Optimal. Leaf size=90 \[ \frac {x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}+\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
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Rubi [A] time = 0.09, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6169, 835, 807, 266, 63, 208} \[ \frac {1}{3} x^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a}+\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 6169
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} x^2 \, dx &=-\operatorname {Subst}\left (\int \frac {1+\frac {x}{a}}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {1}{3} \operatorname {Subst}\left (\int \frac {-\frac {3}{a}-\frac {2 x}{a^2}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\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 {1}{6} \operatorname {Subst}\left (\int \frac {\frac {4}{a^2}+\frac {3 x}{a^3}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}+\frac {\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 {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^3}\\ &=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}+\frac {\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 {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a^3}\\ &=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}+\frac {\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 {\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 {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^2}+\frac {\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 {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 0.67 \[ \frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a^2 x^2+3 a x+4\right )+3 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{6 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.71, size = 84, normalized size = 0.93 \[ \frac {{\left (2 \, a^{3} x^{3} + 5 \, a^{2} x^{2} + 7 \, a x + 4\right )} \sqrt {\frac {a x - 1}{a x + 1}} + 3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{6 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 151, normalized size = 1.68 \[ \frac {1}{6} \, a {\left (\frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{4}} - \frac {3 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{4}} + \frac {2 \, {\left (\frac {4 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - 9 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 173, normalized size = 1.92 \[ \frac {\left (a x -1\right ) \left (3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a +2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a +6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+6 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 166, normalized size = 1.84 \[ -\frac {1}{6} \, a {\left (\frac {2 \, {\left (3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 4 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 9 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{4}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} - \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{4}} + \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 133, normalized size = 1.48 \[ \frac {3\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\left (a\,x-1\right )}{a\,x+1}}+\frac {\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^3} \]
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 \[ \int \frac {x^{2}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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