Optimal. Leaf size=35 \[ -\frac {\cosh (c+d x)}{d (a+i a \sinh (c+d x))}-\frac {i x}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2735, 2648} \[ -\frac {\cosh (c+d x)}{d (a+i a \sinh (c+d x))}-\frac {i x}{a} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rubi steps
\begin {align*} \int \frac {\sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac {i x}{a}+i \int \frac {1}{a+i a \sinh (c+d x)} \, dx\\ &=-\frac {i x}{a}-\frac {\cosh (c+d x)}{d (a+i a \sinh (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 61, normalized size = 1.74 \[ \frac {i \cosh (c+d x) \left (1-\frac {\sinh ^{-1}(\sinh (c+d x)) (\sinh (c+d x)-i)}{\sqrt {\cosh ^2(c+d x)}}\right )}{a d (\sinh (c+d x)-i)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 33, normalized size = 0.94 \[ \frac {-i \, d x e^{\left (d x + c\right )} - d x - 2}{a d e^{\left (d x + c\right )} - i \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 33, normalized size = 0.94 \[ \frac {-\frac {2 i \, {\left (d x + c\right )}}{a} - \frac {4 i}{a {\left (i \, e^{\left (d x + c\right )} + 1\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 67, normalized size = 1.91 \[ \frac {i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d a}-\frac {i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}+\frac {2 i}{d a \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 36, normalized size = 1.03 \[ -\frac {i \, {\left (d x + c\right )}}{a d} - \frac {2}{{\left (a e^{\left (-d x - c\right )} + i \, a\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 27, normalized size = 0.77 \[ -\frac {x\,1{}\mathrm {i}}{a}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 27, normalized size = 0.77 \[ \frac {2 e^{c}}{- i a d e^{c} - a d e^{- d x}} - \frac {i x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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