Optimal. Leaf size=41 \[ \frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{4 a^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6038, 308, 212}
\begin {gather*} -\frac {\tanh ^{-1}(a x)}{4 a^4}+\frac {x}{4 a^3}+\frac {1}{4} x^4 \coth ^{-1}(a x)+\frac {x^3}{12 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 308
Rule 6038
Rubi steps
\begin {align*} \int x^3 \coth ^{-1}(a x) \, dx &=\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \int \frac {x^4}{1-a^2 x^2} \, dx\\ &=\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \int \left (-\frac {1}{a^4}-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{4 a^3}\\ &=\frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{4 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 57, normalized size = 1.39 \begin {gather*} \frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \coth ^{-1}(a x)+\frac {\log (1-a x)}{8 a^4}-\frac {\log (1+a x)}{8 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 46, normalized size = 1.12
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \mathrm {arccoth}\left (a x \right )}{4}+\frac {a^{3} x^{3}}{12}+\frac {a x}{4}+\frac {\ln \left (a x -1\right )}{8}-\frac {\ln \left (a x +1\right )}{8}}{a^{4}}\) | \(46\) |
default | \(\frac {\frac {a^{4} x^{4} \mathrm {arccoth}\left (a x \right )}{4}+\frac {a^{3} x^{3}}{12}+\frac {a x}{4}+\frac {\ln \left (a x -1\right )}{8}-\frac {\ln \left (a x +1\right )}{8}}{a^{4}}\) | \(46\) |
risch | \(\frac {x^{4} \ln \left (a x +1\right )}{8}-\frac {x^{4} \ln \left (a x -1\right )}{8}+\frac {x^{3}}{12 a}+\frac {x}{4 a^{3}}+\frac {\ln \left (-a x +1\right )}{8 a^{4}}-\frac {\ln \left (a x +1\right )}{8 a^{4}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 52, normalized size = 1.27 \begin {gather*} \frac {1}{4} \, x^{4} \operatorname {arcoth}\left (a x\right ) + \frac {1}{24} \, a {\left (\frac {2 \, {\left (a^{2} x^{3} + 3 \, x\right )}}{a^{4}} - \frac {3 \, \log \left (a x + 1\right )}{a^{5}} + \frac {3 \, \log \left (a x - 1\right )}{a^{5}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 43, normalized size = 1.05 \begin {gather*} \frac {2 \, a^{3} x^{3} + 6 \, a x + 3 \, {\left (a^{4} x^{4} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{24 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.24, size = 41, normalized size = 1.00 \begin {gather*} \begin {cases} \frac {x^{4} \operatorname {acoth}{\left (a x \right )}}{4} + \frac {x^{3}}{12 a} + \frac {x}{4 a^{3}} - \frac {\operatorname {acoth}{\left (a x \right )}}{4 a^{4}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{4}}{8} & \text {otherwise} \end {cases} \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. 195 vs.
\(2 (33) = 66\).
time = 0.39, size = 195, normalized size = 4.76 \begin {gather*} \frac {1}{3} \, a {\left (\frac {\frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - \frac {3 \, {\left (a x + 1\right )}}{a x - 1} + 2}{a^{5} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{3}} + \frac {3 \, {\left (\frac {{\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {a x + 1}{a x - 1}\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{a^{5} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 33, normalized size = 0.80 \begin {gather*} \frac {\frac {a\,x}{4}-\frac {\mathrm {acoth}\left (a\,x\right )}{4}+\frac {a^3\,x^3}{12}}{a^4}+\frac {x^4\,\mathrm {acoth}\left (a\,x\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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