Optimal. Leaf size=82 \[ \frac {i \text {Li}_2\left (-i c e^{2 a+2 b x}\right )}{4 b}+\frac {1}{2} i x \log \left (1+i c e^{2 a+2 b x}\right )+x \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac {1}{2} i b x^2 \]
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Rubi [A] time = 0.12, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5190, 2184, 2190, 2279, 2391} \[ \frac {i \text {PolyLog}\left (2,-i c e^{2 a+2 b x}\right )}{4 b}+\frac {1}{2} i x \log \left (1+i c e^{2 a+2 b x}\right )+x \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac {1}{2} i b x^2 \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2279
Rule 2391
Rule 5190
Rubi steps
\begin {align*} \int \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx &=x \cot ^{-1}(c-(i-c) \coth (a+b x))+b \int \frac {x}{i-c e^{2 a+2 b x}} \, dx\\ &=-\frac {1}{2} i b x^2+x \cot ^{-1}(c-(i-c) \coth (a+b x))-(i b c) \int \frac {e^{2 a+2 b x} x}{i-c e^{2 a+2 b x}} \, dx\\ &=-\frac {1}{2} i b x^2+x \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{2} i x \log \left (1+i c e^{2 a+2 b x}\right )-\frac {1}{2} i \int \log \left (1+i c e^{2 a+2 b x}\right ) \, dx\\ &=-\frac {1}{2} i b x^2+x \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{2} i x \log \left (1+i c e^{2 a+2 b x}\right )-\frac {i \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{4 b}\\ &=-\frac {1}{2} i b x^2+x \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{2} i x \log \left (1+i c e^{2 a+2 b x}\right )+\frac {i \text {Li}_2\left (-i c e^{2 a+2 b x}\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 71, normalized size = 0.87 \[ \frac {i \left (2 b x \log \left (1-\frac {i e^{-2 (a+b x)}}{c}\right )-\text {Li}_2\left (\frac {i e^{-2 (a+b x)}}{c}\right )\right )}{4 b}+x \cot ^{-1}(c+(c-i) \coth (a+b x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 186, normalized size = 2.27 \[ \frac {-i \, b^{2} x^{2} + i \, b x \log \left (\frac {{\left (c - i\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c e^{\left (2 \, b x + 2 \, a\right )} - i}\right ) + i \, a^{2} + {\left (i \, b x + i \, a\right )} \log \left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )} + 1\right ) + {\left (i \, b x + i \, a\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )} + 1\right ) - i \, a \log \left (\frac {2 \, c e^{\left (b x + a\right )} + i \, \sqrt {-4 i \, c}}{2 \, c}\right ) - i \, a \log \left (\frac {2 \, c e^{\left (b x + a\right )} - i \, \sqrt {-4 i \, c}}{2 \, c}\right ) + i \, {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right ) + i \, {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arccot}\left ({\left (c - i\right )} \coth \left (b x + a\right ) + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 1351, normalized size = 16.48 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.03, size = 80, normalized size = 0.98 \[ 2 \, b {\left (c - i\right )} {\left (\frac {2 \, x^{2}}{2 i \, c + 2} - \frac {2 \, b x \log \left (i \, c e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-i \, c e^{\left (2 \, b x + 2 \, a\right )}\right )}{-2 \, b^{2} {\left (-i \, c - 1\right )}}\right )} + x \operatorname {arccot}\left ({\left (c - i\right )} \coth \left (b x + a\right ) + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {acot}\left (c+\mathrm {coth}\left (a+b\,x\right )\,\left (c-\mathrm {i}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: CoercionFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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