 - 24*a^2*b*c^2*d^8))/d^6 + ((32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^10 - 8*c^3*d^8))/d^6)*(-(c + d)*(c
- d))^(1/2)*(a*d - b*c)^3)/(d^5 - c^2*d^3)))/(d^5 - c^2*d^3))*1i)/(d^5 - c^2*d^3))/((16*(2*b^9*c^7 + b^9*c^5*
d^2 - 3*a*b^8*c^4*d^3 + 4*a^3*b^6*c^6*d - 2*a^6*b^3*c*d^6 + 3*a^2*b^7*c^3*d^4 + 30*a^2*b^7*c^5*d^2 - a^3*b^6*c
^2*d^5 - 36*a^3*b^6*c^4*d^3 + 18*a^4*b^5*c^3*d^4 - 24*a^4*b^5*c^5*d^2 + 60*a^5*b^4*c^4*d^3 - 76*a^6*b^3*c^3*d^
4 + 48*a^7*b^2*c^2*d^5 - 12*a*b^8*c^6*d - 12*a^8*b*c*d^6))/d^5 + (16*tan(e/2 + (f*x)/2)*(8*b^9*c^8 + 2*b^9*c^4
*d^4 + 8*b^9*c^6*d^2 - 6*a*b^8*c^3*d^5 - 48*a*b^8*c^5*d^3 - 2*a^3*b^6*c*d^7 - 24*a^5*b^4*c*d^7 - 72*a^7*b^2*c*
d^7 + 6*a^2*b^7*c^2*d^6 + 120*a^2*b^7*c^4*d^4 + 288*a^2*b^7*c^6*d^2 - 152*a^3*b^6*c^3*d^5 - 656*a^3*b^6*c^5*d^
3 + 96*a^4*b^5*c^2*d^6 + 912*a^4*b^5*c^4*d^4 - 768*a^5*b^4*c^3*d^5 + 360*a^6*b^3*c^2*d^6 - 72*a*b^8*c^7*d))/d^
6 - ((-(c + d)*(c - d))^(1/2)*(a*d - b*c)^3*((8*(b^6*c^2*d^6 + 4*b^6*c^4*d^4 + 4*b^6*c^6*d^2 - 12*a*b^5*c^3*d^
5 - 24*a*b^5*c^5*d^3 + 12*a^2*b^4*c^2*d^6 + 60*a^2*b^4*c^4*d^4 - 72*a^3*b^3*c^3*d^5 + 36*a^4*b^2*c^2*d^6))/d^5
+ (8*tan(e/2 + (f*x)/2)*(2*b^6*c*d^8 - 4*a^6*c*d^8 + 7*b^6*c^3*d^6 + 4*b^6*c^5*d^4 - 8*b^6*c^7*d^2 - 24*a*b^5
*c^2*d^7 - 36*a*b^5*c^4*d^5 + 48*a*b^5*c^6*d^3 + 24*a^2*b^4*c*d^8 + 72*a^4*b^2*c*d^8 + 24*a^5*b*c^2*d^7 + 108*
a^2*b^4*c^3*d^6 - 120*a^2*b^4*c^5*d^4 - 144*a^3*b^3*c^2*d^7 + 152*a^3*b^3*c^4*d^5 - 96*a^4*b^2*c^3*d^6))/d^6 +
((-(c + d)*(c - d))^(1/2)*(a*d - b*c)^3*((8*tan(e/2 + (f*x)/2)*(8*a^3*c*d^9 - 8*b^3*c^4*d^6 + 24*a*b^2*c^3*d^
7 - 24*a^2*b*c^2*d^8))/d^6 - (8*(2*b^3*c*d^8 - 4*a^3*c^2*d^7 + 2*b^3*c^3*d^6 - 12*a*b^2*c^2*d^7 + 12*a^2*b*c*d
^8))/d^5 + ((32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^10 - 8*c^3*d^8))/d^6)*(-(c + d)*(c - d))^(1/2)*(a*d -
b*c)^3)/(d^5 - c^2*d^3)))/(d^5 - c^2*d^3)))/(d^5 - c^2*d^3) + ((-(c + d)*(c - d))^(1/2)*(a*d - b*c)^3*((8*(b^6
*c^2*d^6 + 4*b^6*c^4*d^4 + 4*b^6*c^6*d^2 - 12*a*b^5*c^3*d^5 - 24*a*b^5*c^5*d^3 + 12*a^2*b^4*c^2*d^6 + 60*a^2*b
^4*c^4*d^4 - 72*a^3*b^3*c^3*d^5 + 36*a^4*b^2*c^2*d^6))/d^5 + (8*tan(e/2 + (f*x)/2)*(2*b^6*c*d^8 - 4*a^6*c*d^8
+ 7*b^6*c^3*d^6 + 4*b^6*c^5*d^4 - 8*b^6*c^7*d^2 - 24*a*b^5*c^2*d^7 - 36*a*b^5*c^4*d^5 + 48*a*b^5*c^6*d^3 + 24*
a^2*b^4*c*d^8 + 72*a^4*b^2*c*d^8 + 24*a^5*b*c^2*d^7 + 108*a^2*b^4*c^3*d^6 - 120*a^2*b^4*c^5*d^4 - 144*a^3*b^3*
c^2*d^7 + 152*a^3*b^3*c^4*d^5 - 96*a^4*b^2*c^3*d^6))/d^6 + ((-(c + d)*(c - d))^(1/2)*(a*d - b*c)^3*((8*(2*b^3*
c*d^8 - 4*a^3*c^2*d^7 + 2*b^3*c^3*d^6 - 12*a*b^2*c^2*d^7 + 12*a^2*b*c*d^8))/d^5 - (8*tan(e/2 + (f*x)/2)*(8*a^3
*c*d^9 - 8*b^3*c^4*d^6 + 24*a*b^2*c^3*d^7 - 24*a^2*b*c^2*d^8))/d^6 + ((32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*
c*d^10 - 8*c^3*d^8))/d^6)*(-(c + d)*(c - d))^(1/2)*(a*d - b*c)^3)/(d^5 - c^2*d^3)))/(d^5 - c^2*d^3)))/(d^5 - c
^2*d^3)))*(-(c + d)*(c - d))^(1/2)*(a*d - b*c)^3*2i)/(f*(d^5 - c^2*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)),x)

[Out]

Timed out

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