Optimal. Leaf size=53 \[ -\frac {3 x}{2 a}+\frac {4 \cos ^3(x)}{3 a}-\frac {4 \cos (x)}{a}+\frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}+\frac {3 \sin (x) \cos (x)}{2 a} \]
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Rubi [A] time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2767, 2748, 2635, 8, 2633} \[ -\frac {3 x}{2 a}+\frac {4 \cos ^3(x)}{3 a}-\frac {4 \cos (x)}{a}+\frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}+\frac {3 \sin (x) \cos (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2767
Rubi steps
\begin {align*} \int \frac {\sin ^4(x)}{a+a \sin (x)} \, dx &=\frac {\cos (x) \sin ^3(x)}{a+a \sin (x)}-\frac {\int \sin ^2(x) (3 a-4 a \sin (x)) \, dx}{a^2}\\ &=\frac {\cos (x) \sin ^3(x)}{a+a \sin (x)}-\frac {3 \int \sin ^2(x) \, dx}{a}+\frac {4 \int \sin ^3(x) \, dx}{a}\\ &=\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^3(x)}{a+a \sin (x)}-\frac {3 \int 1 \, dx}{2 a}-\frac {4 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )}{a}\\ &=-\frac {3 x}{2 a}-\frac {4 \cos (x)}{a}+\frac {4 \cos ^3(x)}{3 a}+\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^3(x)}{a+a \sin (x)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 101, normalized size = 1.91 \[ \frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (-36 x \sin \left (\frac {x}{2}\right )+69 \sin \left (\frac {x}{2}\right )-18 \sin \left (\frac {3 x}{2}\right )+2 \sin \left (\frac {5 x}{2}\right )+\sin \left (\frac {7 x}{2}\right )-3 (12 x+7) \cos \left (\frac {x}{2}\right )-18 \cos \left (\frac {3 x}{2}\right )-2 \cos \left (\frac {5 x}{2}\right )+\cos \left (\frac {7 x}{2}\right )\right )}{24 a (\sin (x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 70, normalized size = 1.32 \[ \frac {2 \, \cos \relax (x)^{4} - \cos \relax (x)^{3} - 3 \, {\left (3 \, x + 5\right )} \cos \relax (x) - 12 \, \cos \relax (x)^{2} + {\left (2 \, \cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} - 9 \, x - 9 \, \cos \relax (x) + 6\right )} \sin \relax (x) - 9 \, x - 6}{6 \, {\left (a \cos \relax (x) + a \sin \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 67, normalized size = 1.26 \[ -\frac {3 \, x}{2 \, a} - \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} - \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{5} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 24 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, x\right ) + 10}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 121, normalized size = 2.28 \[ -\frac {\tan ^{5}\left (\frac {x}{2}\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {8 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\tan \left (\frac {x}{2}\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {10}{3 a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.83, size = 180, normalized size = 3.40 \[ -\frac {\frac {7 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {39 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {24 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {24 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {9 \, \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {9 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + 16}{3 \, {\left (a + \frac {a \sin \relax (x)}{\cos \relax (x) + 1} + \frac {3 \, a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {3 \, a \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {3 \, a \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {3 \, a \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {a \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + \frac {a \sin \relax (x)^{7}}{{\left (\cos \relax (x) + 1\right )}^{7}}\right )}} - \frac {3 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.82, size = 78, normalized size = 1.47 \[ -\frac {3\,x}{2\,a}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+13\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {7\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {16}{3}}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^3\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.91, size = 1221, normalized size = 23.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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