Optimal. Leaf size=17 \[ -\frac {\coth ^{-1}(\coth (a+b x))}{x}+b \log (x) \]
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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}(\coth (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}(\coth (a+b x))}{x^2} \, dx &=-\frac {\coth ^{-1}(\coth (a+b x))}{x}+b \int \frac {1}{x} \, dx\\ &=-\frac {\coth ^{-1}(\coth (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}(\coth (a+b x))}{x}+b \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 21, normalized size = 1.24
method | result | size |
default | \(-\frac {\mathrm {arccoth}\left (\coth \left (b x +a \right )\right )}{x}+b \ln \left (-b x \right )\) | \(21\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{b x +a}\right )}{x}+\frac {-2 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{3}-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}+4 \ln \left (x \right ) x b}{4 x}\) | \(289\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 11, normalized size = 0.65 \begin {gather*} b \log \left (x\right ) - \frac {a}{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 [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (14) = 28\).
time = 3.79, size = 68, normalized size = 4.00 \begin {gather*} \begin {cases} - \frac {\operatorname {acoth}{\left (\coth {\left (b x + \log {\left (- e^{- b x} \right )} \right )} \right )}}{x} & \text {for}\: a = \log {\left (- e^{- b x} \right )} \\- \frac {\operatorname {acoth}{\left (\coth {\left (b x + \log {\left (e^{- b x} \right )} \right )} \right )}}{x} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\b \log {\left (x \right )} - \frac {\operatorname {acoth}{\left (\frac {1}{\tanh {\left (a + b x \right )}} \right )}}{x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 12, normalized size = 0.71 \begin {gather*} b \log \left ({\left | x \right |}\right ) - \frac {a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 17, normalized size = 1.00 \begin {gather*} b\,\ln \left (x\right )-\frac {\mathrm {acoth}\left (\mathrm {coth}\left (a+b\,x\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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