Optimal. Leaf size=101 \[ -\frac {2 (3 A-4 i B) \cosh (x)}{105 (-\sinh (x)+i)}-\frac {2 (4 B+3 i A) \cosh (x)}{105 (-\sinh (x)+i)^2}+\frac {(3 A-4 i B) \cosh (x)}{35 (-\sinh (x)+i)^3}+\frac {(-B+i A) \cosh (x)}{7 (-\sinh (x)+i)^4} \]
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Rubi [A] time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2750, 2650, 2648} \[ -\frac {2 (3 A-4 i B) \cosh (x)}{105 (-\sinh (x)+i)}-\frac {2 (4 B+3 i A) \cosh (x)}{105 (-\sinh (x)+i)^2}+\frac {(3 A-4 i B) \cosh (x)}{35 (-\sinh (x)+i)^3}+\frac {(-B+i A) \cosh (x)}{7 (-\sinh (x)+i)^4} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rule 2750
Rubi steps
\begin {align*} \int \frac {A+B \sinh (x)}{(i-\sinh (x))^4} \, dx &=\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*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 0.62 \[ \frac {\cosh (x) \left ((6 A-8 i B) \sinh ^3(x)+(-32 B-24 i A) \sinh ^2(x)+(-39 A+52 i B) \sinh (x)+36 i A+13 B\right )}{105 (\sinh (x)-i)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 96, normalized size = 0.95 \[ -\frac {280 \, B e^{\left (4 \, x\right )} + {\left (420 \, A - 280 i \, B\right )} e^{\left (3 \, x\right )} + 84 \, {\left (-3 i \, A - 4 \, B\right )} e^{\left (2 \, x\right )} - {\left (84 \, A - 112 i \, B\right )} e^{x} + 12 i \, A + 16 \, B}{105 \, e^{\left (7 \, x\right )} - 735 i \, e^{\left (6 \, x\right )} - 2205 \, e^{\left (5 \, x\right )} + 3675 i \, e^{\left (4 \, x\right )} + 3675 \, e^{\left (3 \, x\right )} - 2205 i \, e^{\left (2 \, x\right )} - 735 \, e^{x} + 105 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 60, normalized size = 0.59 \[ -\frac {280 \, B e^{\left (4 \, x\right )} + 420 \, A e^{\left (3 \, x\right )} - 280 i \, B e^{\left (3 \, x\right )} - 252 i \, A e^{\left (2 \, x\right )} - 336 \, B e^{\left (2 \, x\right )} - 84 \, A e^{x} + 112 i \, B e^{x} + 12 i \, A + 16 \, B}{105 \, {\left (e^{x} - i\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 128, normalized size = 1.27 \[ -\frac {32 i A -24 B}{2 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{4}}+\frac {2 A}{\tanh \left (\frac {x}{2}\right )-i}-\frac {2 \left (-32 i B -36 A \right )}{5 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{5}}-\frac {-6 i A +2 B}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}-\frac {2 \left (10 i B +18 A \right )}{3 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}-\frac {2 \left (8 i B +8 A \right )}{7 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{7}}-\frac {-24 i A +24 B}{3 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 468, normalized size = 4.63 \[ \frac {1}{2} \, B {\left (-\frac {224 i \, e^{\left (-x\right )}}{735 \, e^{\left (-x\right )} - 2205 i \, e^{\left (-2 \, x\right )} - 3675 \, e^{\left (-3 \, x\right )} + 3675 i \, e^{\left (-4 \, x\right )} + 2205 \, e^{\left (-5 \, x\right )} - 735 i \, e^{\left (-6 \, x\right )} - 105 \, e^{\left (-7 \, x\right )} + 105 i} - \frac {672 \, e^{\left (-2 \, x\right )}}{735 \, e^{\left (-x\right )} - 2205 i \, e^{\left (-2 \, x\right )} - 3675 \, e^{\left (-3 \, x\right )} + 3675 i \, e^{\left (-4 \, x\right )} + 2205 \, e^{\left (-5 \, x\right )} - 735 i \, e^{\left (-6 \, x\right )} - 105 \, e^{\left (-7 \, x\right )} + 105 i} + \frac {560 i \, e^{\left (-3 \, x\right )}}{735 \, e^{\left (-x\right )} - 2205 i \, e^{\left (-2 \, x\right )} - 3675 \, e^{\left (-3 \, x\right )} + 3675 i \, e^{\left (-4 \, x\right )} + 2205 \, e^{\left (-5 \, x\right )} - 735 i \, e^{\left (-6 \, x\right )} - 105 \, e^{\left (-7 \, x\right )} + 105 i} + \frac {560 \, e^{\left (-4 \, x\right )}}{735 \, e^{\left (-x\right )} - 2205 i \, e^{\left (-2 \, x\right )} - 3675 \, e^{\left (-3 \, x\right )} + 3675 i \, e^{\left (-4 \, x\right )} + 2205 \, e^{\left (-5 \, x\right )} - 735 i \, e^{\left (-6 \, x\right )} - 105 \, e^{\left (-7 \, x\right )} + 105 i} + \frac {32}{735 \, e^{\left (-x\right )} - 2205 i \, e^{\left (-2 \, x\right )} - 3675 \, e^{\left (-3 \, x\right )} + 3675 i \, e^{\left (-4 \, x\right )} + 2205 \, e^{\left (-5 \, x\right )} - 735 i \, e^{\left (-6 \, x\right )} - 105 \, e^{\left (-7 \, x\right )} + 105 i}\right )} + A {\left (\frac {28 \, e^{\left (-x\right )}}{245 \, e^{\left (-x\right )} - 735 i \, e^{\left (-2 \, x\right )} - 1225 \, e^{\left (-3 \, x\right )} + 1225 i \, e^{\left (-4 \, x\right )} + 735 \, e^{\left (-5 \, x\right )} - 245 i \, e^{\left (-6 \, x\right )} - 35 \, e^{\left (-7 \, x\right )} + 35 i} - \frac {84 i \, e^{\left (-2 \, x\right )}}{245 \, e^{\left (-x\right )} - 735 i \, e^{\left (-2 \, x\right )} - 1225 \, e^{\left (-3 \, x\right )} + 1225 i \, e^{\left (-4 \, x\right )} + 735 \, e^{\left (-5 \, x\right )} - 245 i \, e^{\left (-6 \, x\right )} - 35 \, e^{\left (-7 \, x\right )} + 35 i} - \frac {140 \, e^{\left (-3 \, x\right )}}{245 \, e^{\left (-x\right )} - 735 i \, e^{\left (-2 \, x\right )} - 1225 \, e^{\left (-3 \, x\right )} + 1225 i \, e^{\left (-4 \, x\right )} + 735 \, e^{\left (-5 \, x\right )} - 245 i \, e^{\left (-6 \, x\right )} - 35 \, e^{\left (-7 \, x\right )} + 35 i} + \frac {4 i}{245 \, e^{\left (-x\right )} - 735 i \, e^{\left (-2 \, x\right )} - 1225 \, e^{\left (-3 \, x\right )} + 1225 i \, e^{\left (-4 \, x\right )} + 735 \, e^{\left (-5 \, x\right )} - 245 i \, e^{\left (-6 \, x\right )} - 35 \, e^{\left (-7 \, x\right )} + 35 i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 68, normalized size = 0.67 \[ \frac {\frac {12\,A\,{\mathrm {e}}^{2\,x}}{5}+\frac {B\,16{}\mathrm {i}}{105}-\frac {4\,A}{35}+A\,{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,\left (\frac {16\,B}{15}+\frac {A\,4{}\mathrm {i}}{5}\right )-\frac {B\,{\mathrm {e}}^{2\,x}\,16{}\mathrm {i}}{5}+\frac {8\,B\,{\mathrm {e}}^{3\,x}}{3}+\frac {B\,{\mathrm {e}}^{4\,x}\,8{}\mathrm {i}}{3}}{{\left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 109, normalized size = 1.08 \[ \frac {- 12 i A - 280 B e^{4 x} - 16 B + \left (- 420 A + 280 i B\right ) e^{3 x} + \left (84 A - 112 i B\right ) e^{x} + \left (252 i A + 336 B\right ) e^{2 x}}{105 e^{7 x} - 735 i e^{6 x} - 2205 e^{5 x} + 3675 i e^{4 x} + 3675 e^{3 x} - 2205 i e^{2 x} - 735 e^{x} + 105 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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