Optimal. Leaf size=118 \[ \frac {3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}+\frac {14 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a-\frac {1}{x}\right )}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {11 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
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Rubi [A] time = 1.07, antiderivative size = 118, 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, 651, 271, 264, 266, 51, 63, 208} \[ \frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}+\frac {14 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a^2 \left (a-\frac {1}{x}\right )}+\frac {11 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
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 {4}{a^3 (a-x) \sqrt {1-\frac {x^2}{a^2}}}+\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}}}\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^3}-\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}{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 = 0.09, size = 75, normalized size = 0.64 \[ \frac {33 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a^3 x^3+7 a^2 x^2+19 a x-52\right )}{a x-1}}{6 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 112, normalized size = 0.95 \[ \frac {33 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 33 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 33 \, a x - 52\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, {\left (a^{4} x - a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 171, normalized size = 1.45 \[ \frac {1}{6} \, a {\left (\frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{4}} - \frac {33 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{4}} - \frac {24}{a^{4} \sqrt {\frac {a x - 1}{a x + 1}}} + \frac {2 \, {\left (\frac {52 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - \frac {21 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - 39 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 471, normalized size = 3.99 \[ \frac {9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}+2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}-18 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-4 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a +42 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+42 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a +18 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-10 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-84 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -84 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a +42 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+42 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{6 a^{3} \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 182, normalized size = 1.54 \[ -\frac {1}{6} \, a {\left (\frac {2 \, {\left (\frac {75 \, {\left (a x - 1\right )}}{a x + 1} - \frac {88 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {33 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 12\right )}}{a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 3 \, a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 3 \, a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{4} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{4}} + \frac {33 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 154, normalized size = 1.31 \[ \frac {11\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^3}-\frac {\frac {88\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {11\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {25\,\left (a\,x-1\right )}{a\,x+1}+4}{a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}-3\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}+3\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]
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 \[ \int \frac {x^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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