Optimal. Leaf size=90 \[ \frac {1}{2} x^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {3 x \sqrt {1-\frac {1}{a^2 x^2}}}{a}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}+\frac {9 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2} \]
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Rubi [A] time = 0.84, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6169, 6742, 266, 51, 63, 208, 264, 651} \[ \frac {1}{2} x^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {3 x \sqrt {1-\frac {1}{a^2 x^2}}}{a}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}+\frac {9 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 264
Rule 266
Rule 651
Rule 6169
Rule 6742
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} x \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^3 \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^3 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3}{a x^2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {4}{a^2 x \sqrt {1-\frac {x^2}{a^2}}}-\frac {4}{a^2 (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^2}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{a}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \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^2}\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+4 \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a^2}\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {9 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 66, normalized size = 0.73 \[ \frac {\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (a^2 x^2-5 a x-14\right )}{a x+1}+9 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 75, normalized size = 0.83 \[ \frac {{\left (a^{2} x^{2} - 5 \, a x - 14\right )} \sqrt {\frac {a x - 1}{a x + 1}} + 9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{2 \, a^{2}} \]
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 = 0.05, size = 421, normalized size = 4.68 \[ \frac {\left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}-10 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+10 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+4 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-20 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+20 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-10 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a +10 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 a^{2} \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 151, normalized size = 1.68 \[ -\frac {1}{2} \, a {\left (\frac {2 \, {\left (7 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 5 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{3}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{3}} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{3}} + \frac {8 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 120, normalized size = 1.33 \[ \frac {9\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^2}-\frac {4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a^2}-\frac {5\,\sqrt {\frac {a\,x-1}{a\,x+1}}-7\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a^2+\frac {a^2\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,a^2\,\left (a\,x-1\right )}{a\,x+1}} \]
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 \[ \int x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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