Integrand size = 13, antiderivative size = 58 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {7 x}{2}-\frac {16}{3} i \cosh (x)+\frac {7}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}-\frac {8 \cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))} \]
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Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2844, 3056, 2813} \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {7 x}{2}-\frac {16}{3} i \cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{3 (\sinh (x)+i)^2}-\frac {8 \sinh ^2(x) \cosh (x)}{3 (\sinh (x)+i)}+\frac {7}{2} \sinh (x) \cosh (x) \]
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Rule 2813
Rule 2844
Rule 3056
Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}+\frac {1}{3} \int \frac {\sinh ^2(x) (-3 i+5 \sinh (x))}{i+\sinh (x)} \, dx \\ & = -\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}-\frac {8 \cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))}-\frac {1}{3} i \int (16+21 i \sinh (x)) \sinh (x) \, dx \\ & = -\frac {7 x}{2}-\frac {16}{3} i \cosh (x)+\frac {7}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}-\frac {8 \cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(147\) vs. \(2(58)=116\).
Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.53 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=\frac {5 i \cosh (x)}{6 (1-i \sinh (x))^2}-\frac {31 i \cosh (x)}{6 (1-i \sinh (x))}-\frac {i \sqrt {2} \cosh (x) \sqrt {1+\frac {1}{2} (-1+i \sinh (x))}}{\sqrt {1+i \sinh (x)}}-\frac {7 i \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right ) \cosh (x)}{\sqrt {1-i \sinh (x)} \sqrt {1+i \sinh (x)}}-\frac {\cosh (x) \sinh ^3(x)}{2 (1-i \sinh (x))^2} \]
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Time = 3.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.90
method | result | size |
risch | \(-\frac {7 x}{2}+\frac {{\mathrm e}^{2 x}}{8}-i {\mathrm e}^{x}-i {\mathrm e}^{-x}-\frac {{\mathrm e}^{-2 x}}{8}-\frac {2 i \left (21 i {\mathrm e}^{x}+12 \,{\mathrm e}^{2 x}-11\right )}{3 \left ({\mathrm e}^{x}+i\right )^{3}}\) | \(52\) |
default | \(\frac {\frac {1}{2}+2 i}{\tanh \left (\frac {x}{2}\right )-1}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {6}{\tanh \left (\frac {x}{2}\right )+i}+\frac {\frac {1}{2}-2 i}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}\) | \(98\) |
parallelrisch | \(\frac {-993 i+420 \left (\cosh \left (2 x \right )+4 i \sinh \left (x \right )-3-i \sinh \left (2 x \right )+2 \cosh \left (x \right )\right ) \ln \left (1-\coth \left (x \right )+\operatorname {csch}\left (x \right )\right )+420 \left (-\cosh \left (2 x \right )-4 i \sinh \left (x \right )+3+i \sinh \left (2 x \right )-2 \cosh \left (x \right )\right ) \ln \left (\coth \left (x \right )-\operatorname {csch}\left (x \right )+1\right )+1612 i \cosh \left (x \right )-544 i \cosh \left (2 x \right )-60 i \cosh \left (3 x \right )-15 i \cosh \left (4 x \right )-90 \sinh \left (3 x \right )+15 \sinh \left (4 x \right )-714 \sinh \left (x \right )+966 \sinh \left (2 x \right )}{120 \cosh \left (2 x \right )+480 i \sinh \left (x \right )-360-120 i \sinh \left (2 x \right )+240 \cosh \left (x \right )}\) | \(146\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (42) = 84\).
Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.53 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {21 \, {\left (4 \, x - 3\right )} e^{\left (5 \, x\right )} + 21 \, {\left (12 i \, x + 7 i\right )} e^{\left (4 \, x\right )} - 3 \, {\left (84 \, x + 127\right )} e^{\left (3 \, x\right )} - {\left (84 i \, x + 239 i\right )} e^{\left (2 \, x\right )} - 3 \, e^{\left (7 \, x\right )} + 15 i \, e^{\left (6 \, x\right )} + 15 \, e^{x} - 3 i}{24 \, {\left (e^{\left (5 \, x\right )} + 3 i \, e^{\left (4 \, x\right )} - 3 \, e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\right )}\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.21 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=- \frac {7 x}{2} + \frac {- 24 i e^{2 x} + 42 e^{x} + 22 i}{3 e^{3 x} + 9 i e^{2 x} - 9 e^{x} - 3 i} + \frac {e^{2 x}}{8} - i e^{x} - i e^{- x} - \frac {e^{- 2 x}}{8} \]
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Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {7}{2} \, x + \frac {15 \, e^{\left (-x\right )} + 239 i \, e^{\left (-2 \, x\right )} - 405 \, e^{\left (-3 \, x\right )} - 216 i \, e^{\left (-4 \, x\right )} + 3 i}{8 \, {\left (3 i \, e^{\left (-2 \, x\right )} - 9 \, e^{\left (-3 \, x\right )} - 9 i \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}\right )}} - i \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
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Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.86 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {7}{2} \, x - \frac {{\left (216 i \, e^{\left (4 \, x\right )} - 405 \, e^{\left (3 \, x\right )} - 239 i \, e^{\left (2 \, x\right )} + 15 \, e^{x} - 3 i\right )} e^{\left (-2 \, x\right )}}{24 \, {\left (e^{x} + i\right )}^{3}} + \frac {1}{8} \, e^{\left (2 \, x\right )} - i \, e^{x} \]
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Time = 1.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.67 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=\frac {{\mathrm {e}}^{2\,x}}{8}-{\mathrm {e}}^{-x}\,1{}\mathrm {i}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {7\,x}{2}-{\mathrm {e}}^x\,1{}\mathrm {i}-\frac {-2+\frac {{\mathrm {e}}^x\,8{}\mathrm {i}}{3}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}+\frac {4\,{\mathrm {e}}^x-\frac {{\mathrm {e}}^{2\,x}\,8{}\mathrm {i}}{3}+\frac {8}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {8{}\mathrm {i}}{3\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \]
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