Optimal. Leaf size=22 \[ \frac {x}{a}-\frac {i \cosh (c+d x)}{a d} \]
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Rubi [A] time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2682, 8} \[ \frac {x}{a}-\frac {i \cosh (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2682
Rubi steps
\begin {align*} \int \frac {\cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac {i \cosh (c+d x)}{a d}+\frac {\int 1 \, dx}{a}\\ &=\frac {x}{a}-\frac {i \cosh (c+d x)}{a d}\\ \end {align*}
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Mathematica [B] time = 0.15, size = 139, normalized size = 6.32 \[ \frac {\cosh ^3(c+d x) \left (-i \sqrt {1+i \sinh (c+d x)} \sinh (c+d x)+\sqrt {1+i \sinh (c+d x)}-2 \sqrt {1-i \sinh (c+d x)} \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (c+d x)}}{\sqrt {2}}\right )\right )}{a d \sqrt {1+i \sinh (c+d x)} (\sinh (c+d x)-i) (\sinh (c+d x)+i)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 40, normalized size = 1.82 \[ \frac {{\left (2 \, d x e^{\left (d x + c\right )} - i \, e^{\left (2 \, d x + 2 \, c\right )} - i\right )} e^{\left (-d x - c\right )}}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 41, normalized size = 1.86 \[ \frac {\frac {2 \, {\left (d x + c\right )}}{a} - \frac {i \, e^{\left (d x + c\right )}}{a} - \frac {i \, e^{\left (-d x - c\right )}}{a}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 85, normalized size = 3.86 \[ \frac {i}{d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d a}-\frac {i}{d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 44, normalized size = 2.00 \[ \frac {d x + c}{a d} - \frac {i \, e^{\left (d x + c\right )}}{2 \, a d} - \frac {i \, e^{\left (-d x - c\right )}}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 36, normalized size = 1.64 \[ \frac {x}{a}-\frac {\frac {{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{-c-d\,x}\,1{}\mathrm {i}}{2}}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 80, normalized size = 3.64 \[ \begin {cases} \frac {\left (- 2 i a d e^{2 c} e^{d x} - 2 i a d e^{- d x}\right ) e^{- c}}{4 a^{2} d^{2}} & \text {for}\: 4 a^{2} d^{2} e^{c} \neq 0 \\x \left (\frac {\left (- i e^{2 c} + 2 e^{c} + i\right ) e^{- c}}{2 a} - \frac {1}{a}\right ) & \text {otherwise} \end {cases} + \frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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