Integrand size = 17, antiderivative size = 101 \[ \int \frac {A+B \sinh (x)}{(i-\sinh (x))^4} \, dx=\frac {(i A-B) \cosh (x)}{7 (i-\sinh (x))^4}+\frac {(3 A-4 i B) \cosh (x)}{35 (i-\sinh (x))^3}-\frac {2 (3 i A+4 B) \cosh (x)}{105 (i-\sinh (x))^2}-\frac {2 (3 A-4 i B) \cosh (x)}{105 (i-\sinh (x))} \]
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Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2829, 2729, 2727} \[ \int \frac {A+B \sinh (x)}{(i-\sinh (x))^4} \, dx=-\frac {2 (3 A-4 i B) \cosh (x)}{105 (-\sinh (x)+i)}-\frac {2 (4 B+3 i A) \cosh (x)}{105 (-\sinh (x)+i)^2}+\frac {(3 A-4 i B) \cosh (x)}{35 (-\sinh (x)+i)^3}+\frac {(-B+i A) \cosh (x)}{7 (-\sinh (x)+i)^4} \]
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Rule 2727
Rule 2729
Rule 2829
Rubi steps \begin{align*} \text {integral}& = \frac {(i A-B) \cosh (x)}{7 (i-\sinh (x))^4}+\frac {1}{7} (-3 i A-4 B) \int \frac {1}{(i-\sinh (x))^3} \, dx \\ & = \frac {(i A-B) \cosh (x)}{7 (i-\sinh (x))^4}+\frac {(3 A-4 i B) \cosh (x)}{35 (i-\sinh (x))^3}-\frac {1}{35} (2 (3 A-4 i B)) \int \frac {1}{(i-\sinh (x))^2} \, dx \\ & = \frac {(i A-B) \cosh (x)}{7 (i-\sinh (x))^4}+\frac {(3 A-4 i B) \cosh (x)}{35 (i-\sinh (x))^3}-\frac {2 (3 i A+4 B) \cosh (x)}{105 (i-\sinh (x))^2}+\frac {1}{105} (2 (3 i A+4 B)) \int \frac {1}{i-\sinh (x)} \, dx \\ & = \frac {(i A-B) \cosh (x)}{7 (i-\sinh (x))^4}+\frac {(3 A-4 i B) \cosh (x)}{35 (i-\sinh (x))^3}-\frac {2 (3 i A+4 B) \cosh (x)}{105 (i-\sinh (x))^2}-\frac {2 (3 A-4 i B) \cosh (x)}{105 (i-\sinh (x))} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int \frac {A+B \sinh (x)}{(i-\sinh (x))^4} \, dx=\frac {\cosh (x) \left (36 i A+13 B+(-39 A+52 i B) \sinh (x)+(-24 i A-32 B) \sinh ^2(x)+(6 A-8 i B) \sinh ^3(x)\right )}{105 (-i+\sinh (x))^4} \]
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Time = 1.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.65
method | result | size |
risch | \(-\frac {4 \left (4 B -70 i B \,{\mathrm e}^{3 x}-63 i A \,{\mathrm e}^{2 x}+28 i B \,{\mathrm e}^{x}+3 i A -84 B \,{\mathrm e}^{2 x}+105 A \,{\mathrm e}^{3 x}+70 B \,{\mathrm e}^{4 x}-21 A \,{\mathrm e}^{x}\right )}{105 \left ({\mathrm e}^{x}-i\right )^{7}}\) | \(66\) |
default | \(-\frac {2 \left (8 i B +8 A \right )}{7 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{7}}-\frac {32 i A -24 B}{2 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{4}}-\frac {2 \left (-32 i B -36 A \right )}{5 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{5}}-\frac {-24 i A +24 B}{3 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{6}}-\frac {-6 i A +2 B}{\left (-i+\tanh \left (\frac {x}{2}\right )\right )^{2}}+\frac {2 A}{-i+\tanh \left (\frac {x}{2}\right )}-\frac {2 \left (10 i B +18 A \right )}{3 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{3}}\) | \(128\) |
parallelrisch | \(\frac {\left (1092 i A +476 B \right ) \cosh \left (2 x \right )+\left (-168 i A -14 B \right ) \cosh \left (3 x \right )+\left (-42 i A -21 B \right ) \cosh \left (4 x \right )+\left (-42 i B -336 A \right ) \sinh \left (2 x \right )+\left (152 i B -324 A \right ) \sinh \left (3 x \right )+\left (-5 i B +30 A \right ) \sinh \left (4 x \right )+\left (168 i A +14 B \right ) \cosh \left (x \right )+\left (-560 i B +2100 A \right ) \sinh \left (x \right )-1050 i A -455 B}{-5880 i \sinh \left (x \right )+1470 i \sinh \left (2 x \right )+840 i \sinh \left (3 x \right )-105 i \sinh \left (4 x \right )+2940 \cosh \left (2 x \right )+1470 \cosh \left (x \right )-105 \cosh \left (4 x \right )-630 \cosh \left (3 x \right )-3675}\) | \(162\) |
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Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \sinh (x)}{(i-\sinh (x))^4} \, dx=-\frac {4 \, {\left (70 \, B e^{\left (4 \, x\right )} + 35 \, {\left (3 \, A - 2 i \, B\right )} e^{\left (3 \, x\right )} + 21 \, {\left (-3 i \, A - 4 \, B\right )} e^{\left (2 \, x\right )} - 7 \, {\left (3 \, A - 4 i \, B\right )} e^{x} + 3 i \, A + 4 \, B\right )}}{105 \, {\left (e^{\left (7 \, x\right )} - 7 i \, e^{\left (6 \, x\right )} - 21 \, e^{\left (5 \, x\right )} + 35 i \, e^{\left (4 \, x\right )} + 35 \, e^{\left (3 \, x\right )} - 21 i \, e^{\left (2 \, x\right )} - 7 \, e^{x} + i\right )}} \]
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Time = 0.45 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08 \[ \int \frac {A+B \sinh (x)}{(i-\sinh (x))^4} \, dx=\frac {- 12 i A - 280 B e^{4 x} - 16 B + \left (- 420 A + 280 i B\right ) e^{3 x} + \left (84 A - 112 i B\right ) e^{x} + \left (252 i A + 336 B\right ) e^{2 x}}{105 e^{7 x} - 735 i e^{6 x} - 2205 e^{5 x} + 3675 i e^{4 x} + 3675 e^{3 x} - 2205 i e^{2 x} - 735 e^{x} + 105 i} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (67) = 134\).
Time = 0.21 (sec) , antiderivative size = 469, normalized size of antiderivative = 4.64 \[ \int \frac {A+B \sinh (x)}{(i-\sinh (x))^4} \, dx=\frac {4}{35} \, A {\left (\frac {7 \, e^{\left (-x\right )}}{7 \, e^{\left (-x\right )} - 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} + 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} + i} - \frac {21 i \, e^{\left (-2 \, x\right )}}{7 \, e^{\left (-x\right )} - 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} + 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} + i} - \frac {35 \, e^{\left (-3 \, x\right )}}{7 \, e^{\left (-x\right )} - 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} + 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} + i} + \frac {i}{7 \, e^{\left (-x\right )} - 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} + 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} + i}\right )} - \frac {8}{105} \, B {\left (\frac {14 i \, e^{\left (-x\right )}}{7 \, e^{\left (-x\right )} - 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} + 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} + i} + \frac {42 \, e^{\left (-2 \, x\right )}}{7 \, e^{\left (-x\right )} - 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} + 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} + i} - \frac {35 i \, e^{\left (-3 \, x\right )}}{7 \, e^{\left (-x\right )} - 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} + 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} + i} - \frac {35 \, e^{\left (-4 \, x\right )}}{7 \, e^{\left (-x\right )} - 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} + 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} + i} - \frac {2}{7 \, e^{\left (-x\right )} - 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} + 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} + i}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.59 \[ \int \frac {A+B \sinh (x)}{(i-\sinh (x))^4} \, dx=-\frac {4 \, {\left (70 \, B e^{\left (4 \, x\right )} + 105 \, A e^{\left (3 \, x\right )} - 70 i \, B e^{\left (3 \, x\right )} - 63 i \, A e^{\left (2 \, x\right )} - 84 \, B e^{\left (2 \, x\right )} - 21 \, A e^{x} + 28 i \, B e^{x} + 3 i \, A + 4 \, B\right )}}{105 \, {\left (e^{x} - i\right )}^{7}} \]
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Time = 1.71 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67 \[ \int \frac {A+B \sinh (x)}{(i-\sinh (x))^4} \, dx=\frac {\frac {12\,A\,{\mathrm {e}}^{2\,x}}{5}+\frac {B\,16{}\mathrm {i}}{105}-\frac {4\,A}{35}+A\,{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,\left (\frac {16\,B}{15}+\frac {A\,4{}\mathrm {i}}{5}\right )-\frac {B\,{\mathrm {e}}^{2\,x}\,16{}\mathrm {i}}{5}+\frac {8\,B\,{\mathrm {e}}^{3\,x}}{3}+\frac {B\,{\mathrm {e}}^{4\,x}\,8{}\mathrm {i}}{3}}{{\left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^7} \]
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