Optimal. Leaf size=136 \[ \frac {19 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{8 a^2}+\frac {1}{4} x^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {51 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a^4}-\frac {6 x \sqrt {1-\frac {1}{a^2 x^2}}}{a^3}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^3 \left (a+\frac {1}{x}\right )} \]
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Rubi [A] time = 1.02, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6169, 6742, 266, 51, 63, 208, 271, 264, 651} \[ \frac {1}{4} x^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {19 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{8 a^2}-\frac {6 x \sqrt {1-\frac {1}{a^2 x^2}}}{a^3}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^3 \left (a+\frac {1}{x}\right )}+\frac {51 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 264
Rule 266
Rule 271
Rule 651
Rule 6169
Rule 6742
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} x^3 \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^5 \left (1+\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{x^5 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3}{a x^4 \sqrt {1-\frac {x^2}{a^2}}}+\frac {4}{a^2 x^3 \sqrt {1-\frac {x^2}{a^2}}}-\frac {4}{a^3 x^2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {4}{a^4 x \sqrt {1-\frac {x^2}{a^2}}}-\frac {4}{a^4 (a+x) \sqrt {1-\frac {x^2}{a^2}}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^4}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^4}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^3}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^3 \left (a+\frac {1}{x}\right )}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}} x}{a^3}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{a}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )-\frac {2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a^4}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^3}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a^2}\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^3 \left (a+\frac {1}{x}\right )}-\frac {6 \sqrt {1-\frac {1}{a^2 x^2}} x}{a^3}+\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a^4}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{8 a^2}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2}\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^3 \left (a+\frac {1}{x}\right )}-\frac {6 \sqrt {1-\frac {1}{a^2 x^2}} x}{a^3}+\frac {19 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{8 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^4}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{16 a^4}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2}\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^3 \left (a+\frac {1}{x}\right )}-\frac {6 \sqrt {1-\frac {1}{a^2 x^2}} x}{a^3}+\frac {19 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{8 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {6 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^4}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a^2}\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^3 \left (a+\frac {1}{x}\right )}-\frac {6 \sqrt {1-\frac {1}{a^2 x^2}} x}{a^3}+\frac {19 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{8 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {51 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a^4}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 83, normalized size = 0.61 \[ \frac {51 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a^4 x^4-6 a^3 x^3+11 a^2 x^2-29 a x-80\right )}{a x+1}}{8 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.67, size = 92, normalized size = 0.68 \[ \frac {{\left (2 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 11 \, a^{2} x^{2} - 29 \, a x - 80\right )} \sqrt {\frac {a x - 1}{a x + 1}} + 51 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 51 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{8 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 539, normalized size = 3.96 \[ \frac {\left (2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}+4 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+21 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}-8 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}+2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +42 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}-21 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-16 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -72 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+72 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+21 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -42 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+8 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-144 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +144 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-21 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -72 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+72 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{8 a^{4} \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 223, normalized size = 1.64 \[ -\frac {1}{8} \, a {\left (\frac {2 \, {\left (77 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 149 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 123 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 35 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {4 \, {\left (a x - 1\right )} a^{5}}{a x + 1} - \frac {6 \, {\left (a x - 1\right )}^{2} a^{5}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3} a^{5}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{5}}{{\left (a x + 1\right )}^{4}} - a^{5}} - \frac {51 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{5}} + \frac {51 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{5}} + \frac {32 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 192, normalized size = 1.41 \[ \frac {51\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a^4}-\frac {\frac {35\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {123\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}+\frac {149\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{4}-\frac {77\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{a^4+\frac {6\,a^4\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {4\,a^4\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {a^4\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {4\,a^4\,\left (a\,x-1\right )}{a\,x+1}}-\frac {4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a^4} \]
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 \[ \int x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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