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 = 0.0153436, antiderivative size = 32, normalized size of antiderivative = 1., 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 3777
Rule 8
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*}
Mathematica [A] time = 0.103366, 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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Maple [A] time = 0.043, size = 63, normalized size = 2. \begin{align*}{\frac{1}{ab}\ln \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) +1 \right ) }-2\,{\frac{1}{ab \left ( \tanh \left ( 1/2\,bx+a/2 \right ) -i \right ) }}-{\frac{1}{ab}\ln \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13904, size = 47, normalized size = 1.47 \begin{align*} \frac{b x + a}{a b} - \frac{2 i}{{\left (a e^{\left (-b x - a\right )} + i \, a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59477, size = 80, normalized size = 2.5 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{1}{i \operatorname{csch}{\left (a + b x \right )} - 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17111, size = 42, normalized size = 1.31 \begin{align*} \frac{b x + a}{a b} - \frac{2 i}{a b{\left (e^{\left (b x + a\right )} - i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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