Optimal. Leaf size=268 \[ -a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {a \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}+\frac {a \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}+\frac {a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6306, 52, 65,
246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {a \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{\sqrt {2}}+\frac {a \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{\sqrt {2}}-a \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}}+\frac {a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6306
Rubi steps
\begin {align*} \int \frac {e^{-\frac {1}{2} \coth ^{-1}(a x)}}{x^2} \, dx &=-\text {Subst}\left (\int \frac {\sqrt [4]{1-\frac {x}{a}}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+(2 a) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-\frac {1}{a x}}\right )\\ &=-a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+(2 a) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )\\ &=-a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+a \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )+a \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )\\ &=-a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )-\frac {a \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}-\frac {a \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}\\ &=-a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}+\frac {a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}+\frac {a \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {a \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}\\ &=-a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}+\frac {a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}+\frac {a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 33, normalized size = 0.12 \begin {gather*} -\frac {8}{3} a e^{\frac {3}{2} \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-e^{2 \coth ^{-1}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 186, normalized size = 0.69 \begin {gather*} \frac {1}{4} \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \frac {8 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{\frac {a x - 1}{a x + 1} + 1}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 368, normalized size = 1.37 \begin {gather*} -\frac {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 {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} \sqrt {a^{2} \sqrt {\frac {a x - 1}{a x + 1}} + \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (\frac {a x - 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 {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} \sqrt {a^{2} \sqrt {\frac {a x - 1}{a x + 1}} - \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (\frac {a x - 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 {a x - 1}{a x + 1}} + \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (\frac {a x - 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 {a x - 1}{a x + 1}} - \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {a^{4}}\right ) + 4 \, {\left (a x + 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{\frac {a x - 1}{a x + 1}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 186, normalized size = 0.69 \begin {gather*} \frac {1}{4} \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \frac {8 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{\frac {a x - 1}{a x + 1} + 1}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 88, normalized size = 0.33 \begin {gather*} -\frac {2\,a\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{\frac {a\,x-1}{a\,x+1}+1}-{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )\,1{}\mathrm {i}-{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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