Optimal. Leaf size=75 \[ \frac {2 \cos (c+d x)}{a d}+\frac {\sin ^2(c+d x) \cos (c+d x)}{d (a \sin (c+d x)+a)}-\frac {3 \sin (c+d x) \cos (c+d x)}{2 a d}+\frac {3 x}{2 a} \]
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Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2767, 2734} \[ \frac {2 \cos (c+d x)}{a d}+\frac {\sin ^2(c+d x) \cos (c+d x)}{d (a \sin (c+d x)+a)}-\frac {3 \sin (c+d x) \cos (c+d x)}{2 a d}+\frac {3 x}{2 a} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2767
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\cos (c+d x) \sin ^2(c+d x)}{d (a+a \sin (c+d x))}-\frac {\int \sin (c+d x) (2 a-3 a \sin (c+d x)) \, dx}{a^2}\\ &=\frac {3 x}{2 a}+\frac {2 \cos (c+d x)}{a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {\cos (c+d x) \sin ^2(c+d x)}{d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 117, normalized size = 1.56 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right ) (-\sin (2 (c+d x))+4 \cos (c+d x)+6 c+6 d x-8)+\cos \left (\frac {1}{2} (c+d x)\right ) (-\sin (2 (c+d x))+4 \cos (c+d x)+6 c+6 d x)\right )}{4 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 92, normalized size = 1.23 \[ \frac {\cos \left (d x + c\right )^{3} + 3 \, d x + 3 \, {\left (d x + 1\right )} \cos \left (d x + c\right ) + 2 \, \cos \left (d x + c\right )^{2} + {\left (3 \, d x - \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 2}{2 \, {\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 91, normalized size = 1.21 \[ \frac {\frac {3 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a} + \frac {4}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 163, normalized size = 2.17 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {2}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.67, size = 212, normalized size = 2.83 \[ \frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 4}{a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.29, size = 92, normalized size = 1.23 \[ \frac {3\,x}{2\,a}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4}{a\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.52, size = 1127, normalized size = 15.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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