Optimal. Leaf size=92 \[ \frac {b}{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {1}{2 x^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {b^2 \log (x)}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {b^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2194, 2191,
2188, 29} \begin {gather*} -\frac {b^2 \log (x)}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {b^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {1}{2 x^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {b}{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2188
Rule 2191
Rule 2194
Rubi steps
\begin {align*} \int \frac {1}{x^3 \tanh ^{-1}(\tanh (a+b x))} \, dx &=\frac {1}{2 x^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {b \int \frac {1}{x^2 \tanh ^{-1}(\tanh (a+b x))} \, dx}{-b x+\tanh ^{-1}(\tanh (a+b x))}\\ &=\frac {b}{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {1}{2 x^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {b^2 \int \frac {1}{x \tanh ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {b}{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {1}{2 x^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {b^2 \int \frac {1}{x} \, dx}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {b^3 \int \frac {1}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {b}{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {1}{2 x^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {b^2 \log (x)}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}-\frac {b^2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {b}{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {1}{2 x^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {b^2 \log (x)}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {b^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 66, normalized size = 0.72 \begin {gather*} \frac {-4 b x \tanh ^{-1}(\tanh (a+b x))+\tanh ^{-1}(\tanh (a+b x))^2+b^2 x^2 \left (3-2 \log (x)+2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )\right )}{2 x^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 87, normalized size = 0.95 \[-\frac {b^{2} \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}-\frac {1}{2 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) x^{2}}+\frac {b^{2} \ln \left (x \right )}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {b}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2} x}\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 40, normalized size = 0.43 \begin {gather*} -\frac {b^{2} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} \log \left (x\right )}{a^{3}} + \frac {2 \, b x - a}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 41, normalized size = 0.45 \begin {gather*} -\frac {2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \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^{3} \operatorname {atanh}{\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 [A]
time = 0.40, size = 45, normalized size = 0.49 \begin {gather*} -\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.89, size = 286, normalized size = 3.11 \begin {gather*} \frac {{\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 (2\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+8\,b\,x\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+12\,b^2\,x^2+8\,b\,x\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+b^2\,x^2\,\mathrm {atan}\left (\frac {-\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,1{}\mathrm {i}+\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}}{\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 )\,16{}\mathrm {i}}{x^2\,{\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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