Optimal. Leaf size=89 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right )}{\sqrt{3-2 i} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right )}{\sqrt{3+2 i} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.106433, 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, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right )}{\sqrt{3-2 i} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right )}{\sqrt{3+2 i} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3575
Rule 912
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\tan (c+d x)} \sqrt{-2+3 \tan (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\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{i}{2 (i-x) \sqrt{x} \sqrt{-2+3 x}}+\frac{i}{2 \sqrt{x} (i+x) \sqrt{-2+3 x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{i \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{x} \sqrt{-2+3 x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{i \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (i+x) \sqrt{-2+3 x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{i \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 \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{\tanh ^{-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{\tanh ^{-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 = 0.132056, size = 89, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{-3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right )}{\sqrt{-3+2 i} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right )}{\sqrt{3+2 i} d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.105, size = 480, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \, \tan \left (d x + c\right ) - 2} \sqrt{\tan \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \tan{\left (c + d x \right )} - 2} \sqrt{\tan{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]