Optimal. Leaf size=279 \[ -\frac{e^{3/2} \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{4 \sqrt{2} a^2 d}+\frac{e^{3/2} \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{4 \sqrt{2} a^2 d}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d}+\frac{e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d}-\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{2 \sqrt{2} a^2 d}-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )} \]
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Rubi [A] time = 0.563629, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 15, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3567, 3653, 12, 16, 3476, 329, 297, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac{e^{3/2} \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{4 \sqrt{2} a^2 d}+\frac{e^{3/2} \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{4 \sqrt{2} a^2 d}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d}+\frac{e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d}-\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{2 \sqrt{2} a^2 d}-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )} \]
Antiderivative was successfully verified.
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Rule 3567
Rule 3653
Rule 12
Rule 16
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx &=-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac{\int \frac{\frac{a e^2}{2}-a e^2 \cot (c+d x)-\frac{1}{2} a e^2 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{2 a^2}\\ &=-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac{\int -\frac{2 a^2 e^2 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{4 a^4}-\frac{e^2 \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{4 a}\\ &=-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}+\frac{e^2 \int \frac{\cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{2 a^2}-\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{4 a d}\\ &=-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}+\frac{e \int \sqrt{e \cot (c+d x)} \, dx}{2 a^2}+\frac{e \operatorname{Subst}\left (\int \frac{1}{a+\frac{a x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 a d}\\ &=\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d}-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac{e^2 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{e^2+x^2} \, dx,x,e \cot (c+d x)\right )}{2 a^2 d}\\ &=\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d}-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac{e^2 \operatorname{Subst}\left (\int \frac{x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{a^2 d}\\ &=\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d}-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}+\frac{e^2 \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 a^2 d}-\frac{e^2 \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 a^2 d}\\ &=\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d}-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}+2 x}{-e-\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d}-\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}-2 x}{-e+\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d}-\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{4 a^2 d}-\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{4 a^2 d}\\ &=\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d}-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac{e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d}+\frac{e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d}-\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d}+\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d}\\ &=\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d}+\frac{e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d}-\frac{e^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d}-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac{e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d}+\frac{e^{3/2} \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d}\\ \end{align*}
Mathematica [A] time = 2.75169, size = 312, normalized size = 1.12 \[ -\frac{\sin ^2(c+d x) (e \cot (c+d x))^{3/2} \left (4 \cot ^{\frac{7}{2}}(c+d x)-4 \cot ^{\frac{5}{2}}(c+d x)+4 \cot ^{\frac{3}{2}}(c+d x)-4 \sqrt{\cot (c+d x)}+\sqrt{2} \cos (2 (c+d x)) \csc ^4(c+d x) \log \left (-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right )-\sqrt{2} \cos (2 (c+d x)) \csc ^4(c+d x) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-2 \sqrt{2} \cos (2 (c+d x)) \csc ^4(c+d x) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )+2 \sqrt{2} \cos (2 (c+d x)) \csc ^4(c+d x) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-4 \cos (2 (c+d x)) \csc ^4(c+d x) \tan ^{-1}\left (\sqrt{\cot (c+d x)}\right )\right )}{8 a^2 d \cot ^{\frac{3}{2}}(c+d x) \left (\cot ^2(c+d x)-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 234, normalized size = 0.8 \begin{align*} -{\frac{{e}^{2}\sqrt{2}}{8\,d{a}^{2}}\ln \left ({ \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-{\frac{{e}^{2}\sqrt{2}}{4\,d{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{{e}^{2}\sqrt{2}}{4\,d{a}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-{\frac{{e}^{2}}{2\,d{a}^{2} \left ( e\cot \left ( dx+c \right ) +e \right ) }\sqrt{e\cot \left ( dx+c \right ) }}+{\frac{1}{2\,d{a}^{2}}{e}^{{\frac{3}{2}}}\arctan \left ({\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \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*} \frac{\int \frac{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}{\cot ^{2}{\left (c + d x \right )} + 2 \cot{\left (c + d x \right )} + 1}\, dx}{a^{2}} \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 \frac{\left (e \cot \left (d x + c\right )\right )^{\frac{3}{2}}}{{\left (a \cot \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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