Optimal. Leaf size=73 \[ \frac {i \log \left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
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Rubi [A] time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2660, 616, 31} \[ \frac {i \log \left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rule 2660
Rubi steps
\begin {align*} \int \frac {1}{3+5 i \sinh (c+d x)} \, dx &=-\frac {(2 i) \operatorname {Subst}\left (\int \frac {1}{3+10 x+3 x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{d}\\ &=-\frac {(3 i) \operatorname {Subst}\left (\int \frac {1}{1+3 x} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{4 d}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {1}{9+3 x} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{4 d}\\ &=\frac {i \log \left (3+i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {i \log \left (1+3 i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 81, normalized size = 1.11 \[ \frac {\tan ^{-1}\left (3 \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {i \log (4-5 \cosh (c+d x))}{8 d}+\frac {i \log (5 \cosh (c+d x)+4)}{8 d}+\frac {\tan ^{-1}\left (3 \coth \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 28, normalized size = 0.38 \[ \frac {i \, \log \left (e^{\left (d x + c\right )} - \frac {3}{5} i + \frac {4}{5}\right ) - i \, \log \left (e^{\left (d x + c\right )} - \frac {3}{5} i - \frac {4}{5}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 32, normalized size = 0.44 \[ -\frac {-i \, \log \left (-\left (i - 2\right ) \, e^{\left (d x + c\right )} - 2 i + 1\right ) + i \, \log \left (-\left (2 i - 1\right ) \, e^{\left (d x + c\right )} + i - 2\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 0.58 \[ \frac {i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}{4 d}-\frac {i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}{4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 19, normalized size = 0.26 \[ \frac {\arctan \left (\frac {5}{4} i \, e^{\left (-d x - c\right )} - \frac {3}{4}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 39, normalized size = 0.53 \[ -\frac {\ln \left (-\frac {5}{2}+{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (2-\frac {3}{2}{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{4\,d}+\frac {\ln \left (\frac {5}{2}+{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (2+\frac {3}{2}{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 34, normalized size = 0.47 \[ \frac {\operatorname {RootSum} {\left (400 z^{2} + 1, \left (i \mapsto i \log {\left (16 i i e^{c} + \frac {3 i e^{c}}{5} + e^{- d x} \right )} \right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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