Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x},x\right ) \]
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Rubi [A]
time = 0.06, 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 = {}
\begin {gather*} \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx &=\int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.25, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccoth}\left (1+i d +d \cot \left (b x +a \right )\right )}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acoth}{\left (d \cot {\left (a + b x \right )} + i d + 1 \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {acoth}\left (d\,\mathrm {cot}\left (a+b\,x\right )+1+d\,1{}\mathrm {i}\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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