Optimal. Leaf size=192 \[ -\frac{\sqrt{c} \log \left (\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}+\frac{\sqrt{c} \log \left (\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}+\frac{\sqrt{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right )}{\sqrt{2} b} \]
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Rubi [A] time = 0.11515, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt{c} \log \left (\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}+\frac{\sqrt{c} \log \left (\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}+\frac{\sqrt{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right )}{\sqrt{2} b} \]
Antiderivative was successfully verified.
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Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \sqrt{c \cot (a+b x)} \, dx &=-\frac{c \operatorname{Subst}\left (\int \frac{\sqrt{x}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=-\frac{(2 c) \operatorname{Subst}\left (\int \frac{x^2}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}\\ &=\frac{c \operatorname{Subst}\left (\int \frac{c-x^2}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}-\frac{c \operatorname{Subst}\left (\int \frac{c+x^2}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}\\ &=-\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{c}+2 x}{-c-\sqrt{2} \sqrt{c} x-x^2} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}-\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{c}-2 x}{-c+\sqrt{2} \sqrt{c} x-x^2} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}-\frac{c \operatorname{Subst}\left (\int \frac{1}{c-\sqrt{2} \sqrt{c} x+x^2} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{2 b}-\frac{c \operatorname{Subst}\left (\int \frac{1}{c+\sqrt{2} \sqrt{c} x+x^2} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{2 b}\\ &=-\frac{\sqrt{c} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}+\frac{\sqrt{c} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}-\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}+\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}\\ &=\frac{\sqrt{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}-\frac{\sqrt{c} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}-\frac{\sqrt{c} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}+\frac{\sqrt{c} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}\\ \end{align*}
Mathematica [C] time = 0.0391497, size = 40, normalized size = 0.21 \[ -\frac{2 (c \cot (a+b x))^{3/2} \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(a+b x)\right )}{3 b c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 160, normalized size = 0.8 \begin{align*} -{\frac{c\sqrt{2}}{4\,b}\ln \left ({ \left ( c\cot \left ( bx+a \right ) -\sqrt [4]{{c}^{2}}\sqrt{c\cot \left ( bx+a \right ) }\sqrt{2}+\sqrt{{c}^{2}} \right ) \left ( c\cot \left ( bx+a \right ) +\sqrt [4]{{c}^{2}}\sqrt{c\cot \left ( bx+a \right ) }\sqrt{2}+\sqrt{{c}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{c}^{2}}}}}-{\frac{c\sqrt{2}}{2\,b}\arctan \left ({\sqrt{2}\sqrt{c\cot \left ( bx+a \right ) }{\frac{1}{\sqrt [4]{{c}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{c}^{2}}}}}+{\frac{c\sqrt{2}}{2\,b}\arctan \left ( -{\sqrt{2}\sqrt{c\cot \left ( bx+a \right ) }{\frac{1}{\sqrt [4]{{c}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{c}^{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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{c \cot{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \cot \left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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