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 = 0.101751, antiderivative size = 124, normalized size of antiderivative = 1., 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 2664
Rule 2754
Rule 12
Rule 2657
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*}
Mathematica [B] time = 1.76811, 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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Maple [B] time = 0.052, size = 314, normalized size = 2.5 \begin{align*}{\frac{1053}{32000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-3}}-{\frac{{\frac{99\,i}{8000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-3}}+{\frac{783}{128000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-2}}-{\frac{{\frac{3753\,i}{64000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-2}}-{\frac{39933}{1024000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-1}}-{\frac{{\frac{8361\,i}{256000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-1}}+{\frac{385}{32768\,d}\ln \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) }+{\frac{1053}{32000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-3}}+{\frac{{\frac{99\,i}{8000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-3}}-{\frac{783}{128000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-2}}-{\frac{{\frac{3753\,i}{64000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-2}}-{\frac{39933}{1024000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-1}}+{\frac{{\frac{8361\,i}{256000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-1}}-{\frac{385}{32768\,d}\ln \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.71295, size = 205, normalized size = 1.65 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.17341, size = 946, normalized size = 7.63 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.23034, size = 219, normalized size = 1.77 \begin{align*} \frac{- \frac{385 i e^{- c} e^{5 d x}}{12288 d} - \frac{9625 e^{- 2 c} e^{4 d x}}{36864 d} + \frac{119735 i e^{- 3 c} e^{3 d x}}{165888 d} + \frac{12137 e^{- 4 c} e^{2 d x}}{18432 d} - \frac{2725 i e^{- 5 c} e^{d x}}{12288 d} - \frac{311 e^{- 6 c}}{12288 d}}{e^{6 d x} - 10 i e^{- c} e^{5 d x} - \frac{109 e^{- 2 c} e^{4 d x}}{3} + \frac{1540 i e^{- 3 c} e^{3 d x}}{27} + \frac{109 e^{- 4 c} e^{2 d x}}{3} - 10 i e^{- 5 c} e^{d x} - e^{- 6 c}} + \frac{- \frac{385 \log{\left (e^{d x} - 3 i e^{- c} \right )}}{32768} + \frac{385 \log{\left (e^{d x} - \frac{i e^{- c}}{3} \right )}}{32768}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29281, size = 153, normalized size = 1.23 \begin{align*} \frac{385 \, \log \left (3 \, e^{\left (d x + c\right )} - i\right )}{32768 \, d} - \frac{385 \, \log \left (e^{\left (d x + c\right )} - 3 i\right )}{32768 \, d} - \frac{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}{12288 \, d{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} - 10 i \, e^{\left (d x + c\right )} - 3\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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