Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x},x\right ) \]
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Rubi [A] time = 0.12, 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 = {} \[ \int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx &=\int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, \pi + i \, \log \left (\frac {{\left (c - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right )}{2 \, x}, x\right ) \]
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 \[ \int \frac {\pi - \operatorname {arccot}\left ({\left (i \, c + 1\right )} \cot \left (b x + a\right ) - c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.31, size = 0, normalized size = 0.00 \[ \int \frac {\pi -\mathrm {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\Pi +\mathrm {acot}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (1+c\,1{}\mathrm {i}\right )\right )}{x} \,d x \]
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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