Optimal. Leaf size=89 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{-2 \tan (c+d x)-3}}\right )}{\sqrt{2-3 i} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{-2 \tan (c+d x)-3}}\right )}{\sqrt{2+3 i} d} \]
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Rubi [A] time = 0.101034, antiderivative size = 89, 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, 912, 93, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{-2 \tan (c+d x)-3}}\right )}{\sqrt{2-3 i} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{-2 \tan (c+d x)-3}}\right )}{\sqrt{2+3 i} d} \]
Antiderivative was successfully verified.
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Rule 3575
Rule 912
Rule 93
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3-2 \tan (c+d x)} \sqrt{\tan (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-3-2 x} \sqrt{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{i}{2 \sqrt{-3-2 x} (i-x) \sqrt{x}}+\frac{i}{2 \sqrt{-3-2 x} \sqrt{x} (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{i \operatorname{Subst}\left (\int \frac{1}{\sqrt{-3-2 x} (i-x) \sqrt{x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{i \operatorname{Subst}\left (\int \frac{1}{\sqrt{-3-2 x} \sqrt{x} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{i \operatorname{Subst}\left (\int \frac{1}{i-(3-2 i) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{-3-2 \tan (c+d x)}}\right )}{d}+\frac{i \operatorname{Subst}\left (\int \frac{1}{i+(3+2 i) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{-3-2 \tan (c+d x)}}\right )}{d}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{-3-2 \tan (c+d x)}}\right )}{\sqrt{2-3 i} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{-3-2 \tan (c+d x)}}\right )}{\sqrt{2+3 i} d}\\ \end{align*}
Mathematica [A] time = 0.141591, size = 101, normalized size = 1.13 \[ \frac{\sqrt{-2+3 i} \tanh ^{-1}\left (\frac{\sqrt{-\frac{2}{13}+\frac{3 i}{13}} \sqrt{-2 \tan (c+d x)-3}}{\sqrt{\tan (c+d x)}}\right )-\sqrt{2+3 i} \tan ^{-1}\left (\frac{\sqrt{\frac{2}{13}+\frac{3 i}{13}} \sqrt{-2 \tan (c+d x)-3}}{\sqrt{\tan (c+d x)}}\right )}{\sqrt{13} d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.078, size = 435, normalized size = 4.9 \begin{align*} -{\frac{\sqrt{13}-2+3\,\tan \left ( dx+c \right ) }{2\,d\sqrt{2\,\sqrt{13}+4} \left ( 3+2\,\tan \left ( dx+c \right ) \right ) \left ( 17\,\sqrt{13}-52 \right ) }\sqrt{-3-2\,\tan \left ( dx+c \right ) }\sqrt{-{\frac{\tan \left ( dx+c \right ) \left ( 3+2\,\tan \left ( dx+c \right ) \right ) }{ \left ( \sqrt{13}-2+3\,\tan \left ( dx+c \right ) \right ) ^{2}}}} \left ( 4\,\sqrt{13}\sqrt{2\,\sqrt{13}+4}\sqrt{-4+2\,\sqrt{13}}{\it Artanh} \left ({\frac{ \left ( \sqrt{13}+2 \right ) \left ( 17\,\sqrt{13}-52 \right ) \left ( \sqrt{13}+2-3\,\tan \left ( dx+c \right ) \right ) \sqrt{13}}{351\,\sqrt{-4+2\,\sqrt{13}} \left ( \sqrt{13}-2+3\,\tan \left ( dx+c \right ) \right ) }{\frac{1}{\sqrt{-{\frac{\tan \left ( dx+c \right ) \left ( 3+2\,\tan \left ( dx+c \right ) \right ) }{ \left ( \sqrt{13}-2+3\,\tan \left ( dx+c \right ) \right ) ^{2}}}}}}} \right ) -17\,\sqrt{2\,\sqrt{13}+4}\sqrt{-4+2\,\sqrt{13}}{\it Artanh} \left ({\frac{ \left ( \sqrt{13}+2 \right ) \left ( 17\,\sqrt{13}-52 \right ) \left ( \sqrt{13}+2-3\,\tan \left ( dx+c \right ) \right ) \sqrt{13}}{351\,\sqrt{-4+2\,\sqrt{13}} \left ( \sqrt{13}-2+3\,\tan \left ( dx+c \right ) \right ) }{\frac{1}{\sqrt{-{\frac{\tan \left ( dx+c \right ) \left ( 3+2\,\tan \left ( dx+c \right ) \right ) }{ \left ( \sqrt{13}-2+3\,\tan \left ( dx+c \right ) \right ) ^{2}}}}}}} \right ) -18\,\arctan \left ( 6\,{\frac{\sqrt{13}}{\sqrt{26\,\sqrt{13}+52}}\sqrt{-{\frac{\tan \left ( dx+c \right ) \left ( 3+2\,\tan \left ( dx+c \right ) \right ) }{ \left ( \sqrt{13}-2+3\,\tan \left ( dx+c \right ) \right ) ^{2}}}}} \right ) \sqrt{13}+36\,\arctan \left ( 6\,{\frac{\sqrt{13}}{\sqrt{26\,\sqrt{13}+52}}\sqrt{-{\frac{\tan \left ( dx+c \right ) \left ( 3+2\,\tan \left ( dx+c \right ) \right ) }{ \left ( \sqrt{13}-2+3\,\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{1}{\sqrt{-2 \, \tan \left (d x + c\right ) - 3} \sqrt{\tan \left (d x + c\right )}}\,{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{1}{\sqrt{- 2 \tan{\left (c + d x \right )} - 3} \sqrt{\tan{\left (c + d x \right )}}}\, 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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