Optimal. Leaf size=299 \[ \frac {4 a \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{\frac {1}{a x}+1}}+5 a \left (\frac {1}{a x}+1\right )^{3/4} \sqrt [4]{1-\frac {1}{a x}}+\frac {5 a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}}-\frac {5 a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}}+\frac {5 a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{\sqrt {2}}-\frac {5 a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.24, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6171, 47, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac {4 a \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{\frac {1}{a x}+1}}+5 a \left (\frac {1}{a x}+1\right )^{3/4} \sqrt [4]{1-\frac {1}{a x}}+\frac {5 a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}}-\frac {5 a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}}+\frac {5 a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{\sqrt {2}}-\frac {5 a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 204
Rule 211
Rule 240
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6171
Rubi steps
\begin {align*} \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/4}}{\left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {4 a \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac {1}{a x}}}+5 \operatorname {Subst}\left (\int \frac {\sqrt [4]{1-\frac {x}{a}}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {4 a \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac {1}{a x}}}+5 a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {5}{2} \operatorname {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 {4 a \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac {1}{a x}}}+5 a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-(10 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-\frac {1}{a x}}\right )\\ &=\frac {4 a \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac {1}{a x}}}+5 a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-(10 a) \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 )\\ &=\frac {4 a \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac {1}{a x}}}+5 a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-(5 a) \operatorname {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 )-(5 a) \operatorname {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 {4 a \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac {1}{a x}}}+5 a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{2} (5 a) \operatorname {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} (5 a) \operatorname {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 {(5 a) \operatorname {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 {(5 a) \operatorname {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 {4 a \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac {1}{a x}}}+5 a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {5 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 {5 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 {(5 a) \operatorname {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 {(5 a) \operatorname {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 {4 a \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac {1}{a x}}}+5 a \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {5 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 {5 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 {5 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 {5 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] time = 0.08, size = 31, normalized size = 0.10 \[ 8 a e^{-\frac {1}{2} \coth ^{-1}(a x)} \, _2F_1\left (-\frac {1}{4},2;\frac {3}{4};-e^{2 \coth ^{-1}(a x)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.54, size = 377, normalized size = 1.26 \[ \frac {20 \, \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {a^{4} + \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} \sqrt {a^{2} \sqrt {\frac {a x - 1}{a x + 1}} + \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {a^{4}}}}{a^{4}}\right ) + 20 \, \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \arctan \left (\frac {a^{4} - \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} \sqrt {a^{2} \sqrt {\frac {a x - 1}{a x + 1}} - \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {a^{4}}}}{a^{4}}\right ) - 5 \, \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \log \left (25 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}} + 25 \, \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 25 \, \sqrt {a^{4}}\right ) + 5 \, \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \log \left (25 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}} - 25 \, \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 25 \, \sqrt {a^{4}}\right ) + 4 \, {\left (9 \, a x + 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{4 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 204, normalized size = 0.68 \[ -\frac {1}{4} \, {\left (10 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 10 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 5 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 5 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 32 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \frac {8 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{\frac {a x - 1}{a x + 1} + 1}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 204, normalized size = 0.68 \[ -\frac {1}{4} \, {\left (10 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 10 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 5 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 5 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 32 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \frac {8 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{\frac {a x - 1}{a x + 1} + 1}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 106, normalized size = 0.35 \[ 8\,a\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}+5\,{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )+\frac {2\,a\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{\frac {a\,x-1}{a\,x+1}+1}+{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )\,5{}\mathrm {i} \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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