3.5.49 \(\int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^2} \, dx\) [449]

Optimal. Leaf size=161 \[ -\frac {a^3 (2 c-3 d) x}{d^3}+\frac {2 a^3 (c-d)^2 (2 c+3 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 (c+d) \sqrt {c^2-d^2} f}-\frac {2 a^3 c \cos (e+f x)}{d^2 (c+d) f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{d (c+d) f (c+d \sin (e+f x))} \]

[Out]

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

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Rubi [A]
time = 0.27, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2841, 3047, 3102, 2814, 2739, 632, 210} \begin {gather*} \frac {2 a^3 (c-d)^2 (2 c+3 d) \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) \sqrt {c^2-d^2}}-\frac {a^3 x (2 c-3 d)}{d^3}-\frac {2 a^3 c \cos (e+f x)}{d^2 f (c+d)}+\frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) (c+d \sin (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^3*(2*c - 3*d)*x)/d^3) + (2*a^3*(c - d)^2*(2*c + 3*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d
^3*(c + d)*Sqrt[c^2 - d^2]*f) - (2*a^3*c*Cos[e + f*x])/(d^2*(c + d)*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[
e + f*x]))/(d*(c + d)*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 2841

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c
+ a*d))), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1
)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{
a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1
] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 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 3102

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

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Mathematica [A]
time = 0.47, size = 162, normalized size = 1.01 \begin {gather*} \frac {a^3 (1+\sin (e+f x))^3 \left ((-2 c+3 d) (e+f x)+\frac {2 (c-d)^2 (2 c+3 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d) \sqrt {c^2-d^2}}-d \cos (e+f x)-\frac {(c-d)^2 d \cos (e+f x)}{(c+d) (c+d \sin (e+f x))}\right )}{d^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + Sin[e + f*x])^3*((-2*c + 3*d)*(e + f*x) + (2*(c - d)^2*(2*c + 3*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/S
qrt[c^2 - d^2]])/((c + d)*Sqrt[c^2 - d^2]) - d*Cos[e + f*x] - ((c - d)^2*d*Cos[e + f*x])/((c + d)*(c + d*Sin[e
+ f*x]))))/(d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)

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Maple [A]
time = 0.54, size = 209, normalized size = 1.30

method	result	size
derivativedivides	\(\frac {2 a^{3} \left (-\frac {\frac {d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (2 c -3 d \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}+\frac {\frac {-\frac {d^{2} \left (c^{2}-2 c d +d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) c}-\frac {d \left (c^{2}-2 c d +d^{2}\right )}{c +d}}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (2 c^{3}-c^{2} d -4 c \,d^{2}+3 d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c +d \right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}\right )}{f}\)	\(209\)
default	\(\frac {2 a^{3} \left (-\frac {\frac {d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (2 c -3 d \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}+\frac {\frac {-\frac {d^{2} \left (c^{2}-2 c d +d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) c}-\frac {d \left (c^{2}-2 c d +d^{2}\right )}{c +d}}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (2 c^{3}-c^{2} d -4 c \,d^{2}+3 d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c +d \right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}\right )}{f}\)	\(209\)
risch	\(-\frac {2 a^{3} x c}{d^{3}}+\frac {3 a^{3} x}{d^{2}}-\frac {a^{3} {\mathrm e}^{i \left (f x +e \right )}}{2 d^{2} f}-\frac {a^{3} {\mathrm e}^{-i \left (f x +e \right )}}{2 d^{2} f}+\frac {2 i a^{3} \left (c^{2}-2 c d +d^{2}\right ) \left (i d +c \,{\mathrm e}^{i \left (f x +e \right )}\right )}{d^{3} \left (c +d \right ) f \left (-i d \,{\mathrm e}^{2 i \left (f x +e \right )}+i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )}+\frac {2 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c^{2}}{\left (c +d \right )^{2} f \,d^{3}}+\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c}{\left (c +d \right )^{2} f \,d^{2}}-\frac {3 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right )}{\left (c +d \right )^{2} f d}-\frac {2 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c -\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c^{2}}{\left (c +d \right )^{2} f \,d^{3}}-\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c -\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c}{\left (c +d \right )^{2} f \,d^{2}}+\frac {3 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c -\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right )}{\left (c +d \right )^{2} f d}\)	\(510\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))^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 de

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Fricas [A]
time = 0.40, size = 664, normalized size = 4.12 \begin {gather*} \left [-\frac {2 \, {\left (2 \, a^{3} c^{3} - a^{3} c^{2} d - 3 \, a^{3} c d^{2}\right )} f x + {\left (2 \, a^{3} c^{3} + a^{3} c^{2} d - 3 \, a^{3} c d^{2} + {\left (2 \, a^{3} c^{2} d + a^{3} c d^{2} - 3 \, a^{3} d^{3}\right )} \sin \left (f x + e\right )\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 ) + 2 \, {\left (2 \, a^{3} c^{2} d - a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right ) + 2 \, {\left ({\left (2 \, a^{3} c^{2} d - a^{3} c d^{2} - 3 \, a^{3} d^{3}\right )} f x + {\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c d^{4} + d^{5}\right )} f \sin \left (f x + e\right ) + {\left (c^{2} d^{3} + c d^{4}\right )} f\right )}}, -\frac {{\left (2 \, a^{3} c^{3} - a^{3} c^{2} d - 3 \, a^{3} c d^{2}\right )} f x + {\left (2 \, a^{3} c^{3} + a^{3} c^{2} d - 3 \, a^{3} c d^{2} + {\left (2 \, a^{3} c^{2} d + a^{3} c d^{2} - 3 \, a^{3} d^{3}\right )} \sin \left (f x + e\right )\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 ) + {\left (2 \, a^{3} c^{2} d - a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right ) + {\left ({\left (2 \, a^{3} c^{2} d - a^{3} c d^{2} - 3 \, a^{3} d^{3}\right )} f x + {\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{{\left (c d^{4} + d^{5}\right )} f \sin \left (f x + e\right ) + {\left (c^{2} d^{3} + c d^{4}\right )} f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*(2*a^3*c^3 - a^3*c^2*d - 3*a^3*c*d^2)*f*x + (2*a^3*c^3 + a^3*c^2*d - 3*a^3*c*d^2 + (2*a^3*c^2*d + a^3
*c*d^2 - 3*a^3*d^3)*sin(f*x + e))*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)) + 2*(2*a^3*c^2*d - a^3*c*d^2 + a^3*d^3)*cos(f*x + e) + 2
*((2*a^3*c^2*d - a^3*c*d^2 - 3*a^3*d^3)*f*x + (a^3*c*d^2 + a^3*d^3)*cos(f*x + e))*sin(f*x + e))/((c*d^4 + d^5)
*f*sin(f*x + e) + (c^2*d^3 + c*d^4)*f), -((2*a^3*c^3 - a^3*c^2*d - 3*a^3*c*d^2)*f*x + (2*a^3*c^3 + a^3*c^2*d -
3*a^3*c*d^2 + (2*a^3*c^2*d + a^3*c*d^2 - 3*a^3*d^3)*sin(f*x + e))*sqrt((c - d)/(c + d))*arctan(-(c*sin(f*x +
e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(f*x + e))) + (2*a^3*c^2*d - a^3*c*d^2 + a^3*d^3)*cos(f*x + e) + ((2
*a^3*c^2*d - a^3*c*d^2 - 3*a^3*d^3)*f*x + (a^3*c*d^2 + a^3*d^3)*cos(f*x + e))*sin(f*x + e))/((c*d^4 + d^5)*f*s
in(f*x + e) + (c^2*d^3 + c*d^4)*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))**3/(c+d*sin(f*x+e))**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (161) = 322\).
time = 0.46, size = 395, normalized size = 2.45 \begin {gather*} \frac {\frac {2 \, {\left (2 \, a^{3} c^{3} - a^{3} c^{2} d - 4 \, a^{3} c d^{2} + 3 \, a^{3} 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 d^{3} + d^{4}\right )} \sqrt {c^{2} - d^{2}}} - \frac {2 \, {\left (a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{3} c^{3} - a^{3} c^{2} d + a^{3} c d^{2}\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 )} {\left (c^{2} d^{2} + c d^{3}\right )}} - \frac {{\left (2 \, a^{3} c - 3 \, a^{3} d\right )} {\left (f x + e\right )}}{d^{3}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 12.95, size = 2500, normalized size = 15.53 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*(2*a^3*c^2 + a^3*d^2 - a^3*c*d))/(d^2*(c + d)) + (2*tan(e/2 + (f*x)/2)^2*(2*a^3*c^2 + a^3*d^2 - a^3*c*d)
)/(d^2*(c + d)) + (2*tan(e/2 + (f*x)/2)*(3*a^3*c^2 + a^3*d^2))/(c*d*(c + d)) + (2*tan(e/2 + (f*x)/2)^3*(a^3*c^
2 + a^3*d^2 - 2*a^3*c*d))/(c*d*(c + d)))/(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*a^3*atan(((a^3*(2*c - 3*d)*((32*(9*a^6*c^2*d^6 + 6*a^6*c^3*d^5
- 11*a^6*c^4*d^4 - 4*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(2*c*d^6 + d^7 + c^2*d^5) + (32*tan(e/2 + (f*x)/2)*(9*a^6*
c*d^8 + 36*a^6*c^2*d^7 - 41*a^6*c^3*d^6 - 34*a^6*c^4*d^5 + 34*a^6*c^5*d^4 + 8*a^6*c^6*d^3 - 8*a^6*c^7*d^2))/(2
*c*d^7 + d^8 + c^2*d^6) + (a^3*(2*c - 3*d)*((32*tan(e/2 + (f*x)/2)*(6*a^3*c*d^10 - 2*a^3*c^2*d^9 - 10*a^3*c^3*
d^8 + 2*a^3*c^4*d^7 + 4*a^3*c^5*d^6))/(2*c*d^7 + d^8 + c^2*d^6) - (32*(3*a^3*c*d^9 + a^3*c^2*d^8 - 3*a^3*c^3*d
^7 - a^3*c^4*d^6))/(2*c*d^6 + d^7 + c^2*d^5) + (a^3*((32*(c^2*d^10 + 2*c^3*d^9 + c^4*d^8))/(2*c*d^6 + d^7 + c^
2*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^12 + 6*c^2*d^11 + c^3*d^10 - 4*c^4*d^9 - 2*c^5*d^8))/(2*c*d^7 + d^8 + c
^2*d^6))*(2*c - 3*d)*1i)/d^3)*1i)/d^3))/d^3 + (a^3*(2*c - 3*d)*((32*(9*a^6*c^2*d^6 + 6*a^6*c^3*d^5 - 11*a^6*c^
4*d^4 - 4*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(2*c*d^6 + d^7 + c^2*d^5) + (32*tan(e/2 + (f*x)/2)*(9*a^6*c*d^8 + 36*a
^6*c^2*d^7 - 41*a^6*c^3*d^6 - 34*a^6*c^4*d^5 + 34*a^6*c^5*d^4 + 8*a^6*c^6*d^3 - 8*a^6*c^7*d^2))/(2*c*d^7 + d^8
+ c^2*d^6) + (a^3*(2*c - 3*d)*((32*(3*a^3*c*d^9 + a^3*c^2*d^8 - 3*a^3*c^3*d^7 - a^3*c^4*d^6))/(2*c*d^6 + d^7
+ c^2*d^5) - (32*tan(e/2 + (f*x)/2)*(6*a^3*c*d^10 - 2*a^3*c^2*d^9 - 10*a^3*c^3*d^8 + 2*a^3*c^4*d^7 + 4*a^3*c^5
*d^6))/(2*c*d^7 + d^8 + c^2*d^6) + (a^3*((32*(c^2*d^10 + 2*c^3*d^9 + c^4*d^8))/(2*c*d^6 + d^7 + c^2*d^5) + (32
*tan(e/2 + (f*x)/2)*(3*c*d^12 + 6*c^2*d^11 + c^3*d^10 - 4*c^4*d^9 - 2*c^5*d^8))/(2*c*d^7 + d^8 + c^2*d^6))*(2*
c - 3*d)*1i)/d^3)*1i)/d^3))/d^3)/((64*(4*a^9*c^6 + 27*a^9*c*d^5 - 20*a^9*c^5*d - 63*a^9*c^2*d^4 + 33*a^9*c^3*d
^3 + 19*a^9*c^4*d^2))/(2*c*d^6 + d^7 + c^2*d^5) - (64*tan(e/2 + (f*x)/2)*(40*a^9*c^6*d - 54*a^9*c*d^6 - 16*a^9
*c^7 + 90*a^9*c^2*d^5 + 42*a^9*c^3*d^4 - 130*a^9*c^4*d^3 + 28*a^9*c^5*d^2))/(2*c*d^7 + d^8 + c^2*d^6) + (a^3*(
2*c - 3*d)*((32*(9*a^6*c^2*d^6 + 6*a^6*c^3*d^5 - 11*a^6*c^4*d^4 - 4*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(2*c*d^6 + d
^7 + c^2*d^5) + (32*tan(e/2 + (f*x)/2)*(9*a^6*c*d^8 + 36*a^6*c^2*d^7 - 41*a^6*c^3*d^6 - 34*a^6*c^4*d^5 + 34*a^
6*c^5*d^4 + 8*a^6*c^6*d^3 - 8*a^6*c^7*d^2))/(2*c*d^7 + d^8 + c^2*d^6) + (a^3*(2*c - 3*d)*((32*tan(e/2 + (f*x)/
2)*(6*a^3*c*d^10 - 2*a^3*c^2*d^9 - 10*a^3*c^3*d^8 + 2*a^3*c^4*d^7 + 4*a^3*c^5*d^6))/(2*c*d^7 + d^8 + c^2*d^6)
- (32*(3*a^3*c*d^9 + a^3*c^2*d^8 - 3*a^3*c^3*d^7 - a^3*c^4*d^6))/(2*c*d^6 + d^7 + c^2*d^5) + (a^3*((32*(c^2*d^
10 + 2*c^3*d^9 + c^4*d^8))/(2*c*d^6 + d^7 + c^2*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^12 + 6*c^2*d^11 + c^3*d^1
0 - 4*c^4*d^9 - 2*c^5*d^8))/(2*c*d^7 + d^8 + c^2*d^6))*(2*c - 3*d)*1i)/d^3)*1i)/d^3)*1i)/d^3 - (a^3*(2*c - 3*d
)*((32*(9*a^6*c^2*d^6 + 6*a^6*c^3*d^5 - 11*a^6*c^4*d^4 - 4*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(2*c*d^6 + d^7 + c^2*
d^5) + (32*tan(e/2 + (f*x)/2)*(9*a^6*c*d^8 + 36*a^6*c^2*d^7 - 41*a^6*c^3*d^6 - 34*a^6*c^4*d^5 + 34*a^6*c^5*d^4
+ 8*a^6*c^6*d^3 - 8*a^6*c^7*d^2))/(2*c*d^7 + d^8 + c^2*d^6) + (a^3*(2*c - 3*d)*((32*(3*a^3*c*d^9 + a^3*c^2*d^
8 - 3*a^3*c^3*d^7 - a^3*c^4*d^6))/(2*c*d^6 + d^7 + c^2*d^5) - (32*tan(e/2 + (f*x)/2)*(6*a^3*c*d^10 - 2*a^3*c^2
*d^9 - 10*a^3*c^3*d^8 + 2*a^3*c^4*d^7 + 4*a^3*c^5*d^6))/(2*c*d^7 + d^8 + c^2*d^6) + (a^3*((32*(c^2*d^10 + 2*c^
3*d^9 + c^4*d^8))/(2*c*d^6 + d^7 + c^2*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^12 + 6*c^2*d^11 + c^3*d^10 - 4*c^4
*d^9 - 2*c^5*d^8))/(2*c*d^7 + d^8 + c^2*d^6))*(2*c - 3*d)*1i)/d^3)*1i)/d^3)*1i)/d^3))*(2*c - 3*d))/(d^3*f) - (
a^3*atan(((a^3*(-(c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d)*((32*(9*a^6*c^2*d^6 + 6*a^6*c^3*d^5 - 11*a^6*c^4*d^4 -
4*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(2*c*d^6 + d^7 + c^2*d^5) + (32*tan(e/2 + (f*x)/2)*(9*a^6*c*d^8 + 36*a^6*c^2*
d^7 - 41*a^6*c^3*d^6 - 34*a^6*c^4*d^5 + 34*a^6*c^5*d^4 + 8*a^6*c^6*d^3 - 8*a^6*c^7*d^2))/(2*c*d^7 + d^8 + c^2*
d^6) + (a^3*(-(c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d)*((32*tan(e/2 + (f*x)/2)*(6*a^3*c*d^10 - 2*a^3*c^2*d^9 - 1
0*a^3*c^3*d^8 + 2*a^3*c^4*d^7 + 4*a^3*c^5*d^6))/(2*c*d^7 + d^8 + c^2*d^6) - (32*(3*a^3*c*d^9 + a^3*c^2*d^8 - 3
*a^3*c^3*d^7 - a^3*c^4*d^6))/(2*c*d^6 + d^7 + c^2*d^5) + (a^3*((32*(c^2*d^10 + 2*c^3*d^9 + c^4*d^8))/(2*c*d^6
+ d^7 + c^2*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^12 + 6*c^2*d^11 + c^3*d^10 - 4*c^4*d^9 - 2*c^5*d^8))/(2*c*d^7
+ d^8 + c^2*d^6))*(-(c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d))/(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3)))/(3*c*d^5
+ d^6 + 3*c^2*d^4 + c^3*d^3))*1i)/(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3) + (a^3*(-(c + d)^3*(c - d)^3)^(1/2)*(2
*c + 3*d)*((32*(9*a^6*c^2*d^6 + 6*a^6*c^3*d^5 - 11*a^6*c^4*d^4 - 4*a^6*c^5*d^3 + 4*a^6*c^6*d^2))/(2*c*d^6 + d^
7 + c^2*d^5) + (32*tan(e/2 + (f*x)/2)*(9*a^6*c*d^8 + 36*a^6*c^2*d^7 - 41*a^6*c^3*d^6 - 34*a^6*c^4*d^5 + 34*a^6
*c^5*d^4 + 8*a^6*c^6*d^3 - 8*a^6*c^7*d^2))/(2*c...

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