Optimal. Leaf size=48 \[ \sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )-\sqrt{a+b \cot ^2(x)} \]
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Rubi [A] time = 0.0667223, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3670, 444, 50, 63, 208} \[ \sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )-\sqrt{a+b \cot ^2(x)} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 444
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \cot (x) \sqrt{a+b \cot ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x \sqrt{a+b x^2}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{1+x} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\sqrt{a+b \cot ^2(x)}-\frac{1}{2} (a-b) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x}} \, dx,x,\cot ^2(x)\right )\\ &=-\sqrt{a+b \cot ^2(x)}-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cot ^2(x)}\right )}{b}\\ &=\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )-\sqrt{a+b \cot ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0255438, size = 48, normalized size = 1. \[ \sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )-\sqrt{a+b \cot ^2(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 71, normalized size = 1.5 \begin{align*} -\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}+{b\arctan \left ({\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}}-{a\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 = 2.22684, size = 587, normalized size = 12.23 \begin{align*} \left [\frac{1}{4} \, \sqrt{a - b} \log \left (-2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, a^{2} + b^{2} - 2 \,{\left ({\left (a - b\right )} \cos \left (2 \, x\right )^{2} -{\left (2 \, a - b\right )} \cos \left (2 \, x\right ) + a\right )} \sqrt{a - b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} + 4 \,{\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right ) - \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}, \frac{1}{2} \, \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}}{\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) - \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}\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 + b \cot ^{2}{\left (x \right )}} \cot{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31237, size = 128, normalized size = 2.67 \begin{align*} -\frac{1}{2} \,{\left (\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 )}^{2}\right ) - \frac{4 \, \sqrt{a - b} b}{{\left (\sqrt{a - b} \sin \left (x\right ) - \sqrt{a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - b}\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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