Optimal. Leaf size=116 \[ -\frac {3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}+\frac {14 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a+\frac {1}{x}\right )}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {11 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
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Rubi [A] time = 0.87, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6169, 6742, 271, 264, 266, 51, 63, 208, 651} \[ \frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}+\frac {14 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a+\frac {1}{x}\right )}-\frac {11 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
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^2 \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^4 \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^4 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3}{a x^3 \sqrt {1-\frac {x^2}{a^2}}}+\frac {4}{a^2 x^2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {4}{a^3 x \sqrt {1-\frac {x^2}{a^2}}}+\frac {4}{a^3 (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^3}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^3}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a+\frac {1}{x}\right )}+\frac {4 \sqrt {1-\frac {1}{a^2 x^2}} x}{a^2}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a^3}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 a^2}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a+\frac {1}{x}\right )}+\frac {14 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a^3}-\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}\\ &=\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a+\frac {1}{x}\right )}+\frac {14 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^3}-\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 )}{2 a}\\ &=\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a+\frac {1}{x}\right )}+\frac {14 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 a}+\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {11 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 75, normalized size = 0.65 \[ \frac {\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a^3 x^3-7 a^2 x^2+19 a x+52\right )}{a x+1}-33 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{6 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.40, size = 84, normalized size = 0.72 \[ \frac {{\left (2 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 19 \, a x + 52\right )} \sqrt {\frac {a x - 1}{a x + 1}} - 33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{6 \, a^{3}} \]
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 = 471, normalized size = 4.06 \[ -\frac {\left (9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}-2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}+18 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-4 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -42 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+42 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -18 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+10 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-84 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +84 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -42 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+42 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{6 a^{3} \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 186, normalized size = 1.60 \[ -\frac {1}{6} \, a {\left (\frac {2 \, {\left (39 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 52 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 21 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{4}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{4}} - \frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{4}} - \frac {24 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 156, normalized size = 1.34 \[ \frac {7\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {52\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+13\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\left (a\,x-1\right )}{a\,x+1}}+\frac {4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a^3}-\frac {11\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^3} \]
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 \[ \int x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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