Optimal. Leaf size=117 \[ -\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4}-\frac {3 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^3}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^2}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))} \]
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Rubi [A]
time = 0.05, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 4, 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)}{35 d (1-i \sinh (c+d x))}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^2}-\frac {3 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^3}-\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4} \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))^4} \, dx &=-\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4}+\frac {3}{7} \int \frac {1}{(1-i \sinh (c+d x))^3} \, dx\\ &=-\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4}-\frac {3 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^3}+\frac {6}{35} \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx\\ &=-\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4}-\frac {3 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^3}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^2}+\frac {2}{35} \int \frac {1}{1-i \sinh (c+d x)} \, dx\\ &=-\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4}-\frac {3 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^3}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^2}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 87, normalized size = 0.74 \begin {gather*} \frac {-21 i \cosh \left (\frac {3}{2} (c+d x)\right )+i \cosh \left (\frac {7}{2} (c+d x)\right )+35 \sinh \left (\frac {1}{2} (c+d x)\right )-7 \sinh \left (\frac {5}{2} (c+d x)\right )}{70 d \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.16, size = 121, normalized size = 1.03
method | result | size |
risch | \(-\frac {4 \left (-7 \,{\mathrm e}^{d x +c}+21 i {\mathrm e}^{2 d x +2 c}+35 \,{\mathrm e}^{3 d x +3 c}-i\right )}{35 d \left ({\mathrm e}^{d x +c}+i\right )^{7}}\) | \(51\) |
derivativedivides | \(\frac {\frac {72}{5 \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {16 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {2}{i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {8 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {16}{7 \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {6 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {12}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}}{d}\) | \(121\) |
default | \(\frac {\frac {72}{5 \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {16 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {2}{i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {8 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {16}{7 \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {6 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {12}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}}{d}\) | \(121\) |
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. 372 vs. \(2 (93) = 186\).
time = 0.30, size = 372, normalized size = 3.18 \begin {gather*} \frac {4 \, e^{\left (-d x - c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} - i\right )}} + \frac {12 i \, e^{\left (-2 \, d x - 2 \, c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} - i\right )}} - \frac {4 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (7 \, e^{\left (-d x - c\right )} + 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} - i\right )}} - \frac {4 i}{35 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} - i\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 120, normalized size = 1.03 \begin {gather*} -\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} + 21 i \, e^{\left (2 \, d x + 2 \, c\right )} - 7 \, e^{\left (d x + c\right )} - i\right )}}{35 \, {\left (d e^{\left (7 \, d x + 7 \, c\right )} + 7 i \, d e^{\left (6 \, d x + 6 \, c\right )} - 21 \, d e^{\left (5 \, d x + 5 \, c\right )} - 35 i \, d e^{\left (4 \, d x + 4 \, c\right )} + 35 \, d e^{\left (3 \, d x + 3 \, c\right )} + 21 i \, d e^{\left (2 \, d x + 2 \, c\right )} - 7 \, 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 = 0.29, size = 155, normalized size = 1.32 \begin {gather*} \frac {- 140 e^{3 c} e^{3 d x} - 84 i e^{2 c} e^{2 d x} + 28 e^{c} e^{d x} + 4 i}{35 d e^{7 c} e^{7 d x} + 245 i d e^{6 c} e^{6 d x} - 735 d e^{5 c} e^{5 d x} - 1225 i d e^{4 c} e^{4 d x} + 1225 d e^{3 c} e^{3 d x} + 735 i d e^{2 c} e^{2 d x} - 245 d e^{c} e^{d x} - 35 i d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 47, normalized size = 0.40 \begin {gather*} -\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} + 21 i \, e^{\left (2 \, d x + 2 \, c\right )} - 7 \, e^{\left (d x + c\right )} - i\right )}}{35 \, d {\left (e^{\left (d x + c\right )} + i\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.91, size = 52, normalized size = 0.44 \begin {gather*} -\frac {4\,\left (21\,{\mathrm {e}}^{2\,c+2\,d\,x}-1+{\mathrm {e}}^{c+d\,x}\,7{}\mathrm {i}-{\mathrm {e}}^{3\,c+3\,d\,x}\,35{}\mathrm {i}\right )}{35\,d\,{\left (-1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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