Optimal. Leaf size=356 \[ \frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}+\frac {3}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}+\frac {3 a^3 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{16 \sqrt {2}}-\frac {3 a^3 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{16 \sqrt {2}}-\frac {3 a^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{8 \sqrt {2}}+\frac {3 a^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}}{3 x} \]
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Rubi [A] time = 0.29, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6171, 90, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}+\frac {3}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}}{3 x}+\frac {3 a^3 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{16 \sqrt {2}}-\frac {3 a^3 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{16 \sqrt {2}}-\frac {3 a^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{8 \sqrt {2}}+\frac {3 a^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 204
Rule 297
Rule 331
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6171
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} \coth ^{-1}(a x)}}{x^4} \, dx &=-\operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {x}{a}}}{\sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}{3 x}+\frac {1}{3} a^2 \operatorname {Subst}\left (\int \frac {\left (-1-\frac {x}{2 a}\right ) \sqrt [4]{1+\frac {x}{a}}}{\sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}{3 x}-\frac {1}{8} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+\frac {x}{a}}}{\sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}{3 x}-\frac {1}{16} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}{3 x}+\frac {1}{4} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-\frac {1}{a x}}\right )\\ &=\frac {3}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}{3 x}+\frac {1}{4} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )\\ &=\frac {3}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}{3 x}-\frac {1}{8} \left (3 a^3\right ) \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 {1}{8} \left (3 a^3\right ) \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 {3}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}{3 x}+\frac {1}{16} \left (3 a^3\right ) \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}{16} \left (3 a^3\right ) \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 {\left (3 a^3\right ) \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 )}{16 \sqrt {2}}+\frac {\left (3 a^3\right ) \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 )}{16 \sqrt {2}}\\ &=\frac {3}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}{3 x}+\frac {3 a^3 \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 )}{16 \sqrt {2}}-\frac {3 a^3 \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 )}{16 \sqrt {2}}+\frac {\left (3 a^3\right ) \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 )}{8 \sqrt {2}}-\frac {\left (3 a^3\right ) \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 )}{8 \sqrt {2}}\\ &=\frac {3}{8} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}}{3 x}-\frac {3 a^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {3 a^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {3 a^3 \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 )}{16 \sqrt {2}}-\frac {3 a^3 \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 )}{16 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 93, normalized size = 0.26 \[ \frac {1}{96} a^3 \left (9 \text {RootSum}\left [\text {$\#$1}^4+1\& ,\frac {\coth ^{-1}(a x)-2 \log \left (e^{\frac {1}{2} \coth ^{-1}(a x)}-\text {$\#$1}\right )}{\text {$\#$1}^3}\& \right ]+\frac {8 e^{\frac {1}{2} \coth ^{-1}(a x)} \left (6 e^{2 \coth ^{-1}(a x)}+29 e^{4 \coth ^{-1}(a x)}+9\right )}{\left (e^{2 \coth ^{-1}(a x)}+1\right )^3}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.42, size = 427, normalized size = 1.20 \[ -\frac {36 \, \sqrt {2} {\left (a^{12}\right )}^{\frac {1}{4}} x^{3} \arctan \left (-\frac {a^{12} + \sqrt {2} {\left (a^{12}\right )}^{\frac {1}{4}} a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \sqrt {2} \sqrt {a^{18} \sqrt {\frac {a x - 1}{a x + 1}} + \sqrt {a^{12}} a^{12} + \sqrt {2} {\left (a^{12}\right )}^{\frac {3}{4}} a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} {\left (a^{12}\right )}^{\frac {1}{4}}}{a^{12}}\right ) + 36 \, \sqrt {2} {\left (a^{12}\right )}^{\frac {1}{4}} x^{3} \arctan \left (\frac {a^{12} - \sqrt {2} {\left (a^{12}\right )}^{\frac {1}{4}} a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {2} \sqrt {a^{18} \sqrt {\frac {a x - 1}{a x + 1}} + \sqrt {a^{12}} a^{12} - \sqrt {2} {\left (a^{12}\right )}^{\frac {3}{4}} a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} {\left (a^{12}\right )}^{\frac {1}{4}}}{a^{12}}\right ) + 9 \, \sqrt {2} {\left (a^{12}\right )}^{\frac {1}{4}} x^{3} \log \left (729 \, a^{18} \sqrt {\frac {a x - 1}{a x + 1}} + 729 \, \sqrt {a^{12}} a^{12} + 729 \, \sqrt {2} {\left (a^{12}\right )}^{\frac {3}{4}} a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 9 \, \sqrt {2} {\left (a^{12}\right )}^{\frac {1}{4}} x^{3} \log \left (729 \, a^{18} \sqrt {\frac {a x - 1}{a x + 1}} + 729 \, \sqrt {a^{12}} a^{12} - 729 \, \sqrt {2} {\left (a^{12}\right )}^{\frac {3}{4}} a^{9} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 4 \, {\left (11 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 18 \, a x + 8\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{96 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 271, normalized size = 0.76 \[ \frac {1}{96} \, {\left (18 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 18 \, \sqrt {2} a^{2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 9 \, \sqrt {2} a^{2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 9 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {8 \, {\left (\frac {6 \, {\left (a x - 1\right )} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} + \frac {9 \, {\left (a x - 1\right )}^{2} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} + 29 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}} x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 270, normalized size = 0.76 \[ \frac {1}{96} \, {\left (9 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right )\right )} a^{2} + \frac {8 \, {\left (9 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{4}} + 6 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{4}} + 29 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 168, normalized size = 0.47 \[ \frac {\frac {29\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/4}}{12}+\frac {a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/4}}{2}+\frac {3\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/4}}{4}}{\frac {3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {3\,\left (a\,x-1\right )}{a\,x+1}+1}+\frac {3\,{\left (-1\right )}^{1/4}\,a^3\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8}-\frac {3\,{\left (-1\right )}^{1/4}\,a^3\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8} \]
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 \[ \int \frac {1}{x^{4} \sqrt [4]{\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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