Optimal. Leaf size=193 \[ \frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}-\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{a} \]
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Rubi [A] time = 0.142258, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {6140, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}-\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6140
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx &=\int \frac{1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{a}\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac{2 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}\\ &=\frac{\log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}-\frac{\log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}-\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}+\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac{\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}+\frac{\log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}-\frac{\log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}\\ \end{align*}
Mathematica [C] time = 0.0107849, size = 40, normalized size = 0.21 \[ -\frac{2\ 2^{3/4} \sqrt [4]{1-a x} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{4},\frac{5}{4},\frac{1}{2} (1-a x)\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.21, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \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{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.77645, size = 1183, normalized size = 6.13 \begin{align*} -2 \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a \sqrt{\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} +{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - 1\right ) - 2 \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a \sqrt{-\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} -{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + 1\right ) - \frac{1}{2} \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} +{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) + \frac{1}{2} \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} -{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\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*} \int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \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{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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