3.262 \(\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx\)

Optimal. Leaf size=246 \[ \frac {2 a^3 (c-d)^3 (B c-A d) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^4 f \sqrt {c^2-d^2}}+\frac {a^3 \left (A d (2 c-5 d)-B \left (2 c^2-5 c d+5 d^2\right )\right ) \cos (e+f x)}{2 d^3 f}+\frac {a^3 x \left (A d \left (2 c^2-6 c d+7 d^2\right )-B \left (2 c^3-6 c^2 d+7 c d^2-5 d^3\right )\right )}{2 d^4}+\frac {(-3 A d+3 B c-5 B d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{6 d^2 f}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2}{3 d f} \]

[Out]

1/2*a^3*(A*d*(2*c^2-6*c*d+7*d^2)-B*(2*c^3-6*c^2*d+7*c*d^2-5*d^3))*x/d^4+1/2*a^3*(A*(2*c-5*d)*d-B*(2*c^2-5*c*d+
5*d^2))*cos(f*x+e)/d^3/f-1/3*a*B*cos(f*x+e)*(a+a*sin(f*x+e))^2/d/f+1/6*(-3*A*d+3*B*c-5*B*d)*cos(f*x+e)*(a^3+a^
3*sin(f*x+e))/d^2/f+2*a^3*(c-d)^3*(-A*d+B*c)*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/d^4/f/(c^2-d^2)^
(1/2)

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Rubi [A]  time = 0.90, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2976, 2968, 3023, 2735, 2660, 618, 204} \[ \frac {a^3 \left (A d (2 c-5 d)-B \left (2 c^2-5 c d+5 d^2\right )\right ) \cos (e+f x)}{2 d^3 f}+\frac {2 a^3 (c-d)^3 (B c-A d) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^4 f \sqrt {c^2-d^2}}+\frac {a^3 x \left (A d \left (2 c^2-6 c d+7 d^2\right )-B \left (-6 c^2 d+2 c^3+7 c d^2-5 d^3\right )\right )}{2 d^4}+\frac {(-3 A d+3 B c-5 B d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{6 d^2 f}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2}{3 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(A*d*(2*c^2 - 6*c*d + 7*d^2) - B*(2*c^3 - 6*c^2*d + 7*c*d^2 - 5*d^3))*x)/(2*d^4) + (2*a^3*(c - d)^3*(B*c
- A*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^4*Sqrt[c^2 - d^2]*f) + (a^3*(A*(2*c - 5*d)*d - B*(
2*c^2 - 5*c*d + 5*d^2))*Cos[e + f*x])/(2*d^3*f) - (a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(3*d*f) + ((3*B*c
- 3*A*d - 5*B*d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(6*d^2*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 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&
NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[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+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx &=-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2}{3 d f}+\frac {\int \frac {(a+a \sin (e+f x))^2 (a (2 B c+3 A d)-a (3 B c-3 A d-5 B d) \sin (e+f x))}{c+d \sin (e+f x)} \, dx}{3 d}\\ &=-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2}{3 d f}+\frac {(3 B c-3 A d-5 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 d^2 f}+\frac {\int \frac {(a+a \sin (e+f x)) \left (-3 a^2 (B c (c-3 d)-A d (c+2 d))-3 a^2 \left (A (2 c-5 d) d-B \left (2 c^2-5 c d+5 d^2\right )\right ) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{6 d^2}\\ &=-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2}{3 d f}+\frac {(3 B c-3 A d-5 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 d^2 f}+\frac {\int \frac {-3 a^3 (B c (c-3 d)-A d (c+2 d))+\left (-3 a^3 (B c (c-3 d)-A d (c+2 d))-3 a^3 \left (A (2 c-5 d) d-B \left (2 c^2-5 c d+5 d^2\right )\right )\right ) \sin (e+f x)-3 a^3 \left (A (2 c-5 d) d-B \left (2 c^2-5 c d+5 d^2\right )\right ) \sin ^2(e+f x)}{c+d \sin (e+f x)} \, dx}{6 d^2}\\ &=\frac {a^3 \left (A (2 c-5 d) d-B \left (2 c^2-5 c d+5 d^2\right )\right ) \cos (e+f x)}{2 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2}{3 d f}+\frac {(3 B c-3 A d-5 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 d^2 f}+\frac {\int \frac {-3 a^3 d (B c (c-3 d)-A d (c+2 d))+3 a^3 \left (A d \left (2 c^2-6 c d+7 d^2\right )-B \left (2 c^3-6 c^2 d+7 c d^2-5 d^3\right )\right ) \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{6 d^3}\\ &=\frac {a^3 \left (A d \left (2 c^2-6 c d+7 d^2\right )-B \left (2 c^3-6 c^2 d+7 c d^2-5 d^3\right )\right ) x}{2 d^4}+\frac {a^3 \left (A (2 c-5 d) d-B \left (2 c^2-5 c d+5 d^2\right )\right ) \cos (e+f x)}{2 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2}{3 d f}+\frac {(3 B c-3 A d-5 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 d^2 f}+\frac {\left (a^3 (c-d)^3 (B c-A d)\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{d^4}\\ &=\frac {a^3 \left (A d \left (2 c^2-6 c d+7 d^2\right )-B \left (2 c^3-6 c^2 d+7 c d^2-5 d^3\right )\right ) x}{2 d^4}+\frac {a^3 \left (A (2 c-5 d) d-B \left (2 c^2-5 c d+5 d^2\right )\right ) \cos (e+f x)}{2 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2}{3 d f}+\frac {(3 B c-3 A d-5 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 d^2 f}+\frac {\left (2 a^3 (c-d)^3 (B c-A d)\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^4 f}\\ &=\frac {a^3 \left (A d \left (2 c^2-6 c d+7 d^2\right )-B \left (2 c^3-6 c^2 d+7 c d^2-5 d^3\right )\right ) x}{2 d^4}+\frac {a^3 \left (A (2 c-5 d) d-B \left (2 c^2-5 c d+5 d^2\right )\right ) \cos (e+f x)}{2 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2}{3 d f}+\frac {(3 B c-3 A d-5 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 d^2 f}-\frac {\left (4 a^3 (c-d)^3 (B c-A d)\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^4 f}\\ &=\frac {a^3 \left (A d \left (2 c^2-6 c d+7 d^2\right )-B \left (2 c^3-6 c^2 d+7 c d^2-5 d^3\right )\right ) x}{2 d^4}+\frac {2 a^3 (c-d)^3 (B c-A d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^4 \sqrt {c^2-d^2} f}+\frac {a^3 \left (A (2 c-5 d) d-B \left (2 c^2-5 c d+5 d^2\right )\right ) \cos (e+f x)}{2 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2}{3 d f}+\frac {(3 B c-3 A d-5 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 d^2 f}\\ \end {align*}

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Mathematica [A]  time = 0.98, size = 233, normalized size = 0.95 \[ \frac {a^3 (\sin (e+f x)+1)^3 \left (-3 d \left (4 A d (3 d-c)+B \left (4 c^2-12 c d+15 d^2\right )\right ) \cos (e+f x)+\frac {24 (c-d)^3 (B c-A d) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{\sqrt {c^2-d^2}}+6 (e+f x) \left (A d \left (2 c^2-6 c d+7 d^2\right )+B \left (-2 c^3+6 c^2 d-7 c d^2+5 d^3\right )\right )-3 d^2 (A d-B c+3 B d) \sin (2 (e+f x))+B d^3 \cos (3 (e+f x))\right )}{12 d^4 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + Sin[e + f*x])^3*(6*(A*d*(2*c^2 - 6*c*d + 7*d^2) + B*(-2*c^3 + 6*c^2*d - 7*c*d^2 + 5*d^3))*(e + f*x)
+ (24*(c - d)^3*(B*c - A*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] - 3*d*(4*A*d*(-c
+ 3*d) + B*(4*c^2 - 12*c*d + 15*d^2))*Cos[e + f*x] + B*d^3*Cos[3*(e + f*x)] - 3*d^2*(-(B*c) + A*d + 3*B*d)*Si
n[2*(e + f*x)]))/(12*d^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)

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fricas [A]  time = 0.55, size = 627, normalized size = 2.55 \[ \left [\frac {2 \, B a^{3} d^{3} \cos \left (f x + e\right )^{3} - 3 \, {\left (2 \, B a^{3} c^{3} - 2 \, {\left (A + 3 \, B\right )} a^{3} c^{2} d + {\left (6 \, A + 7 \, B\right )} a^{3} c d^{2} - {\left (7 \, A + 5 \, B\right )} a^{3} d^{3}\right )} f x + 3 \, {\left (B a^{3} c d^{2} - {\left (A + 3 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, {\left (B a^{3} c^{3} - {\left (A + 2 \, B\right )} a^{3} c^{2} d + {\left (2 \, A + B\right )} a^{3} c d^{2} - A a^{3} d^{3}\right )} \sqrt {-\frac {c - d}{c + d}} \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 ({\left (c^{2} + c d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {c - d}{c + d}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) - 6 \, {\left (B a^{3} c^{2} d - {\left (A + 3 \, B\right )} a^{3} c d^{2} + {\left (3 \, A + 4 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )}{6 \, d^{4} f}, \frac {2 \, B a^{3} d^{3} \cos \left (f x + e\right )^{3} - 3 \, {\left (2 \, B a^{3} c^{3} - 2 \, {\left (A + 3 \, B\right )} a^{3} c^{2} d + {\left (6 \, A + 7 \, B\right )} a^{3} c d^{2} - {\left (7 \, A + 5 \, B\right )} a^{3} d^{3}\right )} f x + 3 \, {\left (B a^{3} c d^{2} - {\left (A + 3 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 6 \, {\left (B a^{3} c^{3} - {\left (A + 2 \, B\right )} a^{3} c^{2} d + {\left (2 \, A + B\right )} a^{3} c d^{2} - A a^{3} d^{3}\right )} \sqrt {\frac {c - d}{c + d}} \arctan \left (-\frac {{\left (c \sin \left (f x + e\right ) + d\right )} \sqrt {\frac {c - d}{c + d}}}{{\left (c - d\right )} \cos \left (f x + e\right )}\right ) - 6 \, {\left (B a^{3} c^{2} d - {\left (A + 3 \, B\right )} a^{3} c d^{2} + {\left (3 \, A + 4 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )}{6 \, d^{4} f}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(2*B*a^3*d^3*cos(f*x + e)^3 - 3*(2*B*a^3*c^3 - 2*(A + 3*B)*a^3*c^2*d + (6*A + 7*B)*a^3*c*d^2 - (7*A + 5*B
)*a^3*d^3)*f*x + 3*(B*a^3*c*d^2 - (A + 3*B)*a^3*d^3)*cos(f*x + e)*sin(f*x + e) + 3*(B*a^3*c^3 - (A + 2*B)*a^3*
c^2*d + (2*A + B)*a^3*c*d^2 - A*a^3*d^3)*sqrt(-(c - d)/(c + d))*log(-((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin
(f*x + e) - c^2 - d^2 - 2*((c^2 + c*d)*cos(f*x + e)*sin(f*x + e) + (c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c
+ d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 6*(B*a^3*c^2*d - (A + 3*B)*a^3*c*d^2 + (3*A +
4*B)*a^3*d^3)*cos(f*x + e))/(d^4*f), 1/6*(2*B*a^3*d^3*cos(f*x + e)^3 - 3*(2*B*a^3*c^3 - 2*(A + 3*B)*a^3*c^2*d
+ (6*A + 7*B)*a^3*c*d^2 - (7*A + 5*B)*a^3*d^3)*f*x + 3*(B*a^3*c*d^2 - (A + 3*B)*a^3*d^3)*cos(f*x + e)*sin(f*x
+ e) - 6*(B*a^3*c^3 - (A + 2*B)*a^3*c^2*d + (2*A + B)*a^3*c*d^2 - A*a^3*d^3)*sqrt((c - d)/(c + d))*arctan(-(c*
sin(f*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(f*x + e))) - 6*(B*a^3*c^2*d - (A + 3*B)*a^3*c*d^2 + (3*A
+ 4*B)*a^3*d^3)*cos(f*x + e))/(d^4*f)]

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giac [B]  time = 0.21, size = 617, normalized size = 2.51 \[ -\frac {\frac {3 \, {\left (2 \, B a^{3} c^{3} - 2 \, A a^{3} c^{2} d - 6 \, B a^{3} c^{2} d + 6 \, A a^{3} c d^{2} + 7 \, B a^{3} c d^{2} - 7 \, A a^{3} d^{3} - 5 \, B a^{3} d^{3}\right )} {\left (f x + e\right )}}{d^{4}} - \frac {12 \, {\left (B a^{3} c^{4} - A a^{3} c^{3} d - 3 \, B a^{3} c^{3} d + 3 \, A a^{3} c^{2} d^{2} + 3 \, B a^{3} c^{2} d^{2} - 3 \, A a^{3} c d^{3} - B a^{3} c d^{3} + A a^{3} 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 )}}{\sqrt {c^{2} - d^{2}} d^{4}} + \frac {2 \, {\left (3 \, B a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 \, A a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 9 \, B a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, B a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 6 \, A a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 18 \, B a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 18 \, A a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 18 \, B a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 12 \, B a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, A a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 36 \, B a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, A a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 48 \, B a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, B a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, A a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 9 \, B a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, B a^{3} c^{2} - 6 \, A a^{3} c d - 18 \, B a^{3} c d + 18 \, A a^{3} d^{2} + 22 \, B a^{3} d^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{3} d^{3}}}{6 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*(2*B*a^3*c^3 - 2*A*a^3*c^2*d - 6*B*a^3*c^2*d + 6*A*a^3*c*d^2 + 7*B*a^3*c*d^2 - 7*A*a^3*d^3 - 5*B*a^3*d
^3)*(f*x + e)/d^4 - 12*(B*a^3*c^4 - A*a^3*c^3*d - 3*B*a^3*c^3*d + 3*A*a^3*c^2*d^2 + 3*B*a^3*c^2*d^2 - 3*A*a^3*
c*d^3 - B*a^3*c*d^3 + A*a^3*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)))/(sqrt(c^2 - d^2)*d^4) + 2*(3*B*a^3*c*d*tan(1/2*f*x + 1/2*e)^5 - 3*A*a^3*d^2*tan(1/2*f*x +
1/2*e)^5 - 9*B*a^3*d^2*tan(1/2*f*x + 1/2*e)^5 + 6*B*a^3*c^2*tan(1/2*f*x + 1/2*e)^4 - 6*A*a^3*c*d*tan(1/2*f*x +
1/2*e)^4 - 18*B*a^3*c*d*tan(1/2*f*x + 1/2*e)^4 + 18*A*a^3*d^2*tan(1/2*f*x + 1/2*e)^4 + 18*B*a^3*d^2*tan(1/2*f
*x + 1/2*e)^4 + 12*B*a^3*c^2*tan(1/2*f*x + 1/2*e)^2 - 12*A*a^3*c*d*tan(1/2*f*x + 1/2*e)^2 - 36*B*a^3*c*d*tan(1
/2*f*x + 1/2*e)^2 + 36*A*a^3*d^2*tan(1/2*f*x + 1/2*e)^2 + 48*B*a^3*d^2*tan(1/2*f*x + 1/2*e)^2 - 3*B*a^3*c*d*ta
n(1/2*f*x + 1/2*e) + 3*A*a^3*d^2*tan(1/2*f*x + 1/2*e) + 9*B*a^3*d^2*tan(1/2*f*x + 1/2*e) + 6*B*a^3*c^2 - 6*A*a
^3*c*d - 18*B*a^3*c*d + 18*A*a^3*d^2 + 22*B*a^3*d^2)/((tan(1/2*f*x + 1/2*e)^2 + 1)^3*d^3))/f

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maple [B]  time = 0.48, size = 1357, normalized size = 5.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f*a^3/d^2/(1+tan(1/2*f*x+1/2*e)^2)^3*A*tan(1/2*f*x+1/2*e)^4*c-2/f*a^3/d^3/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1
/2*f*x+1/2*e)^4*c^2+6/f*a^3/d^2/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1/2*f*x+1/2*e)^4*c+4/f*a^3/d^2/(1+tan(1/2*f*x
+1/2*e)^2)^3*A*tan(1/2*f*x+1/2*e)^2*c-4/f*a^3/d^3/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1/2*f*x+1/2*e)^2*c^2+12/f*a
^3/d^2/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1/2*f*x+1/2*e)^2*c+1/f*a^3/d^2/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+
1/2*e)*B*c-16/f*a^3/d/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1/2*f*x+1/2*e)^2-1/f*a^3/d/(1+tan(1/2*f*x+1/2*e)^2)^3*t
an(1/2*f*x+1/2*e)*A+6/f*a^3/d^3*arctan(tan(1/2*f*x+1/2*e))*B*c^2-7/f*a^3/d^2*arctan(tan(1/2*f*x+1/2*e))*B*c-3/
f*a^3/d/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)*B+2/f*a^3/d^2/(1+tan(1/2*f*x+1/2*e)^2)^3*A*c-2/f*a^3/d^3
/(1+tan(1/2*f*x+1/2*e)^2)^3*B*c^2+2/f*a^3/d^3*arctan(tan(1/2*f*x+1/2*e))*A*c^2-6/f*a^3/d^2*arctan(tan(1/2*f*x+
1/2*e))*A*c+6/f*a^3/d^2/(1+tan(1/2*f*x+1/2*e)^2)^3*B*c-2/f*a^3/d^4*arctan(tan(1/2*f*x+1/2*e))*B*c^3+1/f*a^3/d/
(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5*A+3/f*a^3/d/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5*B-
6/f*a^3/d/(1+tan(1/2*f*x+1/2*e)^2)^3*A*tan(1/2*f*x+1/2*e)^4-6/f*a^3/d/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1/2*f*x
+1/2*e)^4-12/f*a^3/d/(1+tan(1/2*f*x+1/2*e)^2)^3*A*tan(1/2*f*x+1/2*e)^2-2/f*a^3/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))*A*c^3+6/f*a^3/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*c^2-6/f*a^3/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*c+2/f*a^3/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*c^4-6/f*a^3/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*c^3+6/f*a^3/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))*B*c^2-2/f*a^3/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))*B*c-1/f*a^3/d^2/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5*B*c+7/f*a^
3/d*arctan(tan(1/2*f*x+1/2*e))*A+5/f*a^3/d*arctan(tan(1/2*f*x+1/2*e))*B-6/f*a^3/d/(1+tan(1/2*f*x+1/2*e)^2)^3*A
-22/3/f*a^3/d/(1+tan(1/2*f*x+1/2*e)^2)^3*B+2/f*a^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))*A

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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+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(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 = 21.76, size = 10256, normalized size = 41.69 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*(9*A*a^3*d^2 + 3*B*a^3*c^2 + 11*B*a^3*d^2 - 3*A*a^3*c*d - 9*B*a^3*c*d))/(3*d^3) - (tan(e/2 + (f*x)/2)^5*
(A*a^3*d - B*a^3*c + 3*B*a^3*d))/d^2 + (4*tan(e/2 + (f*x)/2)^2*(3*A*a^3*d^2 + B*a^3*c^2 + 4*B*a^3*d^2 - A*a^3*
c*d - 3*B*a^3*c*d))/d^3 + (2*tan(e/2 + (f*x)/2)^4*(3*A*a^3*d^2 + B*a^3*c^2 + 3*B*a^3*d^2 - A*a^3*c*d - 3*B*a^3
*c*d))/d^3 + (tan(e/2 + (f*x)/2)*(A*a^3*d - B*a^3*c + 3*B*a^3*d))/d^2)/(f*(3*tan(e/2 + (f*x)/2)^2 + 3*tan(e/2
+ (f*x)/2)^4 + tan(e/2 + (f*x)/2)^6 + 1)) - (atan(((((8*(49*A^2*a^6*c^2*d^9 - 84*A^2*a^6*c^3*d^8 + 64*A^2*a^6*
c^4*d^7 - 24*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 25*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 + 109*B^2*a^6*c^4*d
^7 - 104*B^2*a^6*c^5*d^6 + 64*B^2*a^6*c^6*d^5 - 24*B^2*a^6*c^7*d^4 + 4*B^2*a^6*c^8*d^3 + 70*A*B*a^6*c^2*d^9 -
158*A*B*a^6*c^3*d^8 + 188*A*B*a^6*c^4*d^7 - 128*A*B*a^6*c^5*d^6 + 48*A*B*a^6*c^6*d^5 - 8*A*B*a^6*c^7*d^4))/d^8
+ (8*tan(e/2 + (f*x)/2)*(19*A^2*a^6*c^3*d^9 - 144*A^2*a^6*c^2*d^10 + 116*A^2*a^6*c^4*d^8 - 116*A^2*a^6*c^5*d^
7 + 48*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 - 140*B^2*a^6*c^2*d^10 + 189*B^2*a^6*c^3*d^9 - 114*B^2*a^6*c^4*d^8
- 41*B^2*a^6*c^5*d^7 + 136*B^2*a^6*c^6*d^6 - 116*B^2*a^6*c^7*d^5 + 48*B^2*a^6*c^8*d^4 - 8*B^2*a^6*c^9*d^3 + 94
*A^2*a^6*c*d^11 + 50*B^2*a^6*c*d^11 - 308*A*B*a^6*c^2*d^10 + 258*A*B*a^6*c^3*d^9 + 22*A*B*a^6*c^4*d^8 - 252*A*
B*a^6*c^5*d^7 + 232*A*B*a^6*c^6*d^6 - 96*A*B*a^6*c^7*d^5 + 16*A*B*a^6*c^8*d^4 + 140*A*B*a^6*c*d^11))/d^9 + (((
(32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^13 - 8*c^3*d^11))/d^9)*(B*a^3*c^3*1i + (a^3*d^2*(6*A*c + 7*B*c)*1i
)/2 - (a^3*d^3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2))/d^4 - (8*(14*A*a^3*c*d^11 + 10*B*a^3*c*d
^11 - 16*A*a^3*c^2*d^10 + 2*A*a^3*c^3*d^9 - 14*B*a^3*c^2*d^10 + 6*B*a^3*c^3*d^9 - 2*B*a^3*c^4*d^8))/d^8 + (8*t
an(e/2 + (f*x)/2)*(8*A*a^3*c*d^12 - 24*A*a^3*c^2*d^11 + 24*A*a^3*c^3*d^10 - 8*A*a^3*c^4*d^9 - 8*B*a^3*c^2*d^11
+ 24*B*a^3*c^3*d^10 - 24*B*a^3*c^4*d^9 + 8*B*a^3*c^5*d^8))/d^9)*(B*a^3*c^3*1i + (a^3*d^2*(6*A*c + 7*B*c)*1i)/
2 - (a^3*d^3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2))/d^4)*(B*a^3*c^3*1i + (a^3*d^2*(6*A*c + 7*B
*c)*1i)/2 - (a^3*d^3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2)*1i)/d^4 + (((8*(49*A^2*a^6*c^2*d^9
- 84*A^2*a^6*c^3*d^8 + 64*A^2*a^6*c^4*d^7 - 24*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 25*B^2*a^6*c^2*d^9 - 70*B
^2*a^6*c^3*d^8 + 109*B^2*a^6*c^4*d^7 - 104*B^2*a^6*c^5*d^6 + 64*B^2*a^6*c^6*d^5 - 24*B^2*a^6*c^7*d^4 + 4*B^2*a
^6*c^8*d^3 + 70*A*B*a^6*c^2*d^9 - 158*A*B*a^6*c^3*d^8 + 188*A*B*a^6*c^4*d^7 - 128*A*B*a^6*c^5*d^6 + 48*A*B*a^6
*c^6*d^5 - 8*A*B*a^6*c^7*d^4))/d^8 + (8*tan(e/2 + (f*x)/2)*(19*A^2*a^6*c^3*d^9 - 144*A^2*a^6*c^2*d^10 + 116*A^
2*a^6*c^4*d^8 - 116*A^2*a^6*c^5*d^7 + 48*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 - 140*B^2*a^6*c^2*d^10 + 189*B^2*
a^6*c^3*d^9 - 114*B^2*a^6*c^4*d^8 - 41*B^2*a^6*c^5*d^7 + 136*B^2*a^6*c^6*d^6 - 116*B^2*a^6*c^7*d^5 + 48*B^2*a^
6*c^8*d^4 - 8*B^2*a^6*c^9*d^3 + 94*A^2*a^6*c*d^11 + 50*B^2*a^6*c*d^11 - 308*A*B*a^6*c^2*d^10 + 258*A*B*a^6*c^3
*d^9 + 22*A*B*a^6*c^4*d^8 - 252*A*B*a^6*c^5*d^7 + 232*A*B*a^6*c^6*d^6 - 96*A*B*a^6*c^7*d^5 + 16*A*B*a^6*c^8*d^
4 + 140*A*B*a^6*c*d^11))/d^9 + (((8*(14*A*a^3*c*d^11 + 10*B*a^3*c*d^11 - 16*A*a^3*c^2*d^10 + 2*A*a^3*c^3*d^9 -
14*B*a^3*c^2*d^10 + 6*B*a^3*c^3*d^9 - 2*B*a^3*c^4*d^8))/d^8 + ((32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^13
- 8*c^3*d^11))/d^9)*(B*a^3*c^3*1i + (a^3*d^2*(6*A*c + 7*B*c)*1i)/2 - (a^3*d^3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A
*c^2 + 6*B*c^2)*1i)/2))/d^4 - (8*tan(e/2 + (f*x)/2)*(8*A*a^3*c*d^12 - 24*A*a^3*c^2*d^11 + 24*A*a^3*c^3*d^10 -
8*A*a^3*c^4*d^9 - 8*B*a^3*c^2*d^11 + 24*B*a^3*c^3*d^10 - 24*B*a^3*c^4*d^9 + 8*B*a^3*c^5*d^8))/d^9)*(B*a^3*c^3*
1i + (a^3*d^2*(6*A*c + 7*B*c)*1i)/2 - (a^3*d^3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2))/d^4)*(B*
a^3*c^3*1i + (a^3*d^2*(6*A*c + 7*B*c)*1i)/2 - (a^3*d^3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2)*1
i)/d^4)/((16*(2*B^3*a^9*c^10 - 47*A^3*a^9*c^2*d^8 + 55*A^3*a^9*c^3*d^7 - 21*A^3*a^9*c^4*d^6 - 7*A^3*a^9*c^5*d^
5 + 8*A^3*a^9*c^6*d^4 - 2*A^3*a^9*c^7*d^3 - 15*B^3*a^9*c^3*d^7 + 71*B^3*a^9*c^4*d^6 - 148*B^3*a^9*c^5*d^5 + 18
0*B^3*a^9*c^6*d^4 - 139*B^3*a^9*c^7*d^3 + 67*B^3*a^9*c^8*d^2 + 14*A^3*a^9*c*d^9 - 18*B^3*a^9*c^9*d - 6*A*B^2*a
^9*c^9*d + 10*A^2*B*a^9*c*d^9 + 5*A*B^2*a^9*c^2*d^8 - 53*A*B^2*a^9*c^3*d^7 + 174*A*B^2*a^9*c^4*d^6 - 280*A*B^2
*a^9*c^5*d^5 + 257*A*B^2*a^9*c^6*d^4 - 141*A*B^2*a^9*c^7*d^3 + 44*A*B^2*a^9*c^8*d^2 - 32*A^2*B*a^9*c^2*d^8 + 2
1*A^2*B*a^9*c^3*d^7 + 45*A^2*B*a^9*c^4*d^6 - 97*A^2*B*a^9*c^5*d^5 + 81*A^2*B*a^9*c^6*d^4 - 34*A^2*B*a^9*c^7*d^
3 + 6*A^2*B*a^9*c^8*d^2))/d^8 + (((8*(49*A^2*a^6*c^2*d^9 - 84*A^2*a^6*c^3*d^8 + 64*A^2*a^6*c^4*d^7 - 24*A^2*a^
6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 25*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 + 109*B^2*a^6*c^4*d^7 - 104*B^2*a^6*c^
5*d^6 + 64*B^2*a^6*c^6*d^5 - 24*B^2*a^6*c^7*d^4 + 4*B^2*a^6*c^8*d^3 + 70*A*B*a^6*c^2*d^9 - 158*A*B*a^6*c^3*d^8
+ 188*A*B*a^6*c^4*d^7 - 128*A*B*a^6*c^5*d^6 + 48*A*B*a^6*c^6*d^5 - 8*A*B*a^6*c^7*d^4))/d^8 + (8*tan(e/2 + (f*
x)/2)*(19*A^2*a^6*c^3*d^9 - 144*A^2*a^6*c^2*d^10 + 116*A^2*a^6*c^4*d^8 - 116*A^2*a^6*c^5*d^7 + 48*A^2*a^6*c^6*
d^6 - 8*A^2*a^6*c^7*d^5 - 140*B^2*a^6*c^2*d^10 + 189*B^2*a^6*c^3*d^9 - 114*B^2*a^6*c^4*d^8 - 41*B^2*a^6*c^5*d^
7 + 136*B^2*a^6*c^6*d^6 - 116*B^2*a^6*c^7*d^5 + 48*B^2*a^6*c^8*d^4 - 8*B^2*a^6*c^9*d^3 + 94*A^2*a^6*c*d^11 + 5
0*B^2*a^6*c*d^11 - 308*A*B*a^6*c^2*d^10 + 258*A*B*a^6*c^3*d^9 + 22*A*B*a^6*c^4*d^8 - 252*A*B*a^6*c^5*d^7 + 232
*A*B*a^6*c^6*d^6 - 96*A*B*a^6*c^7*d^5 + 16*A*B*a^6*c^8*d^4 + 140*A*B*a^6*c*d^11))/d^9 + ((((32*c^2*d^3 + (8*ta
n(e/2 + (f*x)/2)*(12*c*d^13 - 8*c^3*d^11))/d^9)*(B*a^3*c^3*1i + (a^3*d^2*(6*A*c + 7*B*c)*1i)/2 - (a^3*d^3*(7*A
+ 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2))/d^4 - (8*(14*A*a^3*c*d^11 + 10*B*a^3*c*d^11 - 16*A*a^3*c^2*
d^10 + 2*A*a^3*c^3*d^9 - 14*B*a^3*c^2*d^10 + 6*B*a^3*c^3*d^9 - 2*B*a^3*c^4*d^8))/d^8 + (8*tan(e/2 + (f*x)/2)*(
8*A*a^3*c*d^12 - 24*A*a^3*c^2*d^11 + 24*A*a^3*c^3*d^10 - 8*A*a^3*c^4*d^9 - 8*B*a^3*c^2*d^11 + 24*B*a^3*c^3*d^1
0 - 24*B*a^3*c^4*d^9 + 8*B*a^3*c^5*d^8))/d^9)*(B*a^3*c^3*1i + (a^3*d^2*(6*A*c + 7*B*c)*1i)/2 - (a^3*d^3*(7*A +
5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2))/d^4)*(B*a^3*c^3*1i + (a^3*d^2*(6*A*c + 7*B*c)*1i)/2 - (a^3*d^
3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2))/d^4 - (((8*(49*A^2*a^6*c^2*d^9 - 84*A^2*a^6*c^3*d^8 +
64*A^2*a^6*c^4*d^7 - 24*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 25*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 + 109*B
^2*a^6*c^4*d^7 - 104*B^2*a^6*c^5*d^6 + 64*B^2*a^6*c^6*d^5 - 24*B^2*a^6*c^7*d^4 + 4*B^2*a^6*c^8*d^3 + 70*A*B*a^
6*c^2*d^9 - 158*A*B*a^6*c^3*d^8 + 188*A*B*a^6*c^4*d^7 - 128*A*B*a^6*c^5*d^6 + 48*A*B*a^6*c^6*d^5 - 8*A*B*a^6*c
^7*d^4))/d^8 + (8*tan(e/2 + (f*x)/2)*(19*A^2*a^6*c^3*d^9 - 144*A^2*a^6*c^2*d^10 + 116*A^2*a^6*c^4*d^8 - 116*A^
2*a^6*c^5*d^7 + 48*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 - 140*B^2*a^6*c^2*d^10 + 189*B^2*a^6*c^3*d^9 - 114*B^2*
a^6*c^4*d^8 - 41*B^2*a^6*c^5*d^7 + 136*B^2*a^6*c^6*d^6 - 116*B^2*a^6*c^7*d^5 + 48*B^2*a^6*c^8*d^4 - 8*B^2*a^6*
c^9*d^3 + 94*A^2*a^6*c*d^11 + 50*B^2*a^6*c*d^11 - 308*A*B*a^6*c^2*d^10 + 258*A*B*a^6*c^3*d^9 + 22*A*B*a^6*c^4*
d^8 - 252*A*B*a^6*c^5*d^7 + 232*A*B*a^6*c^6*d^6 - 96*A*B*a^6*c^7*d^5 + 16*A*B*a^6*c^8*d^4 + 140*A*B*a^6*c*d^11
))/d^9 + (((8*(14*A*a^3*c*d^11 + 10*B*a^3*c*d^11 - 16*A*a^3*c^2*d^10 + 2*A*a^3*c^3*d^9 - 14*B*a^3*c^2*d^10 + 6
*B*a^3*c^3*d^9 - 2*B*a^3*c^4*d^8))/d^8 + ((32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^13 - 8*c^3*d^11))/d^9)*(
B*a^3*c^3*1i + (a^3*d^2*(6*A*c + 7*B*c)*1i)/2 - (a^3*d^3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2)
)/d^4 - (8*tan(e/2 + (f*x)/2)*(8*A*a^3*c*d^12 - 24*A*a^3*c^2*d^11 + 24*A*a^3*c^3*d^10 - 8*A*a^3*c^4*d^9 - 8*B*
a^3*c^2*d^11 + 24*B*a^3*c^3*d^10 - 24*B*a^3*c^4*d^9 + 8*B*a^3*c^5*d^8))/d^9)*(B*a^3*c^3*1i + (a^3*d^2*(6*A*c +
7*B*c)*1i)/2 - (a^3*d^3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2))/d^4)*(B*a^3*c^3*1i + (a^3*d^2*
(6*A*c + 7*B*c)*1i)/2 - (a^3*d^3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2))/d^4 + (16*tan(e/2 + (f
*x)/2)*(8*B^3*a^9*c^11 - 462*A^3*a^9*c^2*d^9 + 926*A^3*a^9*c^3*d^8 - 1034*A^3*a^9*c^4*d^7 + 704*A^3*a^9*c^5*d^
6 - 296*A^3*a^9*c^6*d^5 + 72*A^3*a^9*c^7*d^4 - 8*A^3*a^9*c^8*d^3 - 50*B^3*a^9*c^2*d^9 + 290*B^3*a^9*c^3*d^8 -
788*B^3*a^9*c^4*d^7 + 1332*B^3*a^9*c^5*d^6 - 1546*B^3*a^9*c^6*d^5 + 1274*B^3*a^9*c^7*d^4 - 744*B^3*a^9*c^8*d^3
+ 296*B^3*a^9*c^9*d^2 + 98*A^3*a^9*c*d^10 - 72*B^3*a^9*c^10*d + 50*A*B^2*a^9*c*d^10 - 24*A*B^2*a^9*c^10*d + 1
40*A^2*B*a^9*c*d^10 - 430*A*B^2*a^9*c^2*d^9 + 1524*A*B^2*a^9*c^3*d^8 - 3076*A*B^2*a^9*c^4*d^7 + 4018*A*B^2*a^9
*c^5*d^6 - 3582*A*B^2*a^9*c^6*d^5 + 2192*A*B^2*a^9*c^7*d^4 - 888*A*B^2*a^9*c^8*d^3 + 216*A*B^2*a^9*c^9*d^2 - 8
34*A^2*B*a^9*c^2*d^9 + 2206*A^2*B*a^9*c^3*d^8 - 3398*A^2*B*a^9*c^4*d^7 + 3342*A^2*B*a^9*c^5*d^6 - 2152*A^2*B*a
^9*c^6*d^5 + 888*A^2*B*a^9*c^7*d^4 - 216*A^2*B*a^9*c^8*d^3 + 24*A^2*B*a^9*c^9*d^2))/d^9))*(B*a^3*c^3*1i + (a^3
*d^2*(6*A*c + 7*B*c)*1i)/2 - (a^3*d^3*(7*A + 5*B)*1i)/2 - (a^3*d*(2*A*c^2 + 6*B*c^2)*1i)/2)*2i)/(d^4*f) - (a^3
*atan(((a^3*(A*d - B*c)*(-(c + d)*(c - d)^5)^(1/2)*((8*(49*A^2*a^6*c^2*d^9 - 84*A^2*a^6*c^3*d^8 + 64*A^2*a^6*c
^4*d^7 - 24*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 25*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 + 109*B^2*a^6*c^4*d^
7 - 104*B^2*a^6*c^5*d^6 + 64*B^2*a^6*c^6*d^5 - 24*B^2*a^6*c^7*d^4 + 4*B^2*a^6*c^8*d^3 + 70*A*B*a^6*c^2*d^9 - 1
58*A*B*a^6*c^3*d^8 + 188*A*B*a^6*c^4*d^7 - 128*A*B*a^6*c^5*d^6 + 48*A*B*a^6*c^6*d^5 - 8*A*B*a^6*c^7*d^4))/d^8
+ (8*tan(e/2 + (f*x)/2)*(19*A^2*a^6*c^3*d^9 - 144*A^2*a^6*c^2*d^10 + 116*A^2*a^6*c^4*d^8 - 116*A^2*a^6*c^5*d^7
+ 48*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 - 140*B^2*a^6*c^2*d^10 + 189*B^2*a^6*c^3*d^9 - 114*B^2*a^6*c^4*d^8 -
41*B^2*a^6*c^5*d^7 + 136*B^2*a^6*c^6*d^6 - 116*B^2*a^6*c^7*d^5 + 48*B^2*a^6*c^8*d^4 - 8*B^2*a^6*c^9*d^3 + 94*
A^2*a^6*c*d^11 + 50*B^2*a^6*c*d^11 - 308*A*B*a^6*c^2*d^10 + 258*A*B*a^6*c^3*d^9 + 22*A*B*a^6*c^4*d^8 - 252*A*B
*a^6*c^5*d^7 + 232*A*B*a^6*c^6*d^6 - 96*A*B*a^6*c^7*d^5 + 16*A*B*a^6*c^8*d^4 + 140*A*B*a^6*c*d^11))/d^9 + (a^3
*(A*d - B*c)*(-(c + d)*(c - d)^5)^(1/2)*((8*tan(e/2 + (f*x)/2)*(8*A*a^3*c*d^12 - 24*A*a^3*c^2*d^11 + 24*A*a^3*
c^3*d^10 - 8*A*a^3*c^4*d^9 - 8*B*a^3*c^2*d^11 + 24*B*a^3*c^3*d^10 - 24*B*a^3*c^4*d^9 + 8*B*a^3*c^5*d^8))/d^9 -
(8*(14*A*a^3*c*d^11 + 10*B*a^3*c*d^11 - 16*A*a^3*c^2*d^10 + 2*A*a^3*c^3*d^9 - 14*B*a^3*c^2*d^10 + 6*B*a^3*c^3
*d^9 - 2*B*a^3*c^4*d^8))/d^8 + (a^3*(32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^13 - 8*c^3*d^11))/d^9)*(A*d -
B*c)*(-(c + d)*(c - d)^5)^(1/2))/(c*d^4 + d^5)))/(c*d^4 + d^5))*1i)/(c*d^4 + d^5) + (a^3*(A*d - B*c)*(-(c + d)
*(c - d)^5)^(1/2)*((8*(49*A^2*a^6*c^2*d^9 - 84*A^2*a^6*c^3*d^8 + 64*A^2*a^6*c^4*d^7 - 24*A^2*a^6*c^5*d^6 + 4*A
^2*a^6*c^6*d^5 + 25*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 + 109*B^2*a^6*c^4*d^7 - 104*B^2*a^6*c^5*d^6 + 64*B^2*
a^6*c^6*d^5 - 24*B^2*a^6*c^7*d^4 + 4*B^2*a^6*c^8*d^3 + 70*A*B*a^6*c^2*d^9 - 158*A*B*a^6*c^3*d^8 + 188*A*B*a^6*
c^4*d^7 - 128*A*B*a^6*c^5*d^6 + 48*A*B*a^6*c^6*d^5 - 8*A*B*a^6*c^7*d^4))/d^8 + (8*tan(e/2 + (f*x)/2)*(19*A^2*a
^6*c^3*d^9 - 144*A^2*a^6*c^2*d^10 + 116*A^2*a^6*c^4*d^8 - 116*A^2*a^6*c^5*d^7 + 48*A^2*a^6*c^6*d^6 - 8*A^2*a^6
*c^7*d^5 - 140*B^2*a^6*c^2*d^10 + 189*B^2*a^6*c^3*d^9 - 114*B^2*a^6*c^4*d^8 - 41*B^2*a^6*c^5*d^7 + 136*B^2*a^6
*c^6*d^6 - 116*B^2*a^6*c^7*d^5 + 48*B^2*a^6*c^8*d^4 - 8*B^2*a^6*c^9*d^3 + 94*A^2*a^6*c*d^11 + 50*B^2*a^6*c*d^1
1 - 308*A*B*a^6*c^2*d^10 + 258*A*B*a^6*c^3*d^9 + 22*A*B*a^6*c^4*d^8 - 252*A*B*a^6*c^5*d^7 + 232*A*B*a^6*c^6*d^
6 - 96*A*B*a^6*c^7*d^5 + 16*A*B*a^6*c^8*d^4 + 140*A*B*a^6*c*d^11))/d^9 + (a^3*(A*d - B*c)*(-(c + d)*(c - d)^5)
^(1/2)*((8*(14*A*a^3*c*d^11 + 10*B*a^3*c*d^11 - 16*A*a^3*c^2*d^10 + 2*A*a^3*c^3*d^9 - 14*B*a^3*c^2*d^10 + 6*B*
a^3*c^3*d^9 - 2*B*a^3*c^4*d^8))/d^8 - (8*tan(e/2 + (f*x)/2)*(8*A*a^3*c*d^12 - 24*A*a^3*c^2*d^11 + 24*A*a^3*c^3
*d^10 - 8*A*a^3*c^4*d^9 - 8*B*a^3*c^2*d^11 + 24*B*a^3*c^3*d^10 - 24*B*a^3*c^4*d^9 + 8*B*a^3*c^5*d^8))/d^9 + (a
^3*(32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^13 - 8*c^3*d^11))/d^9)*(A*d - B*c)*(-(c + d)*(c - d)^5)^(1/2))/
(c*d^4 + d^5)))/(c*d^4 + d^5))*1i)/(c*d^4 + d^5))/((16*(2*B^3*a^9*c^10 - 47*A^3*a^9*c^2*d^8 + 55*A^3*a^9*c^3*d
^7 - 21*A^3*a^9*c^4*d^6 - 7*A^3*a^9*c^5*d^5 + 8*A^3*a^9*c^6*d^4 - 2*A^3*a^9*c^7*d^3 - 15*B^3*a^9*c^3*d^7 + 71*
B^3*a^9*c^4*d^6 - 148*B^3*a^9*c^5*d^5 + 180*B^3*a^9*c^6*d^4 - 139*B^3*a^9*c^7*d^3 + 67*B^3*a^9*c^8*d^2 + 14*A^
3*a^9*c*d^9 - 18*B^3*a^9*c^9*d - 6*A*B^2*a^9*c^9*d + 10*A^2*B*a^9*c*d^9 + 5*A*B^2*a^9*c^2*d^8 - 53*A*B^2*a^9*c
^3*d^7 + 174*A*B^2*a^9*c^4*d^6 - 280*A*B^2*a^9*c^5*d^5 + 257*A*B^2*a^9*c^6*d^4 - 141*A*B^2*a^9*c^7*d^3 + 44*A*
B^2*a^9*c^8*d^2 - 32*A^2*B*a^9*c^2*d^8 + 21*A^2*B*a^9*c^3*d^7 + 45*A^2*B*a^9*c^4*d^6 - 97*A^2*B*a^9*c^5*d^5 +
81*A^2*B*a^9*c^6*d^4 - 34*A^2*B*a^9*c^7*d^3 + 6*A^2*B*a^9*c^8*d^2))/d^8 + (16*tan(e/2 + (f*x)/2)*(8*B^3*a^9*c^
11 - 462*A^3*a^9*c^2*d^9 + 926*A^3*a^9*c^3*d^8 - 1034*A^3*a^9*c^4*d^7 + 704*A^3*a^9*c^5*d^6 - 296*A^3*a^9*c^6*
d^5 + 72*A^3*a^9*c^7*d^4 - 8*A^3*a^9*c^8*d^3 - 50*B^3*a^9*c^2*d^9 + 290*B^3*a^9*c^3*d^8 - 788*B^3*a^9*c^4*d^7
+ 1332*B^3*a^9*c^5*d^6 - 1546*B^3*a^9*c^6*d^5 + 1274*B^3*a^9*c^7*d^4 - 744*B^3*a^9*c^8*d^3 + 296*B^3*a^9*c^9*d
^2 + 98*A^3*a^9*c*d^10 - 72*B^3*a^9*c^10*d + 50*A*B^2*a^9*c*d^10 - 24*A*B^2*a^9*c^10*d + 140*A^2*B*a^9*c*d^10
- 430*A*B^2*a^9*c^2*d^9 + 1524*A*B^2*a^9*c^3*d^8 - 3076*A*B^2*a^9*c^4*d^7 + 4018*A*B^2*a^9*c^5*d^6 - 3582*A*B^
2*a^9*c^6*d^5 + 2192*A*B^2*a^9*c^7*d^4 - 888*A*B^2*a^9*c^8*d^3 + 216*A*B^2*a^9*c^9*d^2 - 834*A^2*B*a^9*c^2*d^9
+ 2206*A^2*B*a^9*c^3*d^8 - 3398*A^2*B*a^9*c^4*d^7 + 3342*A^2*B*a^9*c^5*d^6 - 2152*A^2*B*a^9*c^6*d^5 + 888*A^2
*B*a^9*c^7*d^4 - 216*A^2*B*a^9*c^8*d^3 + 24*A^2*B*a^9*c^9*d^2))/d^9 + (a^3*(A*d - B*c)*(-(c + d)*(c - d)^5)^(1
/2)*((8*(49*A^2*a^6*c^2*d^9 - 84*A^2*a^6*c^3*d^8 + 64*A^2*a^6*c^4*d^7 - 24*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5
+ 25*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c^3*d^8 + 109*B^2*a^6*c^4*d^7 - 104*B^2*a^6*c^5*d^6 + 64*B^2*a^6*c^6*d^5 -
24*B^2*a^6*c^7*d^4 + 4*B^2*a^6*c^8*d^3 + 70*A*B*a^6*c^2*d^9 - 158*A*B*a^6*c^3*d^8 + 188*A*B*a^6*c^4*d^7 - 128*
A*B*a^6*c^5*d^6 + 48*A*B*a^6*c^6*d^5 - 8*A*B*a^6*c^7*d^4))/d^8 + (8*tan(e/2 + (f*x)/2)*(19*A^2*a^6*c^3*d^9 - 1
44*A^2*a^6*c^2*d^10 + 116*A^2*a^6*c^4*d^8 - 116*A^2*a^6*c^5*d^7 + 48*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 - 140
*B^2*a^6*c^2*d^10 + 189*B^2*a^6*c^3*d^9 - 114*B^2*a^6*c^4*d^8 - 41*B^2*a^6*c^5*d^7 + 136*B^2*a^6*c^6*d^6 - 116
*B^2*a^6*c^7*d^5 + 48*B^2*a^6*c^8*d^4 - 8*B^2*a^6*c^9*d^3 + 94*A^2*a^6*c*d^11 + 50*B^2*a^6*c*d^11 - 308*A*B*a^
6*c^2*d^10 + 258*A*B*a^6*c^3*d^9 + 22*A*B*a^6*c^4*d^8 - 252*A*B*a^6*c^5*d^7 + 232*A*B*a^6*c^6*d^6 - 96*A*B*a^6
*c^7*d^5 + 16*A*B*a^6*c^8*d^4 + 140*A*B*a^6*c*d^11))/d^9 + (a^3*(A*d - B*c)*(-(c + d)*(c - d)^5)^(1/2)*((8*tan
(e/2 + (f*x)/2)*(8*A*a^3*c*d^12 - 24*A*a^3*c^2*d^11 + 24*A*a^3*c^3*d^10 - 8*A*a^3*c^4*d^9 - 8*B*a^3*c^2*d^11 +
24*B*a^3*c^3*d^10 - 24*B*a^3*c^4*d^9 + 8*B*a^3*c^5*d^8))/d^9 - (8*(14*A*a^3*c*d^11 + 10*B*a^3*c*d^11 - 16*A*a
^3*c^2*d^10 + 2*A*a^3*c^3*d^9 - 14*B*a^3*c^2*d^10 + 6*B*a^3*c^3*d^9 - 2*B*a^3*c^4*d^8))/d^8 + (a^3*(32*c^2*d^3
+ (8*tan(e/2 + (f*x)/2)*(12*c*d^13 - 8*c^3*d^11))/d^9)*(A*d - B*c)*(-(c + d)*(c - d)^5)^(1/2))/(c*d^4 + d^5))
)/(c*d^4 + d^5)))/(c*d^4 + d^5) - (a^3*(A*d - B*c)*(-(c + d)*(c - d)^5)^(1/2)*((8*(49*A^2*a^6*c^2*d^9 - 84*A^2
*a^6*c^3*d^8 + 64*A^2*a^6*c^4*d^7 - 24*A^2*a^6*c^5*d^6 + 4*A^2*a^6*c^6*d^5 + 25*B^2*a^6*c^2*d^9 - 70*B^2*a^6*c
^3*d^8 + 109*B^2*a^6*c^4*d^7 - 104*B^2*a^6*c^5*d^6 + 64*B^2*a^6*c^6*d^5 - 24*B^2*a^6*c^7*d^4 + 4*B^2*a^6*c^8*d
^3 + 70*A*B*a^6*c^2*d^9 - 158*A*B*a^6*c^3*d^8 + 188*A*B*a^6*c^4*d^7 - 128*A*B*a^6*c^5*d^6 + 48*A*B*a^6*c^6*d^5
- 8*A*B*a^6*c^7*d^4))/d^8 + (8*tan(e/2 + (f*x)/2)*(19*A^2*a^6*c^3*d^9 - 144*A^2*a^6*c^2*d^10 + 116*A^2*a^6*c^
4*d^8 - 116*A^2*a^6*c^5*d^7 + 48*A^2*a^6*c^6*d^6 - 8*A^2*a^6*c^7*d^5 - 140*B^2*a^6*c^2*d^10 + 189*B^2*a^6*c^3*
d^9 - 114*B^2*a^6*c^4*d^8 - 41*B^2*a^6*c^5*d^7 + 136*B^2*a^6*c^6*d^6 - 116*B^2*a^6*c^7*d^5 + 48*B^2*a^6*c^8*d^
4 - 8*B^2*a^6*c^9*d^3 + 94*A^2*a^6*c*d^11 + 50*B^2*a^6*c*d^11 - 308*A*B*a^6*c^2*d^10 + 258*A*B*a^6*c^3*d^9 + 2
2*A*B*a^6*c^4*d^8 - 252*A*B*a^6*c^5*d^7 + 232*A*B*a^6*c^6*d^6 - 96*A*B*a^6*c^7*d^5 + 16*A*B*a^6*c^8*d^4 + 140*
A*B*a^6*c*d^11))/d^9 + (a^3*(A*d - B*c)*(-(c + d)*(c - d)^5)^(1/2)*((8*(14*A*a^3*c*d^11 + 10*B*a^3*c*d^11 - 16
*A*a^3*c^2*d^10 + 2*A*a^3*c^3*d^9 - 14*B*a^3*c^2*d^10 + 6*B*a^3*c^3*d^9 - 2*B*a^3*c^4*d^8))/d^8 - (8*tan(e/2 +
(f*x)/2)*(8*A*a^3*c*d^12 - 24*A*a^3*c^2*d^11 + 24*A*a^3*c^3*d^10 - 8*A*a^3*c^4*d^9 - 8*B*a^3*c^2*d^11 + 24*B*
a^3*c^3*d^10 - 24*B*a^3*c^4*d^9 + 8*B*a^3*c^5*d^8))/d^9 + (a^3*(32*c^2*d^3 + (8*tan(e/2 + (f*x)/2)*(12*c*d^13
- 8*c^3*d^11))/d^9)*(A*d - B*c)*(-(c + d)*(c - d)^5)^(1/2))/(c*d^4 + d^5)))/(c*d^4 + d^5)))/(c*d^4 + d^5)))*(A
*d - B*c)*(-(c + d)*(c - d)^5)^(1/2)*2i)/(f*(c*d^4 + d^5))

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

[Out]

Timed out

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