Optimal. Leaf size=33 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]
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Rubi [A] time = 0.0638457, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3670, 444, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 444
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot (x)}{\sqrt{a+b \cot ^2(x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\cot (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x}} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cot ^2(x)}\right )}{b}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b}}\\ \end{align*}
Mathematica [A] time = 0.0134122, size = 33, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 29, normalized size = 0.9 \begin{align*} -{\arctan \left ({\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02038, size = 304, normalized size = 9.21 \begin{align*} \left [\frac{\log \left (-\sqrt{a - b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}{\left (\cos \left (2 \, x\right ) - 1\right )} -{\left (a - b\right )} \cos \left (2 \, x\right ) + a\right )}{2 \, \sqrt{a - b}}, \frac{\sqrt{-a + b} \arctan \left (-\frac{\sqrt{-a + b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{a - b}\right )}{a - b}\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 \frac{\cot{\left (x \right )}}{\sqrt{a + b \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 = 1.46362, size = 82, normalized size = 2.48 \begin{align*} \frac{\log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (\sin \left (x\right )\right )}{2 \, \sqrt{a - b}} - \frac{\log \left ({\left | -\sqrt{a - b} \sin \left (x\right ) + \sqrt{a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} \right |}\right )}{\sqrt{a - b} \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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