Optimal. Leaf size=33 \[ \frac {\tan ^3(x)}{3 a}+\frac {\tanh ^{-1}(\sin (x))}{2 a}-\frac {\tan (x) \sec (x)}{2 a} \]
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Rubi [A] time = 0.08, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2706, 2607, 30, 2611, 3770} \[ \frac {\tan ^3(x)}{3 a}+\frac {\tanh ^{-1}(\sin (x))}{2 a}-\frac {\tan (x) \sec (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 2611
Rule 2706
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tan ^4(x)}{a+a \cos (x)} \, dx &=-\frac {\int \sec (x) \tan ^2(x) \, dx}{a}+\frac {\int \sec ^2(x) \tan ^2(x) \, dx}{a}\\ &=-\frac {\sec (x) \tan (x)}{2 a}+\frac {\int \sec (x) \, dx}{2 a}+\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\tan (x)\right )}{a}\\ &=\frac {\tanh ^{-1}(\sin (x))}{2 a}-\frac {\sec (x) \tan (x)}{2 a}+\frac {\tan ^3(x)}{3 a}\\ \end {align*}
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Mathematica [B] time = 0.13, size = 105, normalized size = 3.18 \[ -\frac {\sec ^3(x) \left (2 (-3 \sin (x)+3 \sin (2 x)+\sin (3 x))+9 \cos (x) \left (\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )+3 \cos (3 x) \left (\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )\right )}{24 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 50, normalized size = 1.52 \[ \frac {3 \, \cos \relax (x)^{3} \log \left (\sin \relax (x) + 1\right ) - 3 \, \cos \relax (x)^{3} \log \left (-\sin \relax (x) + 1\right ) - 2 \, {\left (2 \, \cos \relax (x)^{2} + 3 \, \cos \relax (x) - 2\right )} \sin \relax (x)}{12 \, a \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.71, size = 65, normalized size = 1.97 \[ \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right )}{2 \, a} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right )}{2 \, a} - \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{5} + 8 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 3 \, \tan \left (\frac {1}{2} \, x\right )}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 1\right )}^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 103, normalized size = 3.12 \[ -\frac {1}{3 a \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{a \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{2 a \left (\tan \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )-1\right )}{2 a}-\frac {1}{3 a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{a \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{2 a \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 115, normalized size = 3.48 \[ -\frac {\frac {3 \, \sin \relax (x)}{\cos \relax (x) + 1} - \frac {8 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - \frac {3 \, \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}}}{3 \, {\left (a - \frac {3 \, a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {3 \, a \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {a \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}}\right )}} + \frac {\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}{2 \, a} - \frac {\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} - 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 46, normalized size = 1.39 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^5+\frac {8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{3}-\mathrm {tan}\left (\frac {x}{2}\right )}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2-1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\tan ^{4}{\relax (x )}}{\cos {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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