Optimal. Leaf size=124 \[ -\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {385 i \tan ^{-1}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{16384 d}+\frac {385 x}{32768} \]
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Rubi [A] time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2664, 2754, 12, 2657} \[ -\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {385 i \tan ^{-1}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{16384 d}+\frac {385 x}{32768} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2657
Rule 2664
Rule 2754
Rubi steps
\begin {align*} \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx &=-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {1}{48} \int \frac {-15+6 i \sinh (c+d x)}{(5+3 i \sinh (c+d x))^3} \, dx\\ &=-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}+\frac {\int \frac {186-75 i \sinh (c+d x)}{(5+3 i \sinh (c+d x))^2} \, dx}{1536}\\ &=-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}-\frac {\int -\frac {1155}{5+3 i \sinh (c+d x)} \, dx}{24576}\\ &=-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}+\frac {385 \int \frac {1}{5+3 i \sinh (c+d x)} \, dx}{8192}\\ &=\frac {385 x}{32768}-\frac {385 i \tan ^{-1}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{16384 d}-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}\\ \end {align*}
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Mathematica [B] time = 1.93, size = 308, normalized size = 2.48 \[ \frac {\frac {2 (-298563 i \sinh (c+d x)+89364 i \sinh (2 (c+d x))+8397 i \sinh (3 (c+d x))+166615 \cosh (c+d x)+82530 \cosh (2 (c+d x))-13995 \cosh (3 (c+d x))-235150)}{(3 \sinh (c+d x)-5 i)^3}+\frac {2656-192 i}{\left ((1+2 i) \cosh \left (\frac {1}{2} (c+d x)\right )-(2+i) \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2656+192 i}{\left ((1+2 i) \sinh \left (\frac {1}{2} (c+d x)\right )+(2+i) \cosh \left (\frac {1}{2} (c+d x)\right )\right )^2}-1925 \log (5 \cosh (c+d x)-4 \sinh (c+d x))+1925 \log (4 \sinh (c+d x)+5 \cosh (c+d x))-3850 i \tan ^{-1}\left (\frac {2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )}{\cosh \left (\frac {1}{2} (c+d x)\right )-2 \sinh \left (\frac {1}{2} (c+d x)\right )}\right )+3850 i \tan ^{-1}\left (\frac {2 \sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right )}{\sinh \left (\frac {1}{2} (c+d x)\right )+2 \cosh \left (\frac {1}{2} (c+d x)\right )}\right )}{327680 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 281, normalized size = 2.27 \[ \frac {{\left (31185 \, e^{\left (6 \, d x + 6 \, c\right )} - 311850 i \, e^{\left (5 \, d x + 5 \, c\right )} - 1133055 \, e^{\left (4 \, d x + 4 \, c\right )} + 1778700 i \, e^{\left (3 \, d x + 3 \, c\right )} + 1133055 \, e^{\left (2 \, d x + 2 \, c\right )} - 311850 i \, e^{\left (d x + c\right )} - 31185\right )} \log \left (e^{\left (d x + c\right )} - \frac {1}{3} i\right ) - {\left (31185 \, e^{\left (6 \, d x + 6 \, c\right )} - 311850 i \, e^{\left (5 \, d x + 5 \, c\right )} - 1133055 \, e^{\left (4 \, d x + 4 \, c\right )} + 1778700 i \, e^{\left (3 \, d x + 3 \, c\right )} + 1133055 \, e^{\left (2 \, d x + 2 \, c\right )} - 311850 i \, e^{\left (d x + c\right )} - 31185\right )} \log \left (e^{\left (d x + c\right )} - 3 i\right ) - 83160 i \, e^{\left (5 \, d x + 5 \, c\right )} - 693000 \, e^{\left (4 \, d x + 4 \, c\right )} + 1915760 i \, e^{\left (3 \, d x + 3 \, c\right )} + 1747728 \, e^{\left (2 \, d x + 2 \, c\right )} - 588600 i \, e^{\left (d x + c\right )} - 67176}{2654208 \, d e^{\left (6 \, d x + 6 \, c\right )} - 26542080 i \, d e^{\left (5 \, d x + 5 \, c\right )} - 96436224 \, d e^{\left (4 \, d x + 4 \, c\right )} + 151388160 i \, d e^{\left (3 \, d x + 3 \, c\right )} + 96436224 \, d e^{\left (2 \, d x + 2 \, c\right )} - 26542080 i \, d e^{\left (d x + c\right )} - 2654208 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 109, normalized size = 0.88 \[ -\frac {\frac {8 \, {\left (10395 i \, e^{\left (5 \, d x + 5 \, c\right )} + 86625 \, e^{\left (4 \, d x + 4 \, c\right )} - 239470 i \, e^{\left (3 \, d x + 3 \, c\right )} - 218466 \, e^{\left (2 \, d x + 2 \, c\right )} + 73575 i \, e^{\left (d x + c\right )} + 8397\right )}}{{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} - 10 i \, e^{\left (d x + c\right )} - 3\right )}^{3}} - 1155 \, \log \left (3 \, e^{\left (d x + c\right )} - i\right ) + 1155 \, \log \left (e^{\left (d x + c\right )} - 3 i\right )}{98304 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 314, normalized size = 2.53 \[ \frac {1053}{32000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{3}}+\frac {99 i}{8000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{3}}-\frac {783}{128000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{2}}-\frac {3753 i}{64000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{2}}-\frac {39933}{1024000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}+\frac {8361 i}{256000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}-\frac {385 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}{32768 d}+\frac {1053}{32000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{3}}-\frac {99 i}{8000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{3}}+\frac {783}{128000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{2}}-\frac {3753 i}{64000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{2}}-\frac {39933}{1024000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}-\frac {8361 i}{256000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}+\frac {385 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}{32768 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 152, normalized size = 1.23 \[ -\frac {385 i \, \arctan \left (\frac {3}{4} \, e^{\left (-d x - c\right )} + \frac {5}{4} i\right )}{16384 \, d} - \frac {73575 i \, e^{\left (-d x - c\right )} + 218466 \, e^{\left (-2 \, d x - 2 \, c\right )} - 239470 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 86625 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10395 i \, e^{\left (-5 \, d x - 5 \, c\right )} - 8397}{d {\left (3317760 i \, e^{\left (-d x - c\right )} + 12054528 \, e^{\left (-2 \, d x - 2 \, c\right )} - 18923520 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 12054528 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3317760 i \, e^{\left (-5 \, d x - 5 \, c\right )} + 331776 \, e^{\left (-6 \, d x - 6 \, c\right )} - 331776\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.11, size = 232, normalized size = 1.87 \[ \frac {\frac {1925}{36864\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,385{}\mathrm {i}}{12288\,d}}{1-{\mathrm {e}}^{2\,c+2\,d\,x}+\frac {{\mathrm {e}}^{c+d\,x}\,10{}\mathrm {i}}{3}}+\frac {\frac {41}{486\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,365{}\mathrm {i}}{1458\,d}}{\frac {109\,{\mathrm {e}}^{4\,c+4\,d\,x}}{3}-\frac {109\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3}-{\mathrm {e}}^{6\,c+6\,d\,x}+1+{\mathrm {e}}^{c+d\,x}\,10{}\mathrm {i}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,1540{}\mathrm {i}}{27}+{\mathrm {e}}^{5\,c+5\,d\,x}\,10{}\mathrm {i}}-\frac {385\,\ln \left (-\frac {385\,{\mathrm {e}}^{c+d\,x}}{4}+\frac {1155}{4}{}\mathrm {i}\right )}{32768\,d}+\frac {385\,\ln \left (\frac {3465\,{\mathrm {e}}^{c+d\,x}}{4}-\frac {1155}{4}{}\mathrm {i}\right )}{32768\,d}-\frac {\frac {3461}{31104\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,385{}\mathrm {i}}{10368\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-\frac {118\,{\mathrm {e}}^{2\,c+2\,d\,x}}{9}+1+\frac {{\mathrm {e}}^{c+d\,x}\,20{}\mathrm {i}}{3}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,20{}\mathrm {i}}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotInvertible} \]
Verification of antiderivative is not currently implemented for this CAS.
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