Optimal. Leaf size=61 \[ -\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \log (x)+\frac {1}{2} a^2 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{2 x^2}-\frac {a \coth ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.10, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5917, 5983, 266, 36, 29, 31, 5949} \[ -\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \log (x)+\frac {1}{2} a^2 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{2 x^2}-\frac {a \coth ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 5917
Rule 5949
Rule 5983
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)^2}{x^3} \, dx &=-\frac {\coth ^{-1}(a x)^2}{2 x^2}+a \int \frac {\coth ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\coth ^{-1}(a x)^2}{2 x^2}+a \int \frac {\coth ^{-1}(a x)}{x^2} \, dx+a^3 \int \frac {\coth ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {a \coth ^{-1}(a x)}{x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \coth ^{-1}(a x)}{x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a \coth ^{-1}(a x)}{x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^4 \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a \coth ^{-1}(a x)}{x}+\frac {1}{2} a^2 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 57, normalized size = 0.93 \[ -\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+\frac {\left (a^2 x^2-1\right ) \coth ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac {a \coth ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 79, normalized size = 1.30 \[ -\frac {4 \, a^{2} x^{2} \log \left (a^{2} x^{2} - 1\right ) - 8 \, a^{2} x^{2} \log \relax (x) + 4 \, a x \log \left (\frac {a x + 1}{a x - 1}\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2}}{8 \, x^{2}} \]
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 \[ \int \frac {\operatorname {arcoth}\left (a x\right )^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 164, normalized size = 2.69 \[ -\frac {\mathrm {arccoth}\left (a x \right )^{2}}{2 x^{2}}-\frac {a \,\mathrm {arccoth}\left (a x \right )}{x}-\frac {a^{2} \mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {a^{2} \mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {a^{2} \ln \left (a x -1\right )^{2}}{8}+\frac {a^{2} \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{4}+a^{2} \ln \left (a x \right )-\frac {a^{2} \ln \left (a x -1\right )}{2}-\frac {a^{2} \ln \left (a x +1\right )}{2}-\frac {a^{2} \ln \left (a x +1\right )^{2}}{8}+\frac {a^{2} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{4}-\frac {a^{2} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 96, normalized size = 1.57 \[ \frac {1}{8} \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \relax (x)\right )} a^{2} + \frac {1}{2} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} a \operatorname {arcoth}\left (a x\right ) - \frac {\operatorname {arcoth}\left (a x\right )^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 145, normalized size = 2.38 \[ a^2\,\ln \relax (x)+{\ln \left (\frac {1}{a\,x}+1\right )}^2\,\left (\frac {a^2}{8}-\frac {1}{8\,x^2}\right )+{\ln \left (1-\frac {1}{a\,x}\right )}^2\,\left (\frac {a^2}{8}-\frac {1}{8\,x^2}\right )-\frac {a^2\,\ln \left (a^2\,x^2-1\right )}{2}+\ln \left (1-\frac {1}{a\,x}\right )\,\left (\frac {4\,a\,x-2}{16\,x^2}+\frac {4\,a\,x+2}{16\,x^2}-\ln \left (\frac {1}{a\,x}+1\right )\,\left (\frac {a^2}{4}-\frac {1}{4\,x^2}\right )\right )-\frac {a\,\ln \left (\frac {1}{a\,x}+1\right )}{2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.58, size = 56, normalized size = 0.92 \[ a^{2} \log {\relax (x )} - a^{2} \log {\left (a x + 1 \right )} + \frac {a^{2} \operatorname {acoth}^{2}{\left (a x \right )}}{2} + a^{2} \operatorname {acoth}{\left (a x \right )} - \frac {a \operatorname {acoth}{\left (a x \right )}}{x} - \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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