Optimal. Leaf size=351 \[ -\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {25}{4} a^2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {25 a^2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}+\frac {25 a^2 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}-\frac {25 a^2 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {25 a^2 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6306, 79, 52,
65, 246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {25 a^2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{4 \sqrt {2}}+\frac {25 a^2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{4 \sqrt {2}}-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {5}{2} a^2 \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}-\frac {25}{4} a^2 \left (\frac {1}{a x}+1\right )^{3/4} \sqrt [4]{1-\frac {1}{a x}}-\frac {25 a^2 \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 )}{8 \sqrt {2}}+\frac {25 a^2 \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 )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 79
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6306
Rubi steps
\begin {align*} \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx &=-\text {Subst}\left (\int \frac {x \left (1-\frac {x}{a}\right )^{5/4}}{\left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-(5 a) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/4}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{4} (25 a) \text {Subst}\left (\int \frac {\sqrt [4]{1-\frac {x}{a}}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {25}{4} a^2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{8} (25 a) \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 )\\ &=-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {25}{4} a^2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{2} \left (25 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-\frac {1}{a x}}\right )\\ &=-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {25}{4} a^2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{2} \left (25 a^2\right ) \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 )\\ &=-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {25}{4} a^2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{4} \left (25 a^2\right ) \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 )+\frac {1}{4} \left (25 a^2\right ) \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 )\\ &=-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {25}{4} a^2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{8} \left (25 a^2\right ) \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}{8} \left (25 a^2\right ) \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 {\left (25 a^2\right ) \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 )}{8 \sqrt {2}}-\frac {\left (25 a^2\right ) \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 )}{8 \sqrt {2}}\\ &=-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {25}{4} a^2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {25 a^2 \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 )}{8 \sqrt {2}}+\frac {25 a^2 \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 )}{8 \sqrt {2}}+\frac {\left (25 a^2\right ) \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 )}{4 \sqrt {2}}-\frac {\left (25 a^2\right ) \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 )}{4 \sqrt {2}}\\ &=-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {25}{4} a^2 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {25 a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}+\frac {25 a^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}-\frac {25 a^2 \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 )}{8 \sqrt {2}}+\frac {25 a^2 \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 )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.12, size = 94, normalized size = 0.27 \begin {gather*} a^2 \left (-\frac {e^{-\frac {1}{2} \coth ^{-1}(a x)} \left (16+45 e^{2 \coth ^{-1}(a x)}+25 e^{4 \coth ^{-1}(a x)}\right )}{2 \left (1+e^{2 \coth ^{-1}(a x)}\right )^2}+\frac {25}{16} \text {RootSum}\left [1+\text {$\#$1}^4\&,\frac {\coth ^{-1}(a x)+2 \log \left (e^{-\frac {1}{2} \coth ^{-1}(a x)}-\text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ]\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 247, normalized size = 0.70 \begin {gather*} \frac {1}{16} \, {\left (50 \, \sqrt {2} a \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 ) + 50 \, \sqrt {2} a \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 ) + 25 \, \sqrt {2} a \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 ) - 25 \, \sqrt {2} a \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 ) - 128 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \frac {8 \, {\left (13 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 9 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {2 \, {\left (a x - 1\right )}}{a x + 1} + \frac {{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 405, normalized size = 1.15 \begin {gather*} -\frac {100 \, \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 {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \sqrt {2} {\left (a^{8}\right )}^{\frac {3}{4}} \sqrt {a^{4} \sqrt {\frac {a x - 1}{a x + 1}} + \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {a^{8}}}}{a^{8}}\right ) + 100 \, \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 {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {2} {\left (a^{8}\right )}^{\frac {3}{4}} \sqrt {a^{4} \sqrt {\frac {a x - 1}{a x + 1}} - \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {a^{8}}}}{a^{8}}\right ) - 25 \, \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} x^{2} \log \left (625 \, a^{4} \sqrt {\frac {a x - 1}{a x + 1}} + 625 \, \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 625 \, \sqrt {a^{8}}\right ) + 25 \, \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} x^{2} \log \left (625 \, a^{4} \sqrt {\frac {a x - 1}{a x + 1}} - 625 \, \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 625 \, \sqrt {a^{8}}\right ) + 4 \, {\left (43 \, a^{2} x^{2} + 9 \, a x - 2\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{16 \, x^{2}} \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 {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 243, normalized size = 0.69 \begin {gather*} \frac {1}{16} \, {\left (50 \, \sqrt {2} a \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 ) + 50 \, \sqrt {2} a \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 ) + 25 \, \sqrt {2} a \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 ) - 25 \, \sqrt {2} a \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 ) - 128 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \frac {8 \, {\left (\frac {13 \, {\left (a x - 1\right )} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} + 9 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 153, normalized size = 0.44 \begin {gather*} -8\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-\frac {\frac {9\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{2}+\frac {13\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{2}}{\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2\,\left (a\,x-1\right )}{a\,x+1}+1}-\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )\,25{}\mathrm {i}}{4}-\frac {25\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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