Optimal. Leaf size=223 \[ -\frac{\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}+\frac{3 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt{2} a}-\frac{3 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt{2} a}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt{2} a}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a} \]
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Rubi [A] time = 0.134412, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6125, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}+\frac{3 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt{2} a}-\frac{3 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt{2} a}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt{2} a}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a} \]
Antiderivative was successfully verified.
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Rule 6125
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^{\frac{3}{2} \tanh ^{-1}(a x)} \, dx &=\int \frac{(1+a x)^{3/4}}{(1-a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}+\frac{3}{2} \int \frac{1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}-\frac{6 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{a}\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}-\frac{6 \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{\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}-\frac{3 \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{3 \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{\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}-\frac{3 \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 )}{2 a}-\frac{3 \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 )}{2 a}+\frac{3 \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 )}{2 \sqrt{2} a}+\frac{3 \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 )}{2 \sqrt{2} a}\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}+\frac{3 \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 )}{2 \sqrt{2} a}-\frac{3 \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 )}{2 \sqrt{2} a}-\frac{3 \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 )}{\sqrt{2} a}+\frac{3 \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 )}{\sqrt{2} a}\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}-\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}+\frac{3 \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 )}{2 \sqrt{2} a}-\frac{3 \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 )}{2 \sqrt{2} a}\\ \end{align*}
Mathematica [C] time = 0.0594765, size = 48, normalized size = 0.22 \[ \frac{8 e^{\frac{3}{2} \tanh ^{-1}(a x)} \left (\text{Hypergeometric2F1}\left (\frac{3}{4},2,\frac{7}{4},-e^{2 \tanh ^{-1}(a x)}\right )-\frac{1}{e^{2 \tanh ^{-1}(a x)}+1}\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) ^{{\frac{3}{2}}}\, 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 \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87931, size = 1281, normalized size = 5.74 \begin{align*} -\frac{12 \, \sqrt{2} a \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 ) + 12 \, \sqrt{2} a \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 ) + 3 \, \sqrt{2} a \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 ) - 3 \, \sqrt{2} a \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 ) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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