Integrand size = 15, antiderivative size = 91 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=-\frac {(i A+B) \cosh (x)}{7 (i+\sinh (x))^4}-\frac {(3 A+4 i B) \cosh (x)}{35 (i+\sinh (x))^3}+\frac {2 (3 i A-4 B) \cosh (x)}{105 (i+\sinh (x))^2}+\frac {2 (3 A+4 i B) \cosh (x)}{105 (i+\sinh (x))} \]
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Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 = 67, normalized size of antiderivative = 0.74 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=\frac {\cosh (x) \left (-36 i A+13 B-13 (3 A+4 i B) \sinh (x)+8 i (3 A+4 i B) \sinh ^2(x)+(6 A+8 i B) \sinh ^3(x)\right )}{105 (i+\sinh (x))^4} \]
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Time = 1.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73
method | result | size |
risch | \(-\frac {4 \left (4 B -28 i B \,{\mathrm e}^{x}+63 i A \,{\mathrm e}^{2 x}+70 i B \,{\mathrm e}^{3 x}-3 i A +105 A \,{\mathrm e}^{3 x}+70 B \,{\mathrm e}^{4 x}-84 B \,{\mathrm e}^{2 x}-21 A \,{\mathrm e}^{x}\right )}{105 \left ({\mathrm e}^{x}+i\right )^{7}}\) | \(66\) |
default | \(-\frac {2 \left (-10 i B +18 A \right )}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {6 i A +2 B}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {-32 i A -24 B}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}-\frac {24 i A +24 B}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{6}}+\frac {2 A}{\tanh \left (\frac {x}{2}\right )+i}-\frac {2 \left (32 i B -36 A \right )}{5 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}-\frac {2 \left (-8 i B +8 A \right )}{7 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{7}}\) | \(128\) |
parallelrisch | \(\frac {\left (1092 i A -476 B \right ) \cosh \left (2 x \right )+\left (-168 i A +14 B \right ) \cosh \left (3 x \right )+\left (-42 i A +21 B \right ) \cosh \left (4 x \right )+\left (-42 i B +336 A \right ) \sinh \left (2 x \right )+\left (152 i B +324 A \right ) \sinh \left (3 x \right )+\left (-5 i B -30 A \right ) \sinh \left (4 x \right )+\left (168 i A -14 B \right ) \cosh \left (x \right )+\left (-560 i B -2100 A \right ) \sinh \left (x \right )-1050 i A +455 B}{840 i \sinh \left (3 x \right )-5880 i \sinh \left (x \right )+1470 i \sinh \left (2 x \right )-105 i \sinh \left (4 x \right )-1470 \cosh \left (x \right )+630 \cosh \left (3 x \right )+105 \cosh \left (4 x \right )-2940 \cosh \left (2 x \right )+3675}\) | \(162\) |
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Time = 0.32 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.04 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=-\frac {4 \, {\left (70 \, B e^{\left (4 \, x\right )} + 35 \, {\left (3 \, A + 2 i \, B\right )} e^{\left (3 \, x\right )} + 21 \, {\left (3 i \, A - 4 \, B\right )} e^{\left (2 \, x\right )} - 7 \, {\left (3 \, A + 4 i \, B\right )} e^{x} - 3 i \, A + 4 \, B\right )}}{105 \, {\left (e^{\left (7 \, x\right )} + 7 i \, e^{\left (6 \, x\right )} - 21 \, e^{\left (5 \, x\right )} - 35 i \, e^{\left (4 \, x\right )} + 35 \, e^{\left (3 \, x\right )} + 21 i \, e^{\left (2 \, x\right )} - 7 \, e^{x} - i\right )}} \]
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Time = 0.45 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=\frac {12 i A - 280 B e^{4 x} - 16 B + \left (- 420 A - 280 i B\right ) e^{3 x} + \left (84 A + 112 i B\right ) e^{x} + \left (- 252 i A + 336 B\right ) e^{2 x}}{105 e^{7 x} + 735 i e^{6 x} - 2205 e^{5 x} - 3675 i e^{4 x} + 3675 e^{3 x} + 2205 i e^{2 x} - 735 e^{x} - 105 i} \]
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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 = 5.15 \[ \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 = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.66 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=-\frac {4 \, {\left (70 \, B e^{\left (4 \, x\right )} + 105 \, A e^{\left (3 \, x\right )} + 70 i \, B e^{\left (3 \, x\right )} + 63 i \, A e^{\left (2 \, x\right )} - 84 \, B e^{\left (2 \, x\right )} - 21 \, A e^{x} - 28 i \, B e^{x} - 3 i \, A + 4 \, B\right )}}{105 \, {\left (e^{x} + i\right )}^{7}} \]
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Time = 1.79 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=-\frac {\frac {16\,B}{105}+4\,A\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^x\,\left (\frac {4\,A}{5}+\frac {B\,16{}\mathrm {i}}{15}\right )-\frac {16\,B\,{\mathrm {e}}^{2\,x}}{5}+\frac {8\,B\,{\mathrm {e}}^{4\,x}}{3}-\frac {A\,4{}\mathrm {i}}{35}+\frac {A\,{\mathrm {e}}^{2\,x}\,12{}\mathrm {i}}{5}+\frac {B\,{\mathrm {e}}^{3\,x}\,8{}\mathrm {i}}{3}}{{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^7} \]
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