Optimal. Leaf size=95 \[ \frac{i \tanh ^{-1}\left (\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right )}{\sqrt{2-3 i} d}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right )}{\sqrt{2+3 i} d} \]
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Rubi [A] time = 0.0949088, 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, 208} \[ \frac{i \tanh ^{-1}\left (\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right )}{\sqrt{2-3 i} d}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right )}{\sqrt{2+3 i} d} \]
Antiderivative was successfully verified.
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Rule 3575
Rule 910
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{\tan (c+d x)}}{\sqrt{3+2 \tan (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{3+2 x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{2 (i-x) \sqrt{x} \sqrt{3+2 x}}+\frac{1}{2 \sqrt{x} (i+x) \sqrt{3+2 x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{x} \sqrt{3+2 x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (i+x) \sqrt{3+2 x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac{\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{\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 \tanh ^{-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{i \tanh ^{-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 = 0.117729, size = 95, normalized size = 1. \[ \frac{i \tan ^{-1}\left (\frac{\sqrt{-2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right )}{\sqrt{-2+3 i} d}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right )}{\sqrt{2+3 i} d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 479, normalized size = 5. \begin{align*} \text{result too large to display} \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{2 \, \tan \left (d x + c\right ) + 3}}\,{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{2 \tan{\left (c + d x \right )} + 3}}\, 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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