3.3.57 \(\int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx\) [257]

Optimal. Leaf size=215 \[ \frac {a^2 B x}{d^3}+\frac {a^2 \left (3 A d^3-B \left (2 c^3+4 c^2 d+c d^2-4 d^3\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 (c+d)^2 \sqrt {c^2-d^2} f}+\frac {(B c-A d) \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {a^2 \left (3 A d^2-B \left (2 c^2+3 c d-2 d^2\right )\right ) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))} \]

[Out]

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

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Rubi [A]
time = 0.43, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3054, 3047, 3100, 2814, 2739, 632, 210} \begin {gather*} \frac {a^2 \left (3 A d^3-B \left (2 c^3+4 c^2 d+c d^2-4 d^3\right )\right ) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f (c+d)^2 \sqrt {c^2-d^2}}-\frac {a^2 \left (3 A d^2-B \left (2 c^2+3 c d-2 d^2\right )\right ) \cos (e+f x)}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}+\frac {(B c-A d) \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{2 d f (c+d) (c+d \sin (e+f x))^2}+\frac {a^2 B x}{d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

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 2739

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 2814

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 3047

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 3054

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^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d
*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x
])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n
+ 1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, 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 3100

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m
+ 1)*(a^2 - b^2))), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B +
a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b,
e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.94, size = 226, normalized size = 1.05 \begin {gather*} \frac {a^2 (1+\sin (e+f x))^2 \left (2 B (e+f x)-\frac {2 \left (-3 A d^3+B \left (2 c^3+4 c^2 d+c d^2-4 d^3\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d)^2 \sqrt {c^2-d^2}}-\frac {d (-c+d) (-B c+A d) \cos (e+f x)}{(c+d) (c+d \sin (e+f x))^2}-\frac {d \left (A d (c+4 d)+B \left (-3 c^2-4 c d+2 d^2\right )\right ) \cos (e+f x)}{(c+d)^2 (c+d \sin (e+f x))}\right )}{2 d^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(1 + Sin[e + f*x])^2*(2*B*(e + f*x) - (2*(-3*A*d^3 + B*(2*c^3 + 4*c^2*d + c*d^2 - 4*d^3))*ArcTan[(d + c*T
an[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)^2*Sqrt[c^2 - d^2]) - (d*(-c + d)*(-(B*c) + A*d)*Cos[e + f*x])/((c
+ d)*(c + d*Sin[e + f*x])^2) - (d*(A*d*(c + 4*d) + B*(-3*c^2 - 4*c*d + 2*d^2))*Cos[e + f*x])/((c + d)^2*(c + d
*Sin[e + f*x]))))/(2*d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(430\) vs. \(2(206)=412\).
time = 0.70, size = 431, normalized size = 2.00 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/f*a^2*(1/d^3*((1/2*d^2*(A*c^2*d-4*A*c*d^2-2*A*d^3+B*c^3+4*B*c^2*d)/(c^2+2*c*d+d^2)/c*tan(1/2*f*x+1/2*e)^3-1/
2*d*(4*A*c^3*d^2+A*c^2*d^3+8*A*c*d^4+2*A*d^5-2*B*c^5-4*B*c^4*d-3*B*c^3*d^2-8*B*c^2*d^3+2*B*c*d^4)/(c^2+2*c*d+d
^2)/c^2*tan(1/2*f*x+1/2*e)^2-1/2*d^2*(A*c^2*d+12*A*c*d^2+2*A*d^3-7*B*c^3-12*B*c^2*d+4*B*c*d^2)/c/(c^2+2*c*d+d^
2)*tan(1/2*f*x+1/2*e)-1/2*d*(4*A*c*d^2+A*d^3-2*B*c^3-4*B*c^2*d+B*c*d^2)/(c^2+2*c*d+d^2))/(c*tan(1/2*f*x+1/2*e)
^2+2*d*tan(1/2*f*x+1/2*e)+c)^2+1/2*(3*A*d^3-2*B*c^3-4*B*c^2*d-B*c*d^2+4*B*d^3)/(c^2+2*c*d+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/d^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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))^3,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 de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (212) = 424\).
time = 0.46, size = 1508, normalized size = 7.01 \begin {gather*} \left [\frac {4 \, {\left (B a^{2} c^{4} d^{2} + 2 \, B a^{2} c^{3} d^{3} - 2 \, B a^{2} c d^{5} - B a^{2} d^{6}\right )} f x \cos \left (f x + e\right )^{2} - 4 \, {\left (B a^{2} c^{6} + 2 \, B a^{2} c^{5} d + B a^{2} c^{4} d^{2} - B a^{2} c^{2} d^{4} - 2 \, B a^{2} c d^{5} - B a^{2} d^{6}\right )} f x - {\left (2 \, B a^{2} c^{5} + 4 \, B a^{2} c^{4} d + 3 \, B a^{2} c^{3} d^{2} - 3 \, A a^{2} c^{2} d^{3} + B a^{2} c d^{4} - {\left (3 \, A + 4 \, B\right )} a^{2} d^{5} - {\left (2 \, B a^{2} c^{3} d^{2} + 4 \, B a^{2} c^{2} d^{3} + B a^{2} c d^{4} - {\left (3 \, A + 4 \, B\right )} a^{2} d^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, B a^{2} c^{4} d + 4 \, B a^{2} c^{3} d^{2} + B a^{2} c^{2} d^{3} - {\left (3 \, A + 4 \, B\right )} a^{2} 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 a^{2} c^{5} d + 4 \, B a^{2} c^{4} d^{2} - {\left (4 \, A + 3 \, B\right )} a^{2} c^{3} d^{3} - {\left (A + 4 \, B\right )} a^{2} c^{2} d^{4} + {\left (4 \, A + B\right )} a^{2} c d^{5} + A a^{2} d^{6}\right )} \cos \left (f x + e\right ) - 2 \, {\left (4 \, {\left (B a^{2} c^{5} d + 2 \, B a^{2} c^{4} d^{2} - 2 \, B a^{2} c^{2} d^{4} - B a^{2} c d^{5}\right )} f x + {\left (3 \, B a^{2} c^{4} d^{2} - {\left (A - 4 \, B\right )} a^{2} c^{3} d^{3} - {\left (4 \, A + 5 \, B\right )} a^{2} c^{2} d^{4} + {\left (A - 4 \, B\right )} a^{2} c d^{5} + 2 \, {\left (2 \, A + B\right )} a^{2} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (c^{4} d^{5} + 2 \, c^{3} d^{6} - 2 \, c d^{8} - d^{9}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{5} d^{4} + 2 \, c^{4} d^{5} - 2 \, c^{2} d^{7} - c d^{8}\right )} f \sin \left (f x + e\right ) - {\left (c^{6} d^{3} + 2 \, c^{5} d^{4} + c^{4} d^{5} - c^{2} d^{7} - 2 \, c d^{8} - d^{9}\right )} f\right )}}, \frac {2 \, {\left (B a^{2} c^{4} d^{2} + 2 \, B a^{2} c^{3} d^{3} - 2 \, B a^{2} c d^{5} - B a^{2} d^{6}\right )} f x \cos \left (f x + e\right )^{2} - 2 \, {\left (B a^{2} c^{6} + 2 \, B a^{2} c^{5} d + B a^{2} c^{4} d^{2} - B a^{2} c^{2} d^{4} - 2 \, B a^{2} c d^{5} - B a^{2} d^{6}\right )} f x - {\left (2 \, B a^{2} c^{5} + 4 \, B a^{2} c^{4} d + 3 \, B a^{2} c^{3} d^{2} - 3 \, A a^{2} c^{2} d^{3} + B a^{2} c d^{4} - {\left (3 \, A + 4 \, B\right )} a^{2} d^{5} - {\left (2 \, B a^{2} c^{3} d^{2} + 4 \, B a^{2} c^{2} d^{3} + B a^{2} c d^{4} - {\left (3 \, A + 4 \, B\right )} a^{2} d^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, B a^{2} c^{4} d + 4 \, B a^{2} c^{3} d^{2} + B a^{2} c^{2} d^{3} - {\left (3 \, A + 4 \, B\right )} a^{2} 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 a^{2} c^{5} d + 4 \, B a^{2} c^{4} d^{2} - {\left (4 \, A + 3 \, B\right )} a^{2} c^{3} d^{3} - {\left (A + 4 \, B\right )} a^{2} c^{2} d^{4} + {\left (4 \, A + B\right )} a^{2} c d^{5} + A a^{2} d^{6}\right )} \cos \left (f x + e\right ) - {\left (4 \, {\left (B a^{2} c^{5} d + 2 \, B a^{2} c^{4} d^{2} - 2 \, B a^{2} c^{2} d^{4} - B a^{2} c d^{5}\right )} f x + {\left (3 \, B a^{2} c^{4} d^{2} - {\left (A - 4 \, B\right )} a^{2} c^{3} d^{3} - {\left (4 \, A + 5 \, B\right )} a^{2} c^{2} d^{4} + {\left (A - 4 \, B\right )} a^{2} c d^{5} + 2 \, {\left (2 \, A + B\right )} a^{2} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{4} d^{5} + 2 \, c^{3} d^{6} - 2 \, c d^{8} - d^{9}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{5} d^{4} + 2 \, c^{4} d^{5} - 2 \, c^{2} d^{7} - c d^{8}\right )} f \sin \left (f x + e\right ) - {\left (c^{6} d^{3} + 2 \, c^{5} d^{4} + c^{4} d^{5} - c^{2} d^{7} - 2 \, c d^{8} - d^{9}\right )} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[1/4*(4*(B*a^2*c^4*d^2 + 2*B*a^2*c^3*d^3 - 2*B*a^2*c*d^5 - B*a^2*d^6)*f*x*cos(f*x + e)^2 - 4*(B*a^2*c^6 + 2*B*
a^2*c^5*d + B*a^2*c^4*d^2 - B*a^2*c^2*d^4 - 2*B*a^2*c*d^5 - B*a^2*d^6)*f*x - (2*B*a^2*c^5 + 4*B*a^2*c^4*d + 3*
B*a^2*c^3*d^2 - 3*A*a^2*c^2*d^3 + B*a^2*c*d^4 - (3*A + 4*B)*a^2*d^5 - (2*B*a^2*c^3*d^2 + 4*B*a^2*c^2*d^3 + B*a
^2*c*d^4 - (3*A + 4*B)*a^2*d^5)*cos(f*x + e)^2 + 2*(2*B*a^2*c^4*d + 4*B*a^2*c^3*d^2 + B*a^2*c^2*d^3 - (3*A + 4
*B)*a^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*a^2*c^5*d + 4*B*a^2*c^4*d^2 - (4*A + 3*B)*a^2*c^3*d^3 - (A + 4*B)*a^2*c^2*d^4 + (4*A
+ B)*a^2*c*d^5 + A*a^2*d^6)*cos(f*x + e) - 2*(4*(B*a^2*c^5*d + 2*B*a^2*c^4*d^2 - 2*B*a^2*c^2*d^4 - B*a^2*c*d^
5)*f*x + (3*B*a^2*c^4*d^2 - (A - 4*B)*a^2*c^3*d^3 - (4*A + 5*B)*a^2*c^2*d^4 + (A - 4*B)*a^2*c*d^5 + 2*(2*A + B
)*a^2*d^6)*cos(f*x + e))*sin(f*x + e))/((c^4*d^5 + 2*c^3*d^6 - 2*c*d^8 - d^9)*f*cos(f*x + e)^2 - 2*(c^5*d^4 +
2*c^4*d^5 - 2*c^2*d^7 - c*d^8)*f*sin(f*x + e) - (c^6*d^3 + 2*c^5*d^4 + c^4*d^5 - c^2*d^7 - 2*c*d^8 - d^9)*f),
1/2*(2*(B*a^2*c^4*d^2 + 2*B*a^2*c^3*d^3 - 2*B*a^2*c*d^5 - B*a^2*d^6)*f*x*cos(f*x + e)^2 - 2*(B*a^2*c^6 + 2*B*a
^2*c^5*d + B*a^2*c^4*d^2 - B*a^2*c^2*d^4 - 2*B*a^2*c*d^5 - B*a^2*d^6)*f*x - (2*B*a^2*c^5 + 4*B*a^2*c^4*d + 3*B
*a^2*c^3*d^2 - 3*A*a^2*c^2*d^3 + B*a^2*c*d^4 - (3*A + 4*B)*a^2*d^5 - (2*B*a^2*c^3*d^2 + 4*B*a^2*c^2*d^3 + B*a^
2*c*d^4 - (3*A + 4*B)*a^2*d^5)*cos(f*x + e)^2 + 2*(2*B*a^2*c^4*d + 4*B*a^2*c^3*d^2 + B*a^2*c^2*d^3 - (3*A + 4*
B)*a^2*c*d^4)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - (2*
B*a^2*c^5*d + 4*B*a^2*c^4*d^2 - (4*A + 3*B)*a^2*c^3*d^3 - (A + 4*B)*a^2*c^2*d^4 + (4*A + B)*a^2*c*d^5 + A*a^2*
d^6)*cos(f*x + e) - (4*(B*a^2*c^5*d + 2*B*a^2*c^4*d^2 - 2*B*a^2*c^2*d^4 - B*a^2*c*d^5)*f*x + (3*B*a^2*c^4*d^2
- (A - 4*B)*a^2*c^3*d^3 - (4*A + 5*B)*a^2*c^2*d^4 + (A - 4*B)*a^2*c*d^5 + 2*(2*A + B)*a^2*d^6)*cos(f*x + e))*s
in(f*x + e))/((c^4*d^5 + 2*c^3*d^6 - 2*c*d^8 - d^9)*f*cos(f*x + e)^2 - 2*(c^5*d^4 + 2*c^4*d^5 - 2*c^2*d^7 - c*
d^8)*f*sin(f*x + e) - (c^6*d^3 + 2*c^5*d^4 + c^4*d^5 - c^2*d^7 - 2*c*d^8 - d^9)*f)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (212) = 424\).
time = 0.73, size = 703, normalized size = 3.27 \begin {gather*} \frac {\frac {{\left (f x + e\right )} B a^{2}}{d^{3}} - \frac {{\left (2 \, B a^{2} c^{3} + 4 \, B a^{2} c^{2} d + B a^{2} c d^{2} - 3 \, A a^{2} d^{3} - 4 \, B a^{2} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \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} + 2 \, c d^{4} + d^{5}\right )} \sqrt {c^{2} - d^{2}}} + \frac {B a^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + A a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, B a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, A a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, A a^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, B a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, B a^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 4 \, A a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, B a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - A a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 8 \, B a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 8 \, A a^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, B a^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, A a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, B a^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, B a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, A a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, B a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, A a^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, B a^{2} c^{5} + 4 \, B a^{2} c^{4} d - 4 \, A a^{2} c^{3} d^{2} - B a^{2} c^{3} d^{2} - A a^{2} c^{2} d^{3}}{{\left (c^{4} d^{2} + 2 \, c^{3} d^{3} + c^{2} d^{4}\right )} {\left (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 )}^{2}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

((f*x + e)*B*a^2/d^3 - (2*B*a^2*c^3 + 4*B*a^2*c^2*d + B*a^2*c*d^2 - 3*A*a^2*d^3 - 4*B*a^2*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)))/((c^2*d^3 + 2*c*d^4 + d^5)*s
qrt(c^2 - d^2)) + (B*a^2*c^4*d*tan(1/2*f*x + 1/2*e)^3 + A*a^2*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 + 4*B*a^2*c^3*d^2
*tan(1/2*f*x + 1/2*e)^3 - 4*A*a^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 - 2*A*a^2*c*d^4*tan(1/2*f*x + 1/2*e)^3 + 2*B*
a^2*c^5*tan(1/2*f*x + 1/2*e)^2 + 4*B*a^2*c^4*d*tan(1/2*f*x + 1/2*e)^2 - 4*A*a^2*c^3*d^2*tan(1/2*f*x + 1/2*e)^2
+ 3*B*a^2*c^3*d^2*tan(1/2*f*x + 1/2*e)^2 - A*a^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^2 + 8*B*a^2*c^2*d^3*tan(1/2*f*x
+ 1/2*e)^2 - 8*A*a^2*c*d^4*tan(1/2*f*x + 1/2*e)^2 - 2*B*a^2*c*d^4*tan(1/2*f*x + 1/2*e)^2 - 2*A*a^2*d^5*tan(1/
2*f*x + 1/2*e)^2 + 7*B*a^2*c^4*d*tan(1/2*f*x + 1/2*e) - A*a^2*c^3*d^2*tan(1/2*f*x + 1/2*e) + 12*B*a^2*c^3*d^2*
tan(1/2*f*x + 1/2*e) - 12*A*a^2*c^2*d^3*tan(1/2*f*x + 1/2*e) - 4*B*a^2*c^2*d^3*tan(1/2*f*x + 1/2*e) - 2*A*a^2*
c*d^4*tan(1/2*f*x + 1/2*e) + 2*B*a^2*c^5 + 4*B*a^2*c^4*d - 4*A*a^2*c^3*d^2 - B*a^2*c^3*d^2 - A*a^2*c^2*d^3)/((
c^4*d^2 + 2*c^3*d^3 + c^2*d^4)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2))/f

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Mupad [B]
time = 22.50, size = 2500, normalized size = 11.63 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*B*a^2*atan(((B*a^2*((8*(4*B^2*a^4*c^2*d^6 + 16*B^2*a^4*c^3*d^5 + 24*B^2*a^4*c^4*d^4 + 16*B^2*a^4*c^5*d^3 +
4*B^2*a^4*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(40*B^2*a^4*c^2*
d^7 + 75*B^2*a^4*c^3*d^6 + 24*B^2*a^4*c^4*d^5 - 36*B^2*a^4*c^5*d^4 - 32*B^2*a^4*c^6*d^3 - 8*B^2*a^4*c^7*d^2 -
9*A^2*a^4*c*d^8 - 8*B^2*a^4*c*d^8 + 6*A*B*a^4*c^2*d^7 + 24*A*B*a^4*c^3*d^6 + 12*A*B*a^4*c^4*d^5 - 24*A*B*a^4*c
*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (B*a^2*((8*tan(e/2 + (f*x)/2)*(12*A*a^2*c*d^11 + 1
6*B*a^2*c*d^11 + 24*A*a^2*c^2*d^10 + 12*A*a^2*c^3*d^9 + 28*B*a^2*c^2*d^10 - 8*B*a^2*c^3*d^9 - 44*B*a^2*c^4*d^8
- 32*B*a^2*c^5*d^7 - 8*B*a^2*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) - (8*(4*B*a^2*c*d^1
0 - 6*A*a^2*c^2*d^9 - 12*A*a^2*c^3*d^8 - 6*A*a^2*c^4*d^7 + 8*B*a^2*c^2*d^9 + 6*B*a^2*c^3*d^8 + 4*B*a^2*c^4*d^7
+ 2*B*a^2*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (B*a^2*((8*(4*c^2*d^12 + 16*c^3*d^11
+ 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x
)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 +
d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3)*1i)/d^3))/d^3 + (B*a^2*((8*(4*B^2*a^4*c^2*d^6 + 16*B^2*a^4*c
^3*d^5 + 24*B^2*a^4*c^4*d^4 + 16*B^2*a^4*c^5*d^3 + 4*B^2*a^4*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6
+ c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(40*B^2*a^4*c^2*d^7 + 75*B^2*a^4*c^3*d^6 + 24*B^2*a^4*c^4*d^5 - 36*B^2*a^4*
c^5*d^4 - 32*B^2*a^4*c^6*d^3 - 8*B^2*a^4*c^7*d^2 - 9*A^2*a^4*c*d^8 - 8*B^2*a^4*c*d^8 + 6*A*B*a^4*c^2*d^7 + 24*
A*B*a^4*c^3*d^6 + 12*A*B*a^4*c^4*d^5 - 24*A*B*a^4*c*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) +
(B*a^2*((8*(4*B*a^2*c*d^10 - 6*A*a^2*c^2*d^9 - 12*A*a^2*c^3*d^8 - 6*A*a^2*c^4*d^7 + 8*B*a^2*c^2*d^9 + 6*B*a^2
*c^3*d^8 + 4*B*a^2*c^4*d^7 + 2*B*a^2*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) - (8*tan(e/2
+ (f*x)/2)*(12*A*a^2*c*d^11 + 16*B*a^2*c*d^11 + 24*A*a^2*c^2*d^10 + 12*A*a^2*c^3*d^9 + 28*B*a^2*c^2*d^10 - 8*B
*a^2*c^3*d^9 - 44*B*a^2*c^4*d^8 - 32*B*a^2*c^5*d^7 - 8*B*a^2*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7
+ c^4*d^6) + (B*a^2*((8*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6
*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 -
36*c^5*d^10 - 32*c^6*d^9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3)*1i)/d^3))
/d^3)/((16*(2*B^3*a^6*c^5 + 17*B^3*a^6*c^3*d^2 - 16*B^3*a^6*c*d^4 + 12*B^3*a^6*c^4*d - 24*A*B^2*a^6*c*d^4 + 6*
A*B^2*a^6*c^4*d - 9*A^2*B*a^6*c*d^4 + 12*A*B^2*a^6*c^3*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5)
- (16*tan(e/2 + (f*x)/2)*(28*B^3*a^6*c^2*d^4 - 8*B^3*a^6*c^6 - 8*B^3*a^6*c^3*d^3 - 44*B^3*a^6*c^4*d^2 + 16*B^
3*a^6*c*d^5 - 32*B^3*a^6*c^5*d + 12*A*B^2*a^6*c*d^5 + 24*A*B^2*a^6*c^2*d^4 + 12*A*B^2*a^6*c^3*d^3))/(4*c*d^9 +
d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) - (B*a^2*((8*(4*B^2*a^4*c^2*d^6 + 16*B^2*a^4*c^3*d^5 + 24*B^2*a^4*c^4
*d^4 + 16*B^2*a^4*c^5*d^3 + 4*B^2*a^4*c^6*d^2))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2
+ (f*x)/2)*(40*B^2*a^4*c^2*d^7 + 75*B^2*a^4*c^3*d^6 + 24*B^2*a^4*c^4*d^5 - 36*B^2*a^4*c^5*d^4 - 32*B^2*a^4*c^
6*d^3 - 8*B^2*a^4*c^7*d^2 - 9*A^2*a^4*c*d^8 - 8*B^2*a^4*c*d^8 + 6*A*B*a^4*c^2*d^7 + 24*A*B*a^4*c^3*d^6 + 12*A*
B*a^4*c^4*d^5 - 24*A*B*a^4*c*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (B*a^2*((8*tan(e/2 + (
f*x)/2)*(12*A*a^2*c*d^11 + 16*B*a^2*c*d^11 + 24*A*a^2*c^2*d^10 + 12*A*a^2*c^3*d^9 + 28*B*a^2*c^2*d^10 - 8*B*a^
2*c^3*d^9 - 44*B*a^2*c^4*d^8 - 32*B*a^2*c^5*d^7 - 8*B*a^2*c^6*d^6))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 +
c^4*d^6) - (8*(4*B*a^2*c*d^10 - 6*A*a^2*c^2*d^9 - 12*A*a^2*c^3*d^8 - 6*A*a^2*c^4*d^7 + 8*B*a^2*c^2*d^9 + 6*B*a
^2*c^3*d^8 + 4*B*a^2*c^4*d^7 + 2*B*a^2*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (B*a^2*((
8*(4*c^2*d^12 + 16*c^3*d^11 + 24*c^4*d^10 + 16*c^5*d^9 + 4*c^6*d^8))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c^3*d^6 +
c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(12*c*d^14 + 48*c^2*d^13 + 64*c^3*d^12 + 16*c^4*d^11 - 36*c^5*d^10 - 32*c^6*d
^9 - 8*c^7*d^8))/(4*c*d^9 + d^10 + 6*c^2*d^8 + 4*c^3*d^7 + c^4*d^6))*1i)/d^3)*1i)/d^3)*1i)/d^3 + (B*a^2*((8*(4
*B^2*a^4*c^2*d^6 + 16*B^2*a^4*c^3*d^5 + 24*B^2*a^4*c^4*d^4 + 16*B^2*a^4*c^5*d^3 + 4*B^2*a^4*c^6*d^2))/(4*c*d^8
+ d^9 + 6*c^2*d^7 + 4*c^3*d^6 + c^4*d^5) + (8*tan(e/2 + (f*x)/2)*(40*B^2*a^4*c^2*d^7 + 75*B^2*a^4*c^3*d^6 + 2
4*B^2*a^4*c^4*d^5 - 36*B^2*a^4*c^5*d^4 - 32*B^2*a^4*c^6*d^3 - 8*B^2*a^4*c^7*d^2 - 9*A^2*a^4*c*d^8 - 8*B^2*a^4*
c*d^8 + 6*A*B*a^4*c^2*d^7 + 24*A*B*a^4*c^3*d^6 + 12*A*B*a^4*c^4*d^5 - 24*A*B*a^4*c*d^8))/(4*c*d^9 + d^10 + 6*c
^2*d^8 + 4*c^3*d^7 + c^4*d^6) + (B*a^2*((8*(4*B*a^2*c*d^10 - 6*A*a^2*c^2*d^9 - 12*A*a^2*c^3*d^8 - 6*A*a^2*c^4*
d^7 + 8*B*a^2*c^2*d^9 + 6*B*a^2*c^3*d^8 + 4*B*a^2*c^4*d^7 + 2*B*a^2*c^5*d^6))/(4*c*d^8 + d^9 + 6*c^2*d^7 + 4*c
^3*d^6 + c^4*d^5) - (8*tan(e/2 + (f*x)/2)*(12*A...

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