Optimal. Leaf size=312 \[ \frac{a^2 \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{32 \sqrt{2}}-\frac{a^2 \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{32 \sqrt{2}}-\frac{1}{16} a^2 \tan ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac{a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{16 \sqrt{2}}-\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{16 \sqrt{2}}-\frac{1}{16} a^2 \tanh ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac{(1-a x)^{7/8} (a x+1)^{9/8}}{2 x^2}-\frac{a (1-a x)^{7/8} \sqrt [8]{a x+1}}{8 x} \]
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Rubi [A] time = 0.152656, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6126, 96, 94, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ \frac{a^2 \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{32 \sqrt{2}}-\frac{a^2 \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{32 \sqrt{2}}-\frac{1}{16} a^2 \tan ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac{a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{16 \sqrt{2}}-\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{16 \sqrt{2}}-\frac{1}{16} a^2 \tanh ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac{(1-a x)^{7/8} (a x+1)^{9/8}}{2 x^2}-\frac{a (1-a x)^{7/8} \sqrt [8]{a x+1}}{8 x} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 96
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^3} \, dx &=\int \frac{\sqrt [8]{1+a x}}{x^3 \sqrt [8]{1-a x}} \, dx\\ &=-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{2 x^2}+\frac{1}{8} a \int \frac{\sqrt [8]{1+a x}}{x^2 \sqrt [8]{1-a x}} \, dx\\ &=-\frac{a (1-a x)^{7/8} \sqrt [8]{1+a x}}{8 x}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{2 x^2}+\frac{1}{32} a^2 \int \frac{1}{x \sqrt [8]{1-a x} (1+a x)^{7/8}} \, dx\\ &=-\frac{a (1-a x)^{7/8} \sqrt [8]{1+a x}}{8 x}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{2 x^2}+\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{1}{-1+x^8} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )\\ &=-\frac{a (1-a x)^{7/8} \sqrt [8]{1+a x}}{8 x}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{2 x^2}-\frac{1}{8} a^2 \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}{8} a^2 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )\\ &=-\frac{a (1-a x)^{7/8} \sqrt [8]{1+a x}}{8 x}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{2 x^2}-\frac{1}{16} a^2 \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}{16} a^2 \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}{16} a^2 \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}{16} a^2 \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{a (1-a x)^{7/8} \sqrt [8]{1+a x}}{8 x}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{2 x^2}-\frac{1}{16} a^2 \tan ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac{1}{16} a^2 \tanh ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac{1}{32} a^2 \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}{32} a^2 \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^2 \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 )}{32 \sqrt{2}}+\frac{a^2 \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 )}{32 \sqrt{2}}\\ &=-\frac{a (1-a x)^{7/8} \sqrt [8]{1+a x}}{8 x}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{2 x^2}-\frac{1}{16} a^2 \tan ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )-\frac{1}{16} a^2 \tanh ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac{a^2 \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 )}{32 \sqrt{2}}-\frac{a^2 \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 )}{32 \sqrt{2}}-\frac{a^2 \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 )}{16 \sqrt{2}}+\frac{a^2 \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 )}{16 \sqrt{2}}\\ &=-\frac{a (1-a x)^{7/8} \sqrt [8]{1+a x}}{8 x}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{2 x^2}-\frac{1}{16} a^2 \tan ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac{a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{16 \sqrt{2}}-\frac{a^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )}{16 \sqrt{2}}-\frac{1}{16} a^2 \tanh ^{-1}\left (\frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}\right )+\frac{a^2 \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 )}{32 \sqrt{2}}-\frac{a^2 \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 )}{32 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0209067, size = 73, normalized size = 0.23 \[ -\frac{(1-a x)^{7/8} \left (2 a^2 x^2 \text{Hypergeometric2F1}\left (\frac{7}{8},1,\frac{15}{8},\frac{1-a x}{a x+1}\right )+7 \left (5 a^2 x^2+9 a x+4\right )\right )}{56 x^2 (a x+1)^{7/8}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.37574, size = 1423, normalized size = 4.56 \begin{align*} -\frac{4 \, a^{2} x^{2} \arctan \left (\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}}\right ) + 2 \, a^{2} x^{2} \log \left (\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + 1\right ) - 2 \, a^{2} x^{2} \log \left (\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} - 1\right ) - 4 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \arctan \left (-\frac{a^{8} + \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} - \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} \sqrt{a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}}}{a^{8}}\right ) - 4 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \arctan \left (\frac{a^{8} - \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} \sqrt{a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}}}{a^{8}}\right ) + \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \log \left (a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}\right ) - \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \log \left (a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}\right ) - 8 \,{\left (5 \, a^{2} x^{2} - a x - 4\right )} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}}}{64 \, x^{2}} \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(-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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