Optimal. Leaf size=32 \[ \frac {x}{a}-\frac {\coth (a+b x)}{b (a-i a \text {csch}(a+b x))} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3777, 8} \[ \frac {x}{a}-\frac {\coth (a+b x)}{b (a-i a \text {csch}(a+b x))} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3777
Rubi steps
\begin {align*} \int \frac {1}{a-i a \text {csch}(a+b x)} \, dx &=-\frac {\coth (a+b x)}{b (a-i a \text {csch}(a+b x))}+\frac {\int a \, dx}{a^2}\\ &=\frac {x}{a}-\frac {\coth (a+b x)}{b (a-i a \text {csch}(a+b x))}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 54, normalized size = 1.69 \[ -\frac {2 \sinh \left (\frac {1}{2} (a+b x)\right )}{a b \left (\cosh \left (\frac {1}{2} (a+b x)\right )+i \sinh \left (\frac {1}{2} (a+b x)\right )\right )}+\frac {x}{a}+\frac {1}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 32, normalized size = 1.00 \[ \frac {b x e^{\left (b x + a\right )} - i \, b x - 2 i}{a b e^{\left (b x + a\right )} - i \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 29, normalized size = 0.91 \[ \frac {\frac {b x + a}{a} - \frac {2 i}{a {\left (e^{\left (b x + a\right )} - i\right )}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 63, normalized size = 1.97 \[ -\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b a}+\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b a}-\frac {2}{b a \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 35, normalized size = 1.09 \[ \frac {b x + a}{a b} - \frac {2 i}{{\left (a e^{\left (-b x - a\right )} + i \, a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 26, normalized size = 0.81 \[ \frac {x}{a}-\frac {2{}\mathrm {i}}{a\,b\,\left ({\mathrm {e}}^{a+b\,x}-\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {i \int \frac {1}{\operatorname {csch}{\left (a + b x \right )} + i}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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