Optimal. Leaf size=71 \[ \frac {3 x}{b^3}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2199, 2189,
2188, 29} \begin {gather*} \frac {3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 x}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2188
Rule 2189
Rule 2199
Rubi steps
\begin {align*} \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx &=-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))^2} \, dx}{2 b}\\ &=-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {3 \int \frac {x}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac {3 x}{b^3}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}-\frac {\left (3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^3}\\ &=\frac {3 x}{b^3}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}-\frac {\left (3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^4}\\ &=\frac {3 x}{b^3}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 86, normalized size = 1.21 \begin {gather*} -\frac {b^3 x^3+3 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^3 \left (5+6 \log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )-b x \coth ^{-1}(\tanh (a+b x))^2 \left (11+6 \log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{2 b^4 \coth ^{-1}(\tanh (a+b x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.36, size = 4977, normalized size = 70.10
method | result | size |
risch | \(\text {Expression too large to display}\) | \(4977\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.86, size = 146, normalized size = 2.06 \begin {gather*} \frac {16 \, b^{3} x^{3} - 5 i \, \pi ^{3} + 30 \, \pi ^{2} a + 60 i \, \pi a^{2} - 40 \, a^{3} - 16 \, {\left (i \, \pi b^{2} - 2 \, a b^{2}\right )} x^{2} + 8 \, {\left (\pi ^{2} b + 4 i \, \pi a b - 4 \, a^{2} b\right )} x}{4 \, {\left (4 \, b^{6} x^{2} - \pi ^{2} b^{4} - 4 i \, \pi a b^{4} + 4 \, a^{2} b^{4} - 4 \, {\left (i \, \pi b^{5} - 2 \, a b^{5}\right )} x\right )}} - \frac {3 \, {\left (-i \, \pi + 2 \, a\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 418 vs.
\(2 (67) = 134\).
time = 0.37, size = 418, normalized size = 5.89 \begin {gather*} \frac {32 \, b^{5} x^{5} + 128 \, a b^{4} x^{4} - 5 \, \pi ^{4} a - 40 \, \pi ^{2} a^{3} - 80 \, a^{5} + 8 \, {\left (5 \, \pi ^{2} b^{3} + 12 \, a^{2} b^{3}\right )} x^{3} + 4 \, {\left (11 \, \pi ^{2} a b^{2} - 36 \, a^{3} b^{2}\right )} x^{2} + 2 \, {\left (3 \, \pi ^{4} b - 16 \, \pi ^{2} a^{2} b - 112 \, a^{4} b\right )} x + 6 \, {\left (16 \, \pi b^{4} x^{4} + 64 \, \pi a b^{3} x^{3} + \pi ^{5} + 8 \, \pi ^{3} a^{2} + 16 \, \pi a^{4} + 8 \, {\left (\pi ^{3} b^{2} + 12 \, \pi a^{2} b^{2}\right )} x^{2} + 16 \, {\left (\pi ^{3} a b + 4 \, \pi a^{3} b\right )} x\right )} \arctan \left (-\frac {2 \, b x + 2 \, a - \sqrt {4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) - 3 \, {\left (16 \, a b^{4} x^{4} + 64 \, a^{2} b^{3} x^{3} + \pi ^{4} a + 8 \, \pi ^{2} a^{3} + 16 \, a^{5} + 8 \, {\left (\pi ^{2} a b^{2} + 12 \, a^{3} b^{2}\right )} x^{2} + 16 \, {\left (\pi ^{2} a^{2} b + 4 \, a^{4} b\right )} x\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{2 \, {\left (16 \, b^{8} x^{4} + 64 \, a b^{7} x^{3} + \pi ^{4} b^{4} + 8 \, \pi ^{2} a^{2} b^{4} + 16 \, a^{4} b^{4} + 8 \, {\left (\pi ^{2} b^{6} + 12 \, a^{2} b^{6}\right )} x^{2} + 16 \, {\left (\pi ^{2} a b^{5} + 4 \, a^{3} b^{5}\right )} x\right )}} \end {gather*}
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 \begin {gather*} \int \frac {x^{3}}{\operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.44, size = 123, normalized size = 1.73 \begin {gather*} \frac {12 \, \pi ^{2} b x - 48 i \, \pi a b x - 48 \, a^{2} b x + 5 i \, \pi ^{3} + 30 \, \pi ^{2} a - 60 i \, \pi a^{2} - 40 \, a^{3}}{4 \, {\left (4 \, b^{6} x^{2} + 4 i \, \pi b^{5} x + 8 \, a b^{5} x - \pi ^{2} b^{4} + 4 i \, \pi a b^{4} + 4 \, a^{2} b^{4}\right )}} + \frac {x}{b^{3}} + \frac {3 \, {\left (-i \, \pi - 2 \, a\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.43, size = 620, normalized size = 8.73 \begin {gather*} \frac {x}{b^3}-\frac {x\,\left (3\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-12\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+12\,a^2\right )-\frac {5\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3-8\,a^3-6\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2+12\,a^2\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\right )}{4\,b}}{b^3\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2+x\,\left (8\,a\,b^4-4\,b^4\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\right )+4\,a^2\,b^3+4\,b^5\,x^2-4\,a\,b^3\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}+\frac {\ln \left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\right )\,\left (3\,\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-3\,\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+6\,b\,x\right )}{2\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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