Optimal. Leaf size=83 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{a d \sqrt{e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} (\cot (c+d x)+1)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{\sqrt{2} a d \sqrt{e}} \]
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Rubi [A] time = 0.218891, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3574, 3532, 208, 3634, 63, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{a d \sqrt{e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} (\cot (c+d x)+1)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{\sqrt{2} a d \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 3574
Rule 3532
Rule 208
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx &=\frac{1}{2} \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx+\frac{\int \frac{a-a \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{2 a^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{2 a^2-e x^2} \, dx,x,\frac{a+a \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{d}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} (1+\cot (c+d x))}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{\sqrt{2} a d \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{1}{a+\frac{a x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d e}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{a d \sqrt{e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} (1+\cot (c+d x))}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{\sqrt{2} a d \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.49418, size = 107, normalized size = 1.29 \[ -\frac{\sqrt{\cot (c+d x)} \left (\sqrt{2} \left (\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\log \left (-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right )\right )+4 \tan ^{-1}\left (\sqrt{\cot (c+d x)}\right )\right )}{4 a d \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 365, normalized size = 4.4 \begin{align*} -{\frac{\sqrt{2}}{8\,dae}\sqrt [4]{{e}^{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{\sqrt{2}}{4\,dae}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{\sqrt{2}}{4\,dae}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{\sqrt{2}}{8\,da}\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{\sqrt{2}}{4\,da}\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{\sqrt{2}}{4\,da}\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{1}{da}\arctan \left ({\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \right ){\frac{1}{\sqrt{e}}}} \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 [A] time = 1.43016, size = 819, normalized size = 9.87 \begin{align*} \left [\frac{\sqrt{2} \sqrt{-e} \arctan \left (\frac{\sqrt{2} \sqrt{-e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}{\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \,{\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) - \sqrt{-e} \log \left (\frac{e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) + 2 \, \sqrt{-e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right )}{2 \, a d e}, \frac{\sqrt{2} \sqrt{e} \log \left (\sqrt{2} \sqrt{e} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}{\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} + 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) - 4 \, \sqrt{e} \arctan \left (\frac{\sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt{e}}\right )}{4 \, a d e}\right ] \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{1}{\sqrt{e \cot{\left (c + d x \right )}} \cot{\left (c + d x \right )} + \sqrt{e \cot{\left (c + d x \right )}}}\, dx}{a} \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{1}{{\left (a \cot \left (d x + c\right ) + a\right )} \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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