Optimal. Leaf size=33 \[ a \cos (x) \sqrt {a \sec ^2(x)} \tanh ^{-1}(\sin (x))-a \cot (x) \sqrt {a \sec ^2(x)} \]
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Rubi [A] time = 0.11, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3657, 4125, 2621, 321, 207} \[ a \cos (x) \sqrt {a \sec ^2(x)} \tanh ^{-1}(\sin (x))-a \cot (x) \sqrt {a \sec ^2(x)} \]
Antiderivative was successfully verified.
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Rule 207
Rule 321
Rule 2621
Rule 3657
Rule 4125
Rubi steps
\begin {align*} \int \cot ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx &=\int \cot ^2(x) \left (a \sec ^2(x)\right )^{3/2} \, dx\\ &=\left (a \cos (x) \sqrt {a \sec ^2(x)}\right ) \int \csc ^2(x) \sec (x) \, dx\\ &=-\left (\left (a \cos (x) \sqrt {a \sec ^2(x)}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (x)\right )\right )\\ &=-a \cot (x) \sqrt {a \sec ^2(x)}-\left (a \cos (x) \sqrt {a \sec ^2(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (x)\right )\\ &=a \tanh ^{-1}(\sin (x)) \cos (x) \sqrt {a \sec ^2(x)}-a \cot (x) \sqrt {a \sec ^2(x)}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 27, normalized size = 0.82 \[ -a \cot (x) \sqrt {a \sec ^2(x)} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\sin ^2(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 53, normalized size = 1.61 \[ \frac {a^{\frac {3}{2}} \log \left (2 \, a \tan \relax (x)^{2} + 2 \, \sqrt {a \tan \relax (x)^{2} + a} \sqrt {a} \tan \relax (x) + a\right ) \tan \relax (x) - 2 \, \sqrt {a \tan \relax (x)^{2} + a} a}{2 \, \tan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.75, size = 62, normalized size = 1.88 \[ -\frac {1}{2} \, {\left (\sqrt {a} \log \left ({\left (\sqrt {a} \tan \relax (x) - \sqrt {a \tan \relax (x)^{2} + a}\right )}^{2}\right ) - \frac {4 \, a^{\frac {3}{2}}}{{\left (\sqrt {a} \tan \relax (x) - \sqrt {a \tan \relax (x)^{2} + a}\right )}^{2} - a}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 55, normalized size = 1.67 \[ -\frac {\left (\ln \left (-\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}\right ) \sin \relax (x )-\ln \left (\frac {1-\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}\right ) \sin \relax (x )+1\right ) \left (\cos ^{3}\relax (x )\right ) \left (\frac {a}{\cos \relax (x )^{2}}\right )^{\frac {3}{2}}}{\sin \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.01, size = 134, normalized size = 4.06 \[ -\frac {{\left (4 \, a \cos \relax (x) \sin \left (2 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) \sin \relax (x) - {\left (a \cos \left (2 \, x\right )^{2} + a \sin \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) + {\left (a \cos \left (2 \, x\right )^{2} + a \sin \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) + 4 \, a \sin \relax (x)\right )} \sqrt {a}}{2 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\mathrm {cot}\relax (x)}^2\,{\left (a\,{\mathrm {tan}\relax (x)}^2+a\right )}^{3/2} \,d x \]
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 \[ \int \left (a \left (\tan ^{2}{\relax (x )} + 1\right )\right )^{\frac {3}{2}} \cot ^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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