Optimal. Leaf size=24 \[ \text {Int}\left (\frac {\tanh ^{-1}(d (-\cot (a+b x))-i d+1)}{x},x\right ) \]
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Rubi [A] time = 0.10, 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 = {} \[ \int \frac {\tanh ^{-1}(1-i d-d \cot (a+b x))}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(1-i d-d \cot (a+b x))}{x} \, dx &=\int \frac {\tanh ^{-1}(1-i d-d \cot (a+b x))}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{-1}(1-i d-d \cot (a+b x))}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\log \left (-\frac {d e^{\left (2 i \, b x + 2 i \, a\right )}}{{\left (d + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right )}{2 \, x}, x\right ) \]
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 \[ \int -\frac {\operatorname {artanh}\left (d \cot \left (b x + a\right ) + i \, d - 1\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.53, size = 0, normalized size = 0.00 \[ \int -\frac {\arctanh \left (-1+i d +d \cot \left (b x +a \right )\right )}{x}\, dx \]
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 \[ -i \, b x + \frac {1}{4} \, {\left (i \, \pi - 4 i \, a - 2 \, \log \relax (d)\right )} \log \relax (x) - \frac {1}{2} i \, \int \frac {\arctan \left (-d \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ), -d \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) + 1\right )}{x}\,{d x} + \frac {1}{4} \, \int \frac {\log \left ({\left (d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (d^{2} + 1\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, d \sin \left (2 \, b x + 2 \, a\right ) - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )}{x}\,{d x} \]
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 \[ \int -\frac {\mathrm {atanh}\left (d\,\mathrm {cot}\left (a+b\,x\right )-1+d\,1{}\mathrm {i}\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\operatorname {atanh}{\left (d \cot {\left (a + b x \right )} + i d - 1 \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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