Optimal. Leaf size=78 \[ -\frac{1}{(a-b)^2 \sqrt{a+b \cot ^2(x)}}-\frac{1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )}{(a-b)^{5/2}} \]
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Rubi [A] time = 0.0915714, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3670, 444, 51, 63, 208} \[ -\frac{1}{(a-b)^2 \sqrt{a+b \cot ^2(x)}}-\frac{1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )}{(a-b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 444
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1+x) (a+b x)^{5/2}} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\frac{1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{(1+x) (a+b x)^{3/2}} \, dx,x,\cot ^2(x)\right )}{2 (a-b)}\\ &=-\frac{1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac{1}{(a-b)^2 \sqrt{a+b \cot ^2(x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x}} \, dx,x,\cot ^2(x)\right )}{2 (a-b)^2}\\ &=-\frac{1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac{1}{(a-b)^2 \sqrt{a+b \cot ^2(x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cot ^2(x)}\right )}{(a-b)^2 b}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )}{(a-b)^{5/2}}-\frac{1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac{1}{(a-b)^2 \sqrt{a+b \cot ^2(x)}}\\ \end{align*}
Mathematica [C] time = 0.0390666, size = 47, normalized size = 0.6 \[ -\frac{\text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},\frac{a+b \cot ^2(x)}{a-b}\right )}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 75, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( a-b \right ) ^{2}}\arctan \left ({\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}}-{\frac{1}{ \left ( a-b \right ) ^{2}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}}}-{\frac{1}{3\,a-3\,b} \left ( a+b \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \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 = 2.13093, size = 1416, normalized size = 18.15 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 51.1062, size = 70, normalized size = 0.9 \begin{align*} - \frac{1}{3 \left (a - b\right ) \left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac{3}{2}}} - \frac{1}{\left (a - b\right )^{2} \sqrt{a + b \cot ^{2}{\left (x \right )}}} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b \cot ^{2}{\left (x \right )}}}{\sqrt{- a + b}} \right )}}{\sqrt{- a + b} \left (a - b\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.35537, size = 270, normalized size = 3.46 \begin{align*} \frac{\sqrt{a - b} \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 )}{16 \,{\left (a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right )}} + \frac{{\left (\frac{4 \,{\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \sin \left (x\right )^{2}}{a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}} + \frac{3 \,{\left (a b^{2} - b^{3}\right )}}{a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}}\right )} \sin \left (x\right )}{48 \,{\left (a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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