Optimal. Leaf size=34 \[ \cot (x) \sqrt {a \sec ^2(x)}-\frac {1}{3} \cot (x) \csc ^2(x) \sqrt {a \sec ^2(x)} \]
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Rubi [A] time = 0.10, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3657, 4125, 2606} \[ \cot (x) \sqrt {a \sec ^2(x)}-\frac {1}{3} \cot (x) \csc ^2(x) \sqrt {a \sec ^2(x)} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 3657
Rule 4125
Rubi steps
\begin {align*} \int \cot ^4(x) \sqrt {a+a \tan ^2(x)} \, dx &=\int \cot ^4(x) \sqrt {a \sec ^2(x)} \, dx\\ &=\left (\cos (x) \sqrt {a \sec ^2(x)}\right ) \int \cot ^3(x) \csc (x) \, dx\\ &=-\left (\left (\cos (x) \sqrt {a \sec ^2(x)}\right ) \operatorname {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (x)\right )\right )\\ &=\cot (x) \sqrt {a \sec ^2(x)}-\frac {1}{3} \cot (x) \csc ^2(x) \sqrt {a \sec ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.65 \[ -\frac {1}{3} \cot (x) \left (\csc ^2(x)-3\right ) \sqrt {a \sec ^2(x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 24, normalized size = 0.71 \[ \frac {\sqrt {a \tan \relax (x)^{2} + a} {\left (2 \, \tan \relax (x)^{2} - 1\right )}}{3 \, \tan \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 59, normalized size = 1.74 \[ \frac {4 \, {\left (3 \, {\left (\sqrt {a} \tan \relax (x) - \sqrt {a \tan \relax (x)^{2} + a}\right )}^{2} - a\right )} a^{\frac {5}{2}}}{3 \, {\left ({\left (\sqrt {a} \tan \relax (x) - \sqrt {a \tan \relax (x)^{2} + a}\right )}^{2} - a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 25, normalized size = 0.74 \[ -\frac {\left (3 \left (\cos ^{2}\relax (x )\right )-2\right ) \cos \relax (x ) \sqrt {\frac {a}{\cos \relax (x )^{2}}}}{3 \sin \relax (x )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 29, normalized size = 0.85 \[ \frac {{\left (2 \, \sqrt {a} \tan \relax (x)^{2} - \sqrt {a}\right )} \sqrt {\tan \relax (x)^{2} + 1}}{3 \, \tan \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.85, size = 40, normalized size = 1.18 \[ \frac {\sqrt {2}\,\sqrt {a}\,\left (2\,\sin \left (2\,x\right )-6\,\sin \left (2\,x\right )\,\left (2\,{\cos \relax (x)}^2-1\right )\right )}{24\,\sqrt {2\,{\cos \relax (x)}^2}\,{\left ({\cos \relax (x)}^2-1\right )}^2} \]
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 \[ \int \sqrt {a \left (\tan ^{2}{\relax (x )} + 1\right )} \cot ^{4}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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