Optimal. Leaf size=88 \[ \frac {i \cosh (c+d x)}{5 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{15 d (1+i \sinh (c+d x))^2}+\frac {2 i \cosh (c+d x)}{15 d (1+i \sinh (c+d x))} \]
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Rubi [A]
time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2729, 2727}
\begin {gather*} \frac {2 i \cosh (c+d x)}{15 d (1+i \sinh (c+d x))}+\frac {2 i \cosh (c+d x)}{15 d (1+i \sinh (c+d x))^2}+\frac {i \cosh (c+d x)}{5 d (1+i \sinh (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2729
Rubi steps
\begin {align*} \int \frac {1}{(1+i \sinh (c+d x))^3} \, dx &=\frac {i \cosh (c+d x)}{5 d (1+i \sinh (c+d x))^3}+\frac {2}{5} \int \frac {1}{(1+i \sinh (c+d x))^2} \, dx\\ &=\frac {i \cosh (c+d x)}{5 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{15 d (1+i \sinh (c+d x))^2}+\frac {2}{15} \int \frac {1}{1+i \sinh (c+d x)} \, dx\\ &=\frac {i \cosh (c+d x)}{5 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{15 d (1+i \sinh (c+d x))^2}+\frac {2 i \cosh (c+d x)}{15 d (1+i \sinh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 81, normalized size = 0.92 \begin {gather*} \frac {10-15 \cosh (c+d x)-6 \cosh (2 (c+d x))+\cosh (3 (c+d x))+15 i \sinh (c+d x)-6 i \sinh (2 (c+d x))-i \sinh (3 (c+d x))}{30 d (-i+\sinh (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.42, size = 88, normalized size = 1.00
method | result | size |
risch | \(-\frac {4 i \left (-5 i {\mathrm e}^{d x +c}+10 \,{\mathrm e}^{2 d x +2 c}-1\right )}{15 d \left ({\mathrm e}^{d x +c}-i\right )^{5}}\) | \(40\) |
derivativedivides | \(\frac {-\frac {4 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {16}{3 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {8}{5 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{d}\) | \(88\) |
default | \(\frac {-\frac {4 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {16}{3 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {8}{5 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{d}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 211 vs. \(2 (70) = 140\).
time = 0.29, size = 211, normalized size = 2.40 \begin {gather*} \frac {20 i \, e^{\left (-d x - c\right )}}{-15 \, d {\left (-5 i \, e^{\left (-d x - c\right )} - 10 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} - i \, e^{\left (-5 \, d x - 5 \, c\right )} + 1\right )}} + \frac {40 \, e^{\left (-2 \, d x - 2 \, c\right )}}{-15 \, d {\left (-5 i \, e^{\left (-d x - c\right )} - 10 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} - i \, e^{\left (-5 \, d x - 5 \, c\right )} + 1\right )}} - \frac {4}{-15 \, d {\left (-5 i \, e^{\left (-d x - c\right )} - 10 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} - i \, e^{\left (-5 \, d x - 5 \, c\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 85, normalized size = 0.97 \begin {gather*} -\frac {4 \, {\left (10 i \, e^{\left (2 \, d x + 2 \, c\right )} + 5 \, e^{\left (d x + c\right )} - i\right )}}{15 \, {\left (d e^{\left (5 \, d x + 5 \, c\right )} - 5 i \, d e^{\left (4 \, d x + 4 \, c\right )} - 10 \, d e^{\left (3 \, d x + 3 \, c\right )} + 10 i \, d e^{\left (2 \, d x + 2 \, c\right )} + 5 \, d e^{\left (d x + c\right )} - i \, d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 109, normalized size = 1.24 \begin {gather*} \frac {- 40 i e^{2 c} e^{2 d x} - 20 e^{c} e^{d x} + 4 i}{15 d e^{5 c} e^{5 d x} - 75 i d e^{4 c} e^{4 d x} - 150 d e^{3 c} e^{3 d x} + 150 i d e^{2 c} e^{2 d x} + 75 d e^{c} e^{d x} - 15 i d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 36, normalized size = 0.41 \begin {gather*} -\frac {4 i \, {\left (10 \, e^{\left (2 \, d x + 2 \, c\right )} - 5 i \, e^{\left (d x + c\right )} - 1\right )}}{15 \, d {\left (e^{\left (d x + c\right )} - i\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.69, size = 40, normalized size = 0.45 \begin {gather*} -\frac {\frac {4}{15}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3}+\frac {{\mathrm {e}}^{c+d\,x}\,4{}\mathrm {i}}{3}}{d\,{\left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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