Optimal. Leaf size=92 \[ \frac {1}{2} x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {3 x \sqrt {1-\frac {1}{a^2 x^2}}}{a}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}+\frac {9 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2} \]
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Rubi [A] time = 0.87, antiderivative size = 92, 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, 651, 266, 51, 63, 208, 264} \[ \frac {1}{2} x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {3 x \sqrt {1-\frac {1}{a^2 x^2}}}{a}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}+\frac {9 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2} \]
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 {4}{a^2 (a-x) \sqrt {1-\frac {x^2}{a^2}}}+\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}}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\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 {4 \operatorname {Subst}\left (\int \frac {1}{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 = 0.72 \[ \frac {\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (a^2 x^2+5 a x-14\right )}{a x-1}+9 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 103, normalized size = 1.12 \[ \frac {9 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{3} x^{3} + 6 \, a^{2} x^{2} - 9 \, a x - 14\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a^{3} x - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 140, normalized size = 1.52 \[ \frac {1}{2} \, a {\left (\frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{3}} - \frac {9 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{3}} - \frac {8}{a^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {2 \, {\left (\frac {5 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - 7 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 421, normalized size = 4.58 \[ -\frac {-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}-10 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-10 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+4 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+20 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+20 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-10 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -10 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{2 a^{2} \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 145, normalized size = 1.58 \[ \frac {1}{2} \, a {\left (\frac {2 \, {\left (\frac {15 \, {\left (a x - 1\right )}}{a x + 1} - \frac {9 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 4\right )}}{a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 2 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + a^{3} \sqrt {\frac {a x - 1}{a x + 1}}} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{3}} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 117, normalized size = 1.27 \[ \frac {9\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^2}-\frac {\frac {9\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {15\,\left (a\,x-1\right )}{a\,x+1}+4}{a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}-2\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}+a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]
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 \[ \int \frac {x}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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