Optimal. Leaf size=222 \[ -\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}-\frac{a^2 \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d \sqrt{e}}+\frac{a^2 \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d \sqrt{e}}+\frac{\sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}-\frac{\sqrt{2} a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{d \sqrt{e}} \]
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Rubi [A] time = 0.199634, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {3543, 12, 16, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}-\frac{a^2 \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d \sqrt{e}}+\frac{a^2 \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{\sqrt{2} d \sqrt{e}}+\frac{\sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}-\frac{\sqrt{2} a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{d \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 3543
Rule 12
Rule 16
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(a+a \cot (c+d x))^2}{\sqrt{e \cot (c+d x)}} \, dx &=-\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}+\int \frac{2 a^2 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx\\ &=-\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}+\left (2 a^2\right ) \int \frac{\cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx\\ &=-\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}+\frac{\left (2 a^2\right ) \int \sqrt{e \cot (c+d x)} \, dx}{e}\\ &=-\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{e^2+x^2} \, dx,x,e \cot (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}-\frac{\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}\\ &=-\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}\\ &=-\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d}-\frac{a^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 )}{\sqrt{2} d \sqrt{e}}-\frac{a^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 )}{\sqrt{2} d \sqrt{e}}\\ &=-\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}-\frac{a^2 \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d \sqrt{e}}+\frac{a^2 \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d \sqrt{e}}-\frac{\left (\sqrt{2} a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{\left (\sqrt{2} a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}\\ &=\frac{\sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}-\frac{\sqrt{2} a^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}-\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}-\frac{a^2 \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d \sqrt{e}}+\frac{a^2 \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{\sqrt{2} d \sqrt{e}}\\ \end{align*}
Mathematica [C] time = 0.242883, size = 53, normalized size = 0.24 \[ -\frac{2 a^2 \sqrt{e \cot (c+d x)} \left (2 \cot (c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )+3\right )}{3 d e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 186, normalized size = 0.8 \begin{align*} -2\,{\frac{{a}^{2}\sqrt{e\cot \left ( dx+c \right ) }}{de}}-{\frac{{a}^{2}\sqrt{2}}{2\,d}\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{{a}^{2}\sqrt{2}}{d}\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{{a}^{2}\sqrt{2}}{d}\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}}}}} \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*} a^{2} \left (\int \frac{1}{\sqrt{e \cot{\left (c + d x \right )}}}\, dx + \int \frac{2 \cot{\left (c + d x \right )}}{\sqrt{e \cot{\left (c + d x \right )}}}\, dx + \int \frac{\cot ^{2}{\left (c + d x \right )}}{\sqrt{e \cot{\left (c + d x \right )}}}\, dx\right ) \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 (a \cot \left (d x + c\right ) + a\right )}^{2}}{\sqrt{e \cot \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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