Optimal. Leaf size=75 \[ \frac {3 x^2}{2 b^2}+\frac {3 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac {3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2199, 2190,
2189, 2188, 29} \begin {gather*} \frac {3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}+\frac {3 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac {3 x^2}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2188
Rule 2189
Rule 2190
Rule 2199
Rubi steps
\begin {align*} \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac {x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac {3 \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac {3 x^2}{2 b^2}-\frac {x^3}{b \coth ^{-1}(\tanh (a+b x))}-\frac {\left (3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {x}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac {3 x^2}{2 b^2}+\frac {3 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac {\left (3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^3}\\ &=\frac {3 x^2}{2 b^2}+\frac {3 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac {\left (3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^4}\\ &=\frac {3 x^2}{2 b^2}+\frac {3 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac {3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 83, normalized size = 1.11 \begin {gather*} \frac {x^2}{2 b^2}-\frac {2 x \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^3}{b^4 \coth ^{-1}(\tanh (a+b x))}+\frac {3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4} \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 = 1.46, size = 29109, normalized size = 388.12
method | result | size |
risch | \(\text {Expression too large to display}\) | \(29109\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.66, size = 123, normalized size = 1.64 \begin {gather*} \frac {4 \, b^{3} x^{3} + i \, \pi ^{3} - 6 \, \pi ^{2} a - 12 i \, \pi a^{2} + 8 \, a^{3} - 6 \, {\left (-i \, \pi b^{2} + 2 \, a b^{2}\right )} x^{2} + 4 \, {\left (\pi ^{2} b + 4 i \, \pi a b - 4 \, a^{2} b\right )} x}{4 \, {\left (2 \, b^{5} x - i \, \pi b^{4} + 2 \, a b^{4}\right )}} - \frac {3 \, {\left (\pi ^{2} + 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{4 \, 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. 244 vs.
\(2 (73) = 146\).
time = 0.42, size = 244, normalized size = 3.25 \begin {gather*} \frac {16 \, b^{4} x^{4} - 32 \, a b^{3} x^{3} - 2 \, \pi ^{4} + 32 \, a^{4} + 4 \, {\left (\pi ^{2} b^{2} - 28 \, a^{2} b^{2}\right )} x^{2} - 8 \, {\left (5 \, \pi ^{2} a b + 4 \, a^{3} b\right )} x - 48 \, {\left (4 \, \pi a b^{2} x^{2} + 8 \, \pi a^{2} b x + \pi ^{3} a + 4 \, \pi a^{3}\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 (\pi ^{4} - 16 \, a^{4} + 4 \, {\left (\pi ^{2} b^{2} - 4 \, a^{2} b^{2}\right )} x^{2} + 8 \, {\left (\pi ^{2} a b - 4 \, a^{3} b\right )} x\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{8 \, {\left (4 \, b^{6} x^{2} + 8 \, a b^{5} x + \pi ^{2} b^{4} + 4 \, a^{2} b^{4}\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}^{2}{\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 = 0.41, size = 97, normalized size = 1.29 \begin {gather*} -\frac {i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}}{4 \, {\left (2 \, b^{5} x + i \, \pi b^{4} + 2 \, a b^{4}\right )}} + \frac {x^{2}}{2 \, b^{2}} + \frac {{\left (-i \, \pi - 2 \, a\right )} x}{b^{3}} - \frac {3 \, {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{4 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 490, normalized size = 6.53 \begin {gather*} \frac {x^2}{2\,b^2}+\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\,{\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 )}{4\,b^4}-\frac {{\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 )}{4\,b\,\left (2\,a\,b^3+2\,b^4\,x-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 )\right )}+\frac {x\,\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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