Optimal. Leaf size=212 \[ \frac{c^{5/2} \log \left (\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}-\frac{c^{5/2} \log \left (\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}-\frac{c^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}+\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right )}{\sqrt{2} b}-\frac{2 c (c \cot (a+b x))^{3/2}}{3 b} \]
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Rubi [A] time = 0.142843, antiderivative size = 212, 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, 297, 1162, 617, 204, 1165, 628} \[ \frac{c^{5/2} \log \left (\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}-\frac{c^{5/2} \log \left (\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}-\frac{c^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}+\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right )}{\sqrt{2} b}-\frac{2 c (c \cot (a+b x))^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int (c \cot (a+b x))^{5/2} \, dx &=-\frac{2 c (c \cot (a+b x))^{3/2}}{3 b}-c^2 \int \sqrt{c \cot (a+b x)} \, dx\\ &=-\frac{2 c (c \cot (a+b x))^{3/2}}{3 b}+\frac{c^3 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=-\frac{2 c (c \cot (a+b x))^{3/2}}{3 b}+\frac{\left (2 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}\\ &=-\frac{2 c (c \cot (a+b x))^{3/2}}{3 b}-\frac{c^3 \operatorname{Subst}\left (\int \frac{c-x^2}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}+\frac{c^3 \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 (c \cot (a+b x))^{3/2}}{3 b}+\frac{c^{5/2} \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^{5/2} \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 \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^3 \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 (c \cot (a+b x))^{3/2}}{3 b}+\frac{c^{5/2} \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^{5/2} \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^{5/2} \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^{5/2} \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^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}+\frac{c^{5/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}-\frac{2 c (c \cot (a+b x))^{3/2}}{3 b}+\frac{c^{5/2} \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^{5/2} \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 [C] time = 0.0720708, size = 40, normalized size = 0.19 \[ \frac{2 c (c \cot (a+b x))^{3/2} \left (\text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(a+b x)\right )-1\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 182, normalized size = 0.9 \begin{align*} -{\frac{2\,c}{3\,b} \left ( c\cot \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{{c}^{3}\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}^{3}\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}^{3}\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 \left (c \cot{\left (a + b x \right )}\right )^{\frac{5}{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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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