Optimal. Leaf size=351 \[ -\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {5}{2} a^2 \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}-\frac {25}{4} a^2 \left (\frac {1}{a x}+1\right )^{3/4} \sqrt [4]{1-\frac {1}{a x}}-\frac {25 a^2 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}}+\frac {25 a^2 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}}-\frac {25 a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{4 \sqrt {2}}+\frac {25 a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{4 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, 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 = {6171, 78, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {5}{2} a^2 \left (\frac {1}{a x}+1\right )^{3/4} \left (1-\frac {1}{a x}\right )^{5/4}-\frac {25}{4} a^2 \left (\frac {1}{a x}+1\right )^{3/4} \sqrt [4]{1-\frac {1}{a x}}-\frac {25 a^2 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}}+\frac {25 a^2 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}}-\frac {25 a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{4 \sqrt {2}}+\frac {25 a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 78
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^3} \, dx &=-\operatorname {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) \operatorname {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) \operatorname {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) \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 {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 ) \operatorname {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 ) \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 {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 ) \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}{4} \left (25 a^2\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 {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 ) \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}{8} \left (25 a^2\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 (25 a^2\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 )}{8 \sqrt {2}}-\frac {\left (25 a^2\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 )}{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 ) \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 )}{4 \sqrt {2}}-\frac {\left (25 a^2\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 )}{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*}
________________________________________________________________________________________
Mathematica [C] time = 0.15, size = 101, normalized size = 0.29 \[ -\frac {8}{3} a^2 e^{-\frac {1}{2} \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-e^{2 \coth ^{-1}(a x)}\right )+e^{2 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-e^{2 \coth ^{-1}(a x)}\right )+2 e^{2 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},3;\frac {7}{4};-e^{2 \coth ^{-1}(a x)}\right )+3\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 405, normalized size = 1.15 \[ -\frac {100 \, \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} x^{2} \arctan \left (-\frac {a^{8} + \sqrt {2} {\left (a^{8}\right )}^{\frac {3}{4}} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \sqrt {2} {\left (a^{8}\right )}^{\frac {3}{4}} \sqrt {a^{4} \sqrt {\frac {a x - 1}{a x + 1}} + \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {a^{8}}}}{a^{8}}\right ) + 100 \, \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} x^{2} \arctan \left (\frac {a^{8} - \sqrt {2} {\left (a^{8}\right )}^{\frac {3}{4}} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {2} {\left (a^{8}\right )}^{\frac {3}{4}} \sqrt {a^{4} \sqrt {\frac {a x - 1}{a x + 1}} - \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {a^{8}}}}{a^{8}}\right ) - 25 \, \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} x^{2} \log \left (625 \, a^{4} \sqrt {\frac {a x - 1}{a x + 1}} + 625 \, \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 625 \, \sqrt {a^{8}}\right ) + 25 \, \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} x^{2} \log \left (625 \, a^{4} \sqrt {\frac {a x - 1}{a x + 1}} - 625 \, \sqrt {2} {\left (a^{8}\right )}^{\frac {1}{4}} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 625 \, \sqrt {a^{8}}\right ) + 4 \, {\left (43 \, a^{2} x^{2} + 9 \, a x - 2\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{16 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 243, normalized size = 0.69 \[ \frac {1}{16} \, {\left (50 \, \sqrt {2} a \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 50 \, \sqrt {2} a \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 25 \, \sqrt {2} a \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 25 \, \sqrt {2} a \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 128 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \frac {8 \, {\left (\frac {13 \, {\left (a x - 1\right )} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} + 9 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.37, 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.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 247, normalized size = 0.70 \[ \frac {1}{16} \, {\left (50 \, \sqrt {2} a \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 50 \, \sqrt {2} a \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 25 \, \sqrt {2} a \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 25 \, \sqrt {2} a \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 128 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \frac {8 \, {\left (13 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 9 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {2 \, {\left (a x - 1\right )}}{a x + 1} + \frac {{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 153, normalized size = 0.44 \[ -8\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-\frac {\frac {9\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{2}+\frac {13\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{2}}{\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2\,\left (a\,x-1\right )}{a\,x+1}+1}-\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )\,25{}\mathrm {i}}{4}-\frac {25\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________