Optimal. Leaf size=65 \[ \frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac {b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 65, 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 = {2163, 2160, 2157, 29} \[ \frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac {b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2160
Rule 2163
Rubi steps
\begin {align*} \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))} \, dx &=\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac {b \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))} \, dx}{b x-\coth ^{-1}(\tanh (a+b x))}\\ &=\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac {b \int \frac {1}{x} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {b^2 \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}\\ &=\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac {b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {b \operatorname {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 )^2}\\ &=\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac {b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 0.69 \[ \frac {b x \left (\log \left (\coth ^{-1}(\tanh (a+b x))\right )-\log (x)+1\right )-\coth ^{-1}(\tanh (a+b x))}{x \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 137, normalized size = 2.11 \[ \frac {2 \, {\left (16 \, \pi a b x \arctan \left (-\frac {2 \, b x + 2 \, a - \sqrt {4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) - 2 \, \pi ^{2} a - 8 \, a^{3} - {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right ) + 2 \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x \log \relax (x)\right )}}{{\left (\pi ^{4} + 8 \, \pi ^{2} a^{2} + 16 \, a^{4}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.52, size = 65, normalized size = 1.00 \[ -\frac {4 \, b \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{2} + 4 i \, \pi a - 4 \, a^{2}} + \frac {4 \, b \log \relax (x)}{\pi ^{2} + 4 i \, \pi a - 4 \, a^{2}} + \frac {2}{{\left (i \, \pi - 2 \, a\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.12, size = 220, normalized size = 3.38 \[ \frac {2\,\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-2\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+4\,b\,x+b\,x\,\mathrm {atan}\left (\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}}{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x}\right )\,8{}\mathrm {i}}{x\,{\left (\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2} \]
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 \[ \int \frac {1}{x^{2} \operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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