Optimal. Leaf size=14 \[ -\cot (x) \sqrt{a \sec ^2(x)} \]
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Rubi [A] time = 0.0804953, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3657, 4125, 2606, 8} \[ -\cot (x) \sqrt{a \sec ^2(x)} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4125
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \cot ^2(x) \sqrt{a+a \tan ^2(x)} \, dx &=\int \cot ^2(x) \sqrt{a \sec ^2(x)} \, dx\\ &=\left (\cos (x) \sqrt{a \sec ^2(x)}\right ) \int \cot (x) \csc (x) \, dx\\ &=-\left (\left (\cos (x) \sqrt{a \sec ^2(x)}\right ) \operatorname{Subst}(\int 1 \, dx,x,\csc (x))\right )\\ &=-\cot (x) \sqrt{a \sec ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0141342, size = 14, normalized size = 1. \[ -\cot (x) \sqrt{a \sec ^2(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.083, size = 17, normalized size = 1.2 \begin{align*} -{\frac{\cos \left ( x \right ) }{\sin \left ( x \right ) }\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 = 1.65049, size = 23, normalized size = 1.64 \begin{align*} -\frac{\sqrt{\tan \left (x\right )^{2} + 1} \sqrt{a}}{\tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28554, size = 41, normalized size = 2.93 \begin{align*} -\frac{\sqrt{a \tan \left (x\right )^{2} + a}}{\tan \left (x\right )} \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 ^{2}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1129, size = 43, normalized size = 3.07 \begin{align*} \frac{2 \, a^{\frac{3}{2}}}{{\left (\sqrt{a} \tan \left (x\right ) - \sqrt{a \tan \left (x\right )^{2} + a}\right )}^{2} - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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