Optimal. Leaf size=102 \[ -\frac {2 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {2 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}-\frac {2 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2202, 2194,
2191, 2188, 29} \begin {gather*} -\frac {2 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {2 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}-\frac {2 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2188
Rule 2191
Rule 2194
Rule 2202
Rubi steps
\begin {align*} \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx &=\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}-\frac {(2 b) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^2} \, dx}{-b x+\coth ^{-1}(\tanh (a+b x))}\\ &=-\frac {2 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}-\frac {(2 b) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))} \, dx}{\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}\\ &=-\frac {2 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {(2 b) \int \frac {1}{x} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}-\frac {\left (2 b^2\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}\\ &=-\frac {2 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {2 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}\\ &=-\frac {2 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {2 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}-\frac {2 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 70, normalized size = 0.69 \begin {gather*} \frac {-b^2 x^2+\coth ^{-1}(\tanh (a+b x))^2+2 b x \coth ^{-1}(\tanh (a+b x)) \left (\log (x)-\log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))} \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 = 0.48, size = 27548, normalized size = 270.08 \[\text {output too large to display}\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.65, size = 135, normalized size = 1.32 \begin {gather*} -\frac {16 \, b \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{-i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3}} + \frac {16 \, b \log \left (x\right )}{-i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3}} - \frac {4 \, {\left (i \, \pi - 4 \, b x - 2 \, a\right )}}{2 \, {\left (\pi ^{2} b + 4 i \, \pi a b - 4 \, a^{2} b\right )} x^{2} - {\left (i \, \pi ^{3} - 6 \, \pi ^{2} a - 12 i \, \pi a^{2} + 8 \, a^{3}\right )} x} \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. 480 vs.
\(2 (102) = 204\).
time = 0.36, size = 480, normalized size = 4.71 \begin {gather*} \frac {4 \, {\left (\pi ^{6} + 4 \, \pi ^{4} a^{2} - 16 \, \pi ^{2} a^{4} - 64 \, a^{6} + 8 \, {\left (\pi ^{4} b^{2} - 16 \, a^{4} b^{2}\right )} x^{2} + 4 \, {\left (5 \, \pi ^{4} a b + 8 \, \pi ^{2} a^{3} b - 48 \, a^{5} b\right )} x - 8 \, {\left (4 \, {\left (\pi ^{3} b^{3} - 12 \, \pi a^{2} b^{3}\right )} x^{3} + 8 \, {\left (\pi ^{3} a b^{2} - 12 \, \pi a^{3} b^{2}\right )} x^{2} + {\left (\pi ^{5} b - 8 \, \pi ^{3} a^{2} b - 48 \, \pi a^{4} 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 ) - 4 \, {\left (4 \, {\left (3 \, \pi ^{2} a b^{3} - 4 \, a^{3} b^{3}\right )} x^{3} + 8 \, {\left (3 \, \pi ^{2} a^{2} b^{2} - 4 \, a^{4} b^{2}\right )} x^{2} + {\left (3 \, \pi ^{4} a b + 8 \, \pi ^{2} a^{3} b - 16 \, a^{5} b\right )} x\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right ) + 8 \, {\left (4 \, {\left (3 \, \pi ^{2} a b^{3} - 4 \, a^{3} b^{3}\right )} x^{3} + 8 \, {\left (3 \, \pi ^{2} a^{2} b^{2} - 4 \, a^{4} b^{2}\right )} x^{2} + {\left (3 \, \pi ^{4} a b + 8 \, \pi ^{2} a^{3} b - 16 \, a^{5} b\right )} x\right )} \log \left (x\right )\right )}}{4 \, {\left (\pi ^{6} b^{2} + 12 \, \pi ^{4} a^{2} b^{2} + 48 \, \pi ^{2} a^{4} b^{2} + 64 \, a^{6} b^{2}\right )} x^{3} + 8 \, {\left (\pi ^{6} a b + 12 \, \pi ^{4} a^{3} b + 48 \, \pi ^{2} a^{5} b + 64 \, a^{7} b\right )} x^{2} + {\left (\pi ^{8} + 16 \, \pi ^{6} a^{2} + 96 \, \pi ^{4} a^{4} + 256 \, \pi ^{2} a^{6} + 256 \, a^{8}\right )} x} \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 {1}{x^{2} \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 = 0.42, size = 140, normalized size = 1.37 \begin {gather*} \frac {16 \, b \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{-i \, \pi ^{3} - 6 \, \pi ^{2} a + 12 i \, \pi a^{2} + 8 \, a^{3}} - \frac {16 \, b \log \left (x\right )}{-i \, \pi ^{3} - 6 \, \pi ^{2} a + 12 i \, \pi a^{2} + 8 \, a^{3}} + \frac {8 \, b}{2 \, \pi ^{2} b x - 8 i \, \pi a b x - 8 \, a^{2} b x + i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}} + \frac {4}{\pi ^{2} x - 4 i \, \pi a x - 4 \, a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.80, size = 453, normalized size = 4.44 \begin {gather*} \frac {4\,{\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2-\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\left (8\,\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+b\,x\,\mathrm {atan}\left (\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}}{\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 )\,32{}\mathrm {i}\right )+4\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2-16\,b^2\,x^2+b\,x\,\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\mathrm {atan}\left (\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}}{\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 )\,32{}\mathrm {i}}{x\,\left (\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\right )\,{\left (\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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