Optimal. Leaf size=210 \[ -\frac{c^{3/2} \log \left (\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}+\frac{c^{3/2} \log \left (\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}-\frac{c^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}+\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right )}{\sqrt{2} b}-\frac{2 c \sqrt{c \cot (a+b x)}}{b} \]
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Rubi [A] time = 0.139562, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{c^{3/2} \log \left (\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}+\frac{c^{3/2} \log \left (\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}-\frac{c^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}+\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right )}{\sqrt{2} b}-\frac{2 c \sqrt{c \cot (a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int (c \cot (a+b x))^{3/2} \, dx &=-\frac{2 c \sqrt{c \cot (a+b x)}}{b}-c^2 \int \frac{1}{\sqrt{c \cot (a+b x)}} \, dx\\ &=-\frac{2 c \sqrt{c \cot (a+b x)}}{b}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (c^2+x^2\right )} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=-\frac{2 c \sqrt{c \cot (a+b x)}}{b}+\frac{\left (2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}\\ &=-\frac{2 c \sqrt{c \cot (a+b x)}}{b}+\frac{c^2 \operatorname{Subst}\left (\int \frac{c-x^2}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}+\frac{c^2 \operatorname{Subst}\left (\int \frac{c+x^2}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}\\ &=-\frac{2 c \sqrt{c \cot (a+b x)}}{b}-\frac{c^{3/2} \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^{3/2} \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^2 \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^2 \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{2 c \sqrt{c \cot (a+b x)}}{b}-\frac{c^{3/2} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}+\frac{c^{3/2} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}+\frac{c^{3/2} \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{c^{3/2} \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{c^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}+\frac{c^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}-\frac{2 c \sqrt{c \cot (a+b x)}}{b}-\frac{c^{3/2} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}+\frac{c^{3/2} \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 [A] time = 0.191154, size = 159, normalized size = 0.76 \[ -\frac{(c \cot (a+b x))^{3/2} \left (8 \sqrt{\cot (a+b x)}+\sqrt{2} \log \left (\cot (a+b x)-\sqrt{2} \sqrt{\cot (a+b x)}+1\right )-\sqrt{2} \log \left (\cot (a+b x)+\sqrt{2} \sqrt{\cot (a+b x)}+1\right )+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (a+b x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (a+b x)}+1\right )\right )}{4 b \cot ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 176, normalized size = 0.8 \begin{align*} -2\,{\frac{c\sqrt{c\cot \left ( bx+a \right ) }}{b}}-{\frac{c\sqrt{2}}{2\,b}\sqrt [4]{{c}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{c\cot \left ( bx+a \right ) }{\frac{1}{\sqrt [4]{{c}^{2}}}}}+1 \right ) }+{\frac{c\sqrt{2}}{4\,b}\sqrt [4]{{c}^{2}}\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{c\sqrt{2}}{2\,b}\sqrt [4]{{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{c\cot \left ( bx+a \right ) }{\frac{1}{\sqrt [4]{{c}^{2}}}}}+1 \right ) } \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 \left (c \cot{\left (a + b x \right )}\right )^{\frac{3}{2}}\, 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 \left (c \cot \left (b x + a\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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