Optimal. Leaf size=90 \[ -\frac {a^2}{12 x^2}-\frac {a \coth ^{-1}(a x)}{6 x^3}-\frac {a^3 \coth ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{4 x^4}+\frac {2}{3} a^4 \log (x)-\frac {1}{3} a^4 \log \left (1-a^2 x^2\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6038, 6130,
272, 46, 36, 29, 31, 6096} \begin {gather*} \frac {2}{3} a^4 \log (x)+\frac {1}{4} a^4 \coth ^{-1}(a x)^2-\frac {a^3 \coth ^{-1}(a x)}{2 x}-\frac {a^2}{12 x^2}-\frac {1}{3} a^4 \log \left (1-a^2 x^2\right )-\frac {\coth ^{-1}(a x)^2}{4 x^4}-\frac {a \coth ^{-1}(a x)}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 272
Rule 6038
Rule 6096
Rule 6130
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)^2}{x^5} \, dx &=-\frac {\coth ^{-1}(a x)^2}{4 x^4}+\frac {1}{2} a \int \frac {\coth ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\coth ^{-1}(a x)^2}{4 x^4}+\frac {1}{2} a \int \frac {\coth ^{-1}(a x)}{x^4} \, dx+\frac {1}{2} a^3 \int \frac {\coth ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \coth ^{-1}(a x)}{6 x^3}-\frac {\coth ^{-1}(a x)^2}{4 x^4}+\frac {1}{6} a^2 \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^3 \int \frac {\coth ^{-1}(a x)}{x^2} \, dx+\frac {1}{2} a^5 \int \frac {\coth ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {a \coth ^{-1}(a x)}{6 x^3}-\frac {a^3 \coth ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{4 x^4}+\frac {1}{12} a^2 \text {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{2} a^4 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \coth ^{-1}(a x)}{6 x^3}-\frac {a^3 \coth ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{4 x^4}+\frac {1}{12} a^2 \text {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{4} a^4 \text {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a^2}{12 x^2}-\frac {a \coth ^{-1}(a x)}{6 x^3}-\frac {a^3 \coth ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{4 x^4}+\frac {1}{6} a^4 \log (x)-\frac {1}{12} a^4 \log \left (1-a^2 x^2\right )+\frac {1}{4} a^4 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{4} a^6 \text {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{12 x^2}-\frac {a \coth ^{-1}(a x)}{6 x^3}-\frac {a^3 \coth ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{4 x^4}+\frac {2}{3} a^4 \log (x)-\frac {1}{3} a^4 \log \left (1-a^2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 82, normalized size = 0.91 \begin {gather*} -\frac {a^2}{12 x^2}-\frac {a \left (1+3 a^2 x^2\right ) \coth ^{-1}(a x)}{6 x^3}+\frac {\left (-1+a^4 x^4\right ) \coth ^{-1}(a x)^2}{4 x^4}+\frac {2}{3} a^4 \log (x)-\frac {1}{3} a^4 \log \left (1-a^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs.
\(2(76)=152\).
time = 0.08, size = 158, normalized size = 1.76
method | result | size |
derivativedivides | \(a^{4} \left (-\frac {\mathrm {arccoth}\left (a x \right )^{2}}{4 a^{4} x^{4}}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\mathrm {arccoth}\left (a x \right )}{6 a^{3} x^{3}}-\frac {\mathrm {arccoth}\left (a x \right )}{2 a x}-\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{16}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (a x +1\right )^{2}}{16}-\frac {\ln \left (a x +1\right )}{3}-\frac {1}{12 a^{2} x^{2}}+\frac {2 \ln \left (a x \right )}{3}-\frac {\ln \left (a x -1\right )}{3}\right )\) | \(158\) |
default | \(a^{4} \left (-\frac {\mathrm {arccoth}\left (a x \right )^{2}}{4 a^{4} x^{4}}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\mathrm {arccoth}\left (a x \right )}{6 a^{3} x^{3}}-\frac {\mathrm {arccoth}\left (a x \right )}{2 a x}-\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{16}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (a x +1\right )^{2}}{16}-\frac {\ln \left (a x +1\right )}{3}-\frac {1}{12 a^{2} x^{2}}+\frac {2 \ln \left (a x \right )}{3}-\frac {\ln \left (a x -1\right )}{3}\right )\) | \(158\) |
risch | \(\frac {\left (a^{4} x^{4}-1\right ) \ln \left (a x +1\right )^{2}}{16 x^{4}}-\frac {\left (3 x^{4} \ln \left (a x -1\right ) a^{4}+6 a^{3} x^{3}+2 a x -3 \ln \left (a x -1\right )\right ) \ln \left (a x +1\right )}{24 x^{4}}+\frac {3 a^{4} x^{4} \ln \left (a x -1\right )^{2}+32 \ln \left (x \right ) a^{4} x^{4}-16 \ln \left (a^{2} x^{2}-1\right ) a^{4} x^{4}+12 x^{3} \ln \left (a x -1\right ) a^{3}-4 a^{2} x^{2}+4 x \ln \left (a x -1\right ) a -3 \ln \left (a x -1\right )^{2}}{48 x^{4}}\) | \(162\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 154 vs.
\(2 (76) = 152\).
time = 0.25, size = 154, normalized size = 1.71 \begin {gather*} \frac {1}{48} \, {\left (32 \, a^{2} \log \left (x\right ) - \frac {3 \, a^{2} x^{2} \log \left (a x + 1\right )^{2} + 3 \, a^{2} x^{2} \log \left (a x - 1\right )^{2} + 16 \, a^{2} x^{2} \log \left (a x - 1\right ) - 2 \, {\left (3 \, a^{2} x^{2} \log \left (a x - 1\right ) - 8 \, a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 4}{x^{2}}\right )} a^{2} + \frac {1}{12} \, {\left (3 \, a^{3} \log \left (a x + 1\right ) - 3 \, a^{3} \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} + 1\right )}}{x^{3}}\right )} a \operatorname {arcoth}\left (a x\right ) - \frac {\operatorname {arcoth}\left (a x\right )^{2}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 97, normalized size = 1.08 \begin {gather*} -\frac {16 \, a^{4} x^{4} \log \left (a^{2} x^{2} - 1\right ) - 32 \, a^{4} x^{4} \log \left (x\right ) + 4 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (3 \, a^{3} x^{3} + a x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{48 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 90, normalized size = 1.00 \begin {gather*} \frac {2 a^{4} \log {\left (x \right )}}{3} - \frac {2 a^{4} \log {\left (a x + 1 \right )}}{3} + \frac {a^{4} \operatorname {acoth}^{2}{\left (a x \right )}}{4} + \frac {2 a^{4} \operatorname {acoth}{\left (a x \right )}}{3} - \frac {a^{3} \operatorname {acoth}{\left (a x \right )}}{2 x} - \frac {a^{2}}{12 x^{2}} - \frac {a \operatorname {acoth}{\left (a x \right )}}{6 x^{3}} - \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 319 vs.
\(2 (76) = 152\).
time = 0.39, size = 319, normalized size = 3.54 \begin {gather*} \frac {1}{6} \, {\left (4 \, a^{3} \log \left (\frac {a x + 1}{a x - 1} + 1\right ) - 4 \, a^{3} \log \left (\frac {a x + 1}{a x - 1}\right ) + \frac {2 \, {\left (a x + 1\right )} a^{3}}{{\left (a x - 1\right )} {\left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {2 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}} + \frac {3 \, {\left (\frac {{\left (a x + 1\right )}^{3} a^{3}}{{\left (a x - 1\right )}^{3}} + \frac {{\left (a x + 1\right )} a^{3}}{a x - 1}\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2}}{\frac {{\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {4 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {4 \, {\left (a x + 1\right )}}{a x - 1} + 1} + \frac {2 \, {\left (\frac {3 \, {\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} + \frac {3 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + 2 \, a^{3}\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {3 \, {\left (a x + 1\right )}}{a x - 1} + 1}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.55, size = 196, normalized size = 2.18 \begin {gather*} \frac {2\,a^4\,\ln \left (x\right )}{3}+{\ln \left (\frac {1}{a\,x}+1\right )}^2\,\left (\frac {a^4}{16}-\frac {1}{16\,x^4}\right )+{\ln \left (1-\frac {1}{a\,x}\right )}^2\,\left (\frac {a^4}{16}-\frac {1}{16\,x^4}\right )+\ln \left (1-\frac {1}{a\,x}\right )\,\left (\frac {24\,a^3\,x^3-12\,a^2\,x^2+8\,a\,x-6}{192\,x^4}+\frac {24\,a^3\,x^3+12\,a^2\,x^2+8\,a\,x+6}{192\,x^4}-\ln \left (\frac {1}{a\,x}+1\right )\,\left (\frac {a^4}{8}-\frac {1}{8\,x^4}\right )\right )-\frac {a^4\,\ln \left (a^2\,x^2-1\right )}{3}-\frac {a^2}{12\,x^2}-\frac {a\,\ln \left (\frac {1}{a\,x}+1\right )\,\left (\frac {a^2\,x^2}{4}+\frac {1}{12}\right )}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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