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 = 0.0996947, antiderivative size = 34, normalized size of antiderivative = 1., 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 3657
Rule 4125
Rule 2606
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*}
Mathematica [A] time = 0.0235239, 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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Maple [A] time = 0.084, size = 25, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 3\, \left ( \cos \left ( x \right ) \right ) ^{2}-2 \right ) \cos \left ( x \right ) }{3\, \left ( \sin \left ( x \right ) \right ) ^{3}}\sqrt{{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64409, size = 39, normalized size = 1.15 \begin{align*} \frac{{\left (2 \, \sqrt{a} \tan \left (x\right )^{2} - \sqrt{a}\right )} \sqrt{\tan \left (x\right )^{2} + 1}}{3 \, \tan \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29801, size = 70, normalized size = 2.06 \begin{align*} \frac{\sqrt{a \tan \left (x\right )^{2} + a}{\left (2 \, \tan \left (x\right )^{2} - 1\right )}}{3 \, \tan \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\tan ^{2}{\left (x \right )} + 1\right )} \cot ^{4}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09745, size = 80, normalized size = 2.35 \begin{align*} \frac{4 \,{\left (3 \,{\left (\sqrt{a} \tan \left (x\right ) - \sqrt{a \tan \left (x\right )^{2} + a}\right )}^{2} - a\right )} a^{\frac{5}{2}}}{3 \,{\left ({\left (\sqrt{a} \tan \left (x\right ) - \sqrt{a \tan \left (x\right )^{2} + a}\right )}^{2} - a\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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