Optimal. Leaf size=142 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {1}{2} x^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}+\frac {x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{4 a} \]
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Rubi [A] time = 0.06, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6171, 96, 94, 93, 212, 206, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {1}{2} x^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}+\frac {x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{4 a} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 203
Rule 206
Rule 212
Rule 6171
Rubi steps
\begin {align*} \int e^{\frac {1}{2} \coth ^{-1}(a x)} x \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt [4]{1+\frac {x}{a}}}{x^3 \sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4} x^2-\frac {\operatorname {Subst}\left (\int \frac {\sqrt [4]{1+\frac {x}{a}}}{x^2 \sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{4 a}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{4 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4} x^2-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{8 a^2}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{4 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4} x^2-\frac {\operatorname {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 )}{2 a^2}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{4 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4} x^2+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{4 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4} x^2+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 66, normalized size = 0.46 \[ \frac {\frac {2 e^{\frac {1}{2} \coth ^{-1}(a x)} \left (5 e^{2 \coth ^{-1}(a x)}-1\right )}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}+\tan ^{-1}\left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+\tanh ^{-1}\left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{4 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 93, normalized size = 0.65 \[ \frac {2 \, {\left (2 \, a^{2} x^{2} + 5 \, a x + 3\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}} - 2 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) + \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 139, normalized size = 0.98 \[ -\frac {1}{8} \, a {\left (\frac {2 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{3}} - \frac {\log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} + \frac {\log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{3}} + \frac {4 \, {\left (\frac {{\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - 5 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 149, normalized size = 1.05 \[ \frac {1}{8} \, a {\left (\frac {4 \, {\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{4}} - 5 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{3}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} - \frac {2 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{3}} + \frac {\log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} - \frac {\log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 120, normalized size = 0.85 \[ \frac {\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/4}}{2}-\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/4}}{2}}{a^2+\frac {a^2\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,a^2\,\left (a\,x-1\right )}{a\,x+1}}-\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4\,a^2}+\frac {\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4\,a^2} \]
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 \[ \int \frac {x}{\sqrt [4]{\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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