Optimal. Leaf size=17 \[ -\frac {\coth ^{-1}(\tanh (a+b x))}{x}+b \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2199, 29}
\begin {gather*} b \log (x)-\frac {\coth ^{-1}(\tanh (a+b x))}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2199
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))}{x^2} \, dx &=-\frac {\coth ^{-1}(\tanh (a+b x))}{x}+b \int \frac {1}{x} \, dx\\ &=-\frac {\coth ^{-1}(\tanh (a+b x))}{x}+b \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.06 \begin {gather*} b-\frac {\coth ^{-1}(\tanh (a+b x))}{x}+b \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 21, normalized size = 1.24
method | result | size |
default | \(-\frac {\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}{x}+b \ln \left (-b x \right )\) | \(21\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{b x +a}\right )}{x}+\frac {-i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}-2 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}-2 i \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}+2 i \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+2 i \pi +i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )+4 \ln \left (x \right ) x b}{4 x}\) | \(339\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 17, normalized size = 1.00 \begin {gather*} b \log \left (x\right ) - \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 13, normalized size = 0.76 \begin {gather*} \frac {b x \log \left (x\right ) - a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 14, normalized size = 0.82 \begin {gather*} b \log {\left (x \right )} - \frac {\operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (17) = 34\).
time = 0.38, size = 70, normalized size = 4.12 \begin {gather*} b \log \left ({\left | x \right |}\right ) - \frac {\log \left (-\frac {\frac {e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} + 1}{\frac {e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - 1}\right )}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 17, normalized size = 1.00 \begin {gather*} b\,\ln \left (x\right )-\frac {\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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