Optimal. Leaf size=22 \[ \text{CannotIntegrate}\left (\frac{\tanh ^{-1}(d \cot (a+b x)+i d+1)}{x},x\right ) \]
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Rubi [A] time = 0.0858041, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [A] time = 0.954385, size = 0, normalized size = 0. \[ \int \frac{\tanh ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.395, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Artanh} \left ( 1+id+d\cot \left ( bx+a \right ) \right ) }{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -i \, b x + \frac{1}{4} \,{\left (-i \, \pi - 4 i \, a - 2 \, \log \left (-d\right )\right )} \log \left (x\right ) + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (-\frac{{\left ({\left (d - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{d}\right )}{2 \, x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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