Optimal. Leaf size=95 \[ \frac{i \tan ^{-1}\left (\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{-3 \tan (c+d x)-2}}\right )}{\sqrt{3-2 i} d}-\frac{i \tan ^{-1}\left (\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{-3 \tan (c+d x)-2}}\right )}{\sqrt{3+2 i} d} \]
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Rubi [A] time = 0.0984915, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3575, 910, 93, 205} \[ \frac{i \tan ^{-1}\left (\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{-3 \tan (c+d x)-2}}\right )}{\sqrt{3-2 i} d}-\frac{i \tan ^{-1}\left (\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{-3 \tan (c+d x)-2}}\right )}{\sqrt{3+2 i} d} \]
Antiderivative was successfully verified.
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Rule 3575
Rule 910
Rule 93
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{\tan (c+d x)}}{\sqrt{-2-3 \tan (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{-2-3 x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{2 \sqrt{-2-3 x} (i-x) \sqrt{x}}+\frac{1}{2 \sqrt{-2-3 x} \sqrt{x} (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-2-3 x} (i-x) \sqrt{x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-2-3 x} \sqrt{x} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{i-(2-3 i) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{-2-3 \tan (c+d x)}}\right )}{d}-\frac{\operatorname{Subst}\left (\int \frac{1}{i+(2+3 i) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{-2-3 \tan (c+d x)}}\right )}{d}\\ &=\frac{i \tan ^{-1}\left (\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{-2-3 \tan (c+d x)}}\right )}{\sqrt{3-2 i} d}-\frac{i \tan ^{-1}\left (\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{-2-3 \tan (c+d x)}}\right )}{\sqrt{3+2 i} d}\\ \end{align*}
Mathematica [A] time = 0.102066, size = 103, normalized size = 1.08 \[ -\frac{i \left (\sqrt{3+2 i} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{13}+\frac{2 i}{13}} \sqrt{-3 \tan (c+d x)-2}}{\sqrt{\tan (c+d x)}}\right )+\sqrt{-3+2 i} \tanh ^{-1}\left (\frac{\sqrt{-\frac{3}{13}+\frac{2 i}{13}} \sqrt{-3 \tan (c+d x)-2}}{\sqrt{\tan (c+d x)}}\right )\right )}{\sqrt{13} d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 434, normalized size = 4.6 \begin{align*}{\frac{\sqrt{13}-3+2\,\tan \left ( dx+c \right ) }{2\,d \left ( 2+3\,\tan \left ( dx+c \right ) \right ) \sqrt{2\,\sqrt{13}+6} \left ( 11\,\sqrt{13}-39 \right ) }\sqrt{-2-3\,\tan \left ( dx+c \right ) }\sqrt{-{\frac{\tan \left ( dx+c \right ) \left ( 2+3\,\tan \left ( dx+c \right ) \right ) }{ \left ( \sqrt{13}-3+2\,\tan \left ( dx+c \right ) \right ) ^{2}}}} \left ( \sqrt{13}\sqrt{2\,\sqrt{13}+6}\sqrt{-6+2\,\sqrt{13}}{\it Artanh} \left ({\frac{ \left ( \sqrt{13}+3 \right ) \left ( 11\,\sqrt{13}-39 \right ) \left ( \sqrt{13}+3-2\,\tan \left ( dx+c \right ) \right ) \sqrt{13}}{ \left ( 52\,\sqrt{13}-156+104\,\tan \left ( dx+c \right ) \right ) \sqrt{-6+2\,\sqrt{13}}}{\frac{1}{\sqrt{-{\frac{\tan \left ( dx+c \right ) \left ( 2+3\,\tan \left ( dx+c \right ) \right ) }{ \left ( \sqrt{13}-3+2\,\tan \left ( dx+c \right ) \right ) ^{2}}}}}}} \right ) -3\,\sqrt{2\,\sqrt{13}+6}\sqrt{-6+2\,\sqrt{13}}{\it Artanh} \left ({\frac{ \left ( \sqrt{13}+3 \right ) \left ( 11\,\sqrt{13}-39 \right ) \left ( \sqrt{13}+3-2\,\tan \left ( dx+c \right ) \right ) \sqrt{13}}{ \left ( 52\,\sqrt{13}-156+104\,\tan \left ( dx+c \right ) \right ) \sqrt{-6+2\,\sqrt{13}}}{\frac{1}{\sqrt{-{\frac{\tan \left ( dx+c \right ) \left ( 2+3\,\tan \left ( dx+c \right ) \right ) }{ \left ( \sqrt{13}-3+2\,\tan \left ( dx+c \right ) \right ) ^{2}}}}}}} \right ) +12\,\arctan \left ( 4\,{\frac{\sqrt{13}}{\sqrt{26\,\sqrt{13}+78}}\sqrt{-{\frac{\tan \left ( dx+c \right ) \left ( 2+3\,\tan \left ( dx+c \right ) \right ) }{ \left ( \sqrt{13}-3+2\,\tan \left ( dx+c \right ) \right ) ^{2}}}}} \right ) \sqrt{13}-44\,\arctan \left ( 4\,{\frac{\sqrt{13}}{\sqrt{26\,\sqrt{13}+78}}\sqrt{-{\frac{\tan \left ( dx+c \right ) \left ( 2+3\,\tan \left ( dx+c \right ) \right ) }{ \left ( \sqrt{13}-3+2\,\tan \left ( dx+c \right ) \right ) ^{2}}}}} \right ) \right ){\frac{1}{\sqrt{\tan \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\tan \left (d x + c\right )}}{\sqrt{-3 \, \tan \left (d x + c\right ) - 2}}\,{d x} \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 \frac{\sqrt{\tan{\left (c + d x \right )}}}{\sqrt{- 3 \tan{\left (c + d x \right )} - 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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