Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\tan ^{-1}(c-(-c+i) \coth (a+b x))}{x},x\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\tan ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx &=\int \frac {\tan ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 4.27, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {i \, \log \left (-\frac {{\left (c e^{\left (2 \, b x + 2 \, a\right )} - i\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{c - i}\right )}{2 \, x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left ({\left (c - i\right )} \coth \left (b x + a\right ) + c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.90, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (c -\left (i-c \right ) \coth \left (b x +a \right )\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -i \, b x - \frac {1}{4} \, {\left (2 \, \pi + 4 i \, a - 2 \, \arctan \relax (c) + i \, \log \left (c^{2} + 1\right )\right )} \log \relax (x) - \frac {1}{2} \, \int \frac {\arctan \left (c e^{\left (2 \, b x + 2 \, a\right )}\right )}{x}\,{d x} + \frac {1}{4} i \, \int \frac {\log \left (c^{2} e^{\left (4 \, b x + 4 \, a\right )} + 1\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {atan}\left (c+\mathrm {coth}\left (a+b\,x\right )\,\left (c-\mathrm {i}\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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