Integrand size = 14, antiderivative size = 131 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^3} \, dx=\frac {43 i \log \left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {5 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}-\frac {45 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))} \]
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Time = 0.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2743, 2833, 12, 2739, 630, 31} \[ \int \frac {1}{(3+5 i \sinh (c+d x))^3} \, dx=-\frac {45 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))}+\frac {5 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}+\frac {43 i \log \left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d} \]
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Rule 12
Rule 31
Rule 630
Rule 2739
Rule 2743
Rule 2833
Rubi steps \begin{align*} \text {integral}& = \frac {5 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}+\frac {1}{32} \int \frac {-6+5 i \sinh (c+d x)}{(3+5 i \sinh (c+d x))^2} \, dx \\ & = \frac {5 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}-\frac {45 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))}+\frac {1}{512} \int \frac {43}{3+5 i \sinh (c+d x)} \, dx \\ & = \frac {5 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}-\frac {45 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))}+\frac {43}{512} \int \frac {1}{3+5 i \sinh (c+d x)} \, dx \\ & = \frac {5 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}-\frac {45 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))}-\frac {(43 i) \text {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 )}{256 d} \\ & = \frac {5 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}-\frac {45 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))}-\frac {(129 i) \text {Subst}\left (\int \frac {1}{1+3 x} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{2048 d}+\frac {(129 i) \text {Subst}\left (\int \frac {1}{9+3 x} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{2048 d} \\ & = \frac {43 i \log \left (3+i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 i \log \left (1+3 i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {5 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}-\frac {45 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.56 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^3} \, dx=\frac {86 \arctan \left (3 \coth \left (\frac {1}{2} (c+d x)\right )\right )+86 \arctan \left (3 \tanh \left (\frac {1}{2} (c+d x)\right )\right )-43 i \log (4-5 \cosh (c+d x))+43 i \log (4+5 \cosh (c+d x))-\frac {80 i}{\left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {80 i}{\left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}+\left (-\frac {120}{3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )}-\frac {360}{\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )}\right ) \sinh \left (\frac {1}{2} (c+d x)\right )}{4096 d} \]
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Time = 1.78 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {i \left (-387 i {\mathrm e}^{2 d x +2 c}+215 \,{\mathrm e}^{3 d x +3 c}+225 i-325 \,{\mathrm e}^{d x +c}\right )}{256 d \left (5 \,{\mathrm e}^{2 d x +2 c}-5-6 i {\mathrm e}^{d x +c}\right )^{2}}-\frac {43 i \ln \left ({\mathrm e}^{d x +c}-\frac {4}{5}-\frac {3 i}{5}\right )}{2048 d}+\frac {43 i \ln \left ({\mathrm e}^{d x +c}+\frac {4}{5}-\frac {3 i}{5}\right )}{2048 d}\) | \(100\) |
derivativedivides | \(\frac {\frac {25 i}{128 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{2}}+\frac {43 i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}{2048}+\frac {15}{512 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}-\frac {43 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}{2048}-\frac {25 i}{1152 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}-\frac {155}{4608 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}}{d}\) | \(110\) |
default | \(\frac {\frac {25 i}{128 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{2}}+\frac {43 i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}{2048}+\frac {15}{512 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}-\frac {43 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}{2048}-\frac {25 i}{1152 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}-\frac {155}{4608 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}}{d}\) | \(110\) |
parallelrisch | \(\frac {38700 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-9 i\right )-38700 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+\left (9675 i \cosh \left (2 d x +2 c \right )+22059 i+23220 \sinh \left (d x +c \right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {i}{3}\right )+\left (-9675 i \cosh \left (2 d x +2 c \right )-22059 i-23220 \sinh \left (d x +c \right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )-3960 i \cosh \left (d x +c \right )+5500 i \cosh \left (2 d x +2 c \right )-9460 i+13200 \sinh \left (d x +c \right )+8100 \sinh \left (2 d x +2 c \right )}{18432 d \left (43-25 \cosh \left (2 d x +2 c \right )+60 i \sinh \left (d x +c \right )\right )}\) | \(183\) |
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Time = 0.32 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^3} \, dx=-\frac {43 \, {\left (-25 i \, e^{\left (4 \, d x + 4 \, c\right )} - 60 \, e^{\left (3 \, d x + 3 \, c\right )} + 86 i \, e^{\left (2 \, d x + 2 \, c\right )} + 60 \, e^{\left (d x + c\right )} - 25 i\right )} \log \left (e^{\left (d x + c\right )} - \frac {3}{5} i + \frac {4}{5}\right ) + 43 \, {\left (25 i \, e^{\left (4 \, d x + 4 \, c\right )} + 60 \, e^{\left (3 \, d x + 3 \, c\right )} - 86 i \, e^{\left (2 \, d x + 2 \, c\right )} - 60 \, e^{\left (d x + c\right )} + 25 i\right )} \log \left (e^{\left (d x + c\right )} - \frac {3}{5} i - \frac {4}{5}\right ) + 1720 i \, e^{\left (3 \, d x + 3 \, c\right )} + 3096 \, e^{\left (2 \, d x + 2 \, c\right )} - 2600 i \, e^{\left (d x + c\right )} - 1800}{2048 \, {\left (25 \, d e^{\left (4 \, d x + 4 \, c\right )} - 60 i \, d e^{\left (3 \, d x + 3 \, c\right )} - 86 \, d e^{\left (2 \, d x + 2 \, c\right )} + 60 i \, d e^{\left (d x + c\right )} + 25 \, d\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^3} \, dx=\frac {- 215 i e^{3 c} e^{3 d x} - 387 e^{2 c} e^{2 d x} + 325 i e^{c} e^{d x} + 225}{6400 d e^{4 c} e^{4 d x} - 15360 i d e^{3 c} e^{3 d x} - 22016 d e^{2 c} e^{2 d x} + 15360 i d e^{c} e^{d x} + 6400 d} + \frac {\operatorname {RootSum} {\left (4194304 z^{2} + 1849, \left ( i \mapsto i \log {\left (\frac {\left (- 8192 i i - 129 i\right ) e^{- c}}{215} + e^{d x} \right )} \right )\right )}}{d} \]
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Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^3} \, dx=-\frac {43 i \, \log \left (\frac {5 \, e^{\left (-d x - c\right )} + 3 i - 4}{5 \, e^{\left (-d x - c\right )} + 3 i + 4}\right )}{2048 \, d} - \frac {-325 i \, e^{\left (-d x - c\right )} - 387 \, e^{\left (-2 \, d x - 2 \, c\right )} + 215 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 225}{-256 \, d {\left (60 i \, e^{\left (-d x - c\right )} + 86 \, e^{\left (-2 \, d x - 2 \, c\right )} - 60 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 25 \, e^{\left (-4 \, d x - 4 \, c\right )} - 25\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^3} \, dx=-\frac {\frac {8 \, {\left (-215 i \, e^{\left (3 \, d x + 3 \, c\right )} - 387 \, e^{\left (2 \, d x + 2 \, c\right )} + 325 i \, e^{\left (d x + c\right )} + 225\right )}}{{\left (-5 i \, e^{\left (2 \, d x + 2 \, c\right )} - 6 \, e^{\left (d x + c\right )} + 5 i\right )}^{2}} - 43 i \, \log \left (-\left (i - 2\right ) \, e^{\left (d x + c\right )} - 2 i + 1\right ) + 43 i \, \log \left (-\left (2 i - 1\right ) \, e^{\left (d x + c\right )} + i - 2\right )}{2048 \, d} \]
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Time = 1.83 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^3} \, dx=\frac {\frac {129}{6400\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,43{}\mathrm {i}}{1280\,d}}{1-{\mathrm {e}}^{2\,c+2\,d\,x}+\frac {{\mathrm {e}}^{c+d\,x}\,6{}\mathrm {i}}{5}}-\frac {\ln \left (-\frac {215}{4}+{\mathrm {e}}^{c+d\,x}\,\left (43-\frac {129}{4}{}\mathrm {i}\right )\right )\,43{}\mathrm {i}}{2048\,d}+\frac {\ln \left (\frac {215}{4}+{\mathrm {e}}^{c+d\,x}\,\left (43+\frac {129}{4}{}\mathrm {i}\right )\right )\,43{}\mathrm {i}}{2048\,d}-\frac {-\frac {3}{200\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,7{}\mathrm {i}}{1000\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-\frac {86\,{\mathrm {e}}^{2\,c+2\,d\,x}}{25}+1+\frac {{\mathrm {e}}^{c+d\,x}\,12{}\mathrm {i}}{5}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,12{}\mathrm {i}}{5}} \]
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