Optimal. Leaf size=271 \[ -\frac{(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}+\frac{a \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{4 \sqrt{2}}-\frac{a \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{4 \sqrt{2}}-\frac{1}{2} a \tan ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt{2}}-\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{2 \sqrt{2}}-\frac{1}{2} a \tanh ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right ) \]
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Rubi [A] time = 0.129204, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.929, Rules used = {6126, 94, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ -\frac{(1-a x)^{7/8} \sqrt [8]{a x+1}}{x}+\frac{a \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{4 \sqrt{2}}-\frac{a \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{4 \sqrt{2}}-\frac{1}{2} a \tan ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt{2}}-\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{2 \sqrt{2}}-\frac{1}{2} a \tanh ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right ) \]
Antiderivative was successfully verified.
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Rule 6126
Rule 94
Rule 93
Rule 214
Rule 212
Rule 206
Rule 203
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{4} \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac{\sqrt [8]{1+a x}}{x^2 \sqrt [8]{1-a x}} \, dx\\ &=-\frac{(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}+\frac{1}{4} a \int \frac{1}{x \sqrt [8]{1-a x} (1+a x)^{7/8}} \, dx\\ &=-\frac{(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}+(2 a) \operatorname{Subst}\left (\int \frac{1}{-1+x^8} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )\\ &=-\frac{(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-a \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-a \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )\\ &=-\frac{(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )\\ &=-\frac{(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-\frac{1}{2} a \tan ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac{1}{2} a \tanh ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{4 \sqrt{2}}+\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{4 \sqrt{2}}\\ &=-\frac{(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-\frac{1}{2} a \tan ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac{1}{2} a \tanh ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac{a \log \left (1-\frac{\sqrt{2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}+\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{4 \sqrt{2}}-\frac{a \log \left (1+\frac{\sqrt{2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}+\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{4 \sqrt{2}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt{2}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt{2}}\\ &=-\frac{(1-a x)^{7/8} \sqrt [8]{1+a x}}{x}-\frac{1}{2} a \tan ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt{2}}-\frac{a \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{2 \sqrt{2}}-\frac{1}{2} a \tanh ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac{a \log \left (1-\frac{\sqrt{2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}+\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{4 \sqrt{2}}-\frac{a \log \left (1+\frac{\sqrt{2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}+\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{4 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.016594, size = 58, normalized size = 0.21 \[ -\frac{(1-a x)^{7/8} \left (2 a x \text{Hypergeometric2F1}\left (\frac{7}{8},1,\frac{15}{8},\frac{1-a x}{a x+1}\right )+7 a x+7\right )}{7 x (a x+1)^{7/8}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt [4]{{(ax+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{\left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92928, size = 1359, normalized size = 5.01 \begin{align*} -\frac{4 \, a x \arctan \left (\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}}\right ) + 2 \, a x \log \left (\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + 1\right ) - 2 \, a x \log \left (\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} - 1\right ) - 4 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \arctan \left (-\frac{a^{4} + \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} \sqrt{a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{4}}}}{a^{4}}\right ) - 4 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \arctan \left (\frac{a^{4} - \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} \sqrt{a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{4}}}}{a^{4}}\right ) + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \log \left (a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{4}}\right ) - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \log \left (a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{4}}\right ) - 8 \,{\left (a x - 1\right )} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}}}{8 \, x} \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 [4]{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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