Integrand size = 14, antiderivative size = 124 \[ \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx=\frac {385 x}{32768}-\frac {385 i \arctan \left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{16384 d}-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))} \] Output:
385/32768*x-385/16384*I*arctan(cosh(d*x+c)/(3+I*sinh(d*x+c)))/d-1/16*I*cos h(d*x+c)/d/(5+3*I*sinh(d*x+c))^3-25/512*I*cosh(d*x+c)/d/(5+3*I*sinh(d*x+c) )^2-311/8192*I*cosh(d*x+c)/d/(5+3*I*sinh(d*x+c))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(308\) vs. \(2(124)=248\).
Time = 1.38 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.48 \[ \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx=\frac {-3850 i \arctan \left (\frac {2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )}{\cosh \left (\frac {1}{2} (c+d x)\right )-2 \sinh \left (\frac {1}{2} (c+d x)\right )}\right )+3850 i \arctan \left (\frac {\cosh \left (\frac {1}{2} (c+d x)\right )+2 \sinh \left (\frac {1}{2} (c+d x)\right )}{2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )}\right )-1925 \log (5 \cosh (c+d x)-4 \sinh (c+d x))+1925 \log (5 \cosh (c+d x)+4 \sinh (c+d x))+\frac {2656-192 i}{\left ((1+2 i) \cosh \left (\frac {1}{2} (c+d x)\right )-(2+i) \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2656+192 i}{\left ((2+i) \cosh \left (\frac {1}{2} (c+d x)\right )+(1+2 i) \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 (-235150+166615 \cosh (c+d x)+82530 \cosh (2 (c+d x))-13995 \cosh (3 (c+d x))-298563 i \sinh (c+d x)+89364 i \sinh (2 (c+d x))+8397 i \sinh (3 (c+d x)))}{(-5 i+3 \sinh (c+d x))^3}}{327680 d} \] Input:
Integrate[(5 + (3*I)*Sinh[c + d*x])^(-4),x]
Output:
((-3850*I)*ArcTan[(2*Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2])/(Cosh[(c + d*x )/2] - 2*Sinh[(c + d*x)/2])] + (3850*I)*ArcTan[(Cosh[(c + d*x)/2] + 2*Sinh [(c + d*x)/2])/(2*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2])] - 1925*Log[5*Cos h[c + d*x] - 4*Sinh[c + d*x]] + 1925*Log[5*Cosh[c + d*x] + 4*Sinh[c + d*x] ] + (2656 - 192*I)/((1 + 2*I)*Cosh[(c + d*x)/2] - (2 + I)*Sinh[(c + d*x)/2 ])^2 + (2656 + 192*I)/((2 + I)*Cosh[(c + d*x)/2] + (1 + 2*I)*Sinh[(c + d*x )/2])^2 + (2*(-235150 + 166615*Cosh[c + d*x] + 82530*Cosh[2*(c + d*x)] - 1 3995*Cosh[3*(c + d*x)] - (298563*I)*Sinh[c + d*x] + (89364*I)*Sinh[2*(c + d*x)] + (8397*I)*Sinh[3*(c + d*x)]))/(-5*I + 3*Sinh[c + d*x])^3)/(327680*d )
Time = 0.55 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {3042, 3143, 27, 3042, 3233, 25, 3042, 3233, 27, 3042, 3136}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(5+3 \sin (i c+i d x))^4}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle -\frac {1}{48} \int -\frac {3 (5-2 i \sinh (c+d x))}{(3 i \sinh (c+d x)+5)^3}dx-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \int \frac {5-2 i \sinh (c+d x)}{(3 i \sinh (c+d x)+5)^3}dx-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \int \frac {5-2 \sin (i c+i d x)}{(3 \sin (i c+i d x)+5)^3}dx-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{16} \left (-\frac {1}{32} \int -\frac {62-25 i \sinh (c+d x)}{(3 i \sinh (c+d x)+5)^2}dx-\frac {25 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}\right )-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \int \frac {62-25 i \sinh (c+d x)}{(3 i \sinh (c+d x)+5)^2}dx-\frac {25 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}\right )-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \int \frac {62-25 \sin (i c+i d x)}{(3 \sin (i c+i d x)+5)^2}dx-\frac {25 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}\right )-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (-\frac {1}{16} \int -\frac {385}{3 i \sinh (c+d x)+5}dx-\frac {311 i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))}\right )-\frac {25 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}\right )-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \int \frac {1}{3 i \sinh (c+d x)+5}dx-\frac {311 i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))}\right )-\frac {25 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}\right )-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \int \frac {1}{3 \sin (i c+i d x)+5}dx-\frac {311 i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))}\right )-\frac {25 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}\right )-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 3136 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \left (\frac {x}{4}-\frac {i \arctan \left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d}\right )-\frac {311 i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))}\right )-\frac {25 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}\right )-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}\) |
Input:
Int[(5 + (3*I)*Sinh[c + d*x])^(-4),x]
Output:
(((385*(x/4 - ((I/2)*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/d))/16 - (((311*I)/16)*Cosh[c + d*x])/(d*(5 + (3*I)*Sinh[c + d*x])))/32 - (((25*I) /32)*Cosh[c + d*x])/(d*(5 + (3*I)*Sinh[c + d*x])^2))/16 - ((I/16)*Cosh[c + d*x])/(d*(5 + (3*I)*Sinh[c + d*x])^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[ a^2 - b^2, 2]}, Simp[x/q, x] + Simp[(2/(d*q))*ArcTan[b*(Cos[c + d*x]/(a + q + b*Sin[c + d*x]))], x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2, 0] && PosQ[a]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2))), x] + Simp [1/((n + 1)*(a^2 - b^2)) Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Time = 0.58 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(-\frac {i \left (-86625 i {\mathrm e}^{4 d x +4 c}+10395 \,{\mathrm e}^{5 d x +5 c}+218466 i {\mathrm e}^{2 d x +2 c}-239470 \,{\mathrm e}^{3 d x +3 c}-8397 i+73575 \,{\mathrm e}^{d x +c}\right )}{12288 d \left (3 \,{\mathrm e}^{2 d x +2 c}-3-10 i {\mathrm e}^{d x +c}\right )^{3}}-\frac {385 \ln \left ({\mathrm e}^{d x +c}-3 i\right )}{32768 d}+\frac {385 \ln \left (-\frac {i}{3}+{\mathrm e}^{d x +c}\right )}{32768 d}\) | \(119\) |
| derivativedivides | \(\frac {\frac {\frac {1053}{32000}-\frac {99 i}{8000}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{3}}+\frac {\frac {783}{128000}-\frac {3753 i}{64000}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{2}}+\frac {-\frac {39933}{1024000}-\frac {8361 i}{256000}}{5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i}+\frac {385 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}{32768}+\frac {\frac {1053}{32000}+\frac {99 i}{8000}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{3}}+\frac {-\frac {783}{128000}-\frac {3753 i}{64000}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{2}}+\frac {-\frac {39933}{1024000}+\frac {8361 i}{256000}}{5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i}-\frac {385 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}{32768}}{d}\) | \(168\) |
| default | \(\frac {\frac {\frac {1053}{32000}-\frac {99 i}{8000}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{3}}+\frac {\frac {783}{128000}-\frac {3753 i}{64000}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{2}}+\frac {-\frac {39933}{1024000}-\frac {8361 i}{256000}}{5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i}+\frac {385 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}{32768}+\frac {\frac {1053}{32000}+\frac {99 i}{8000}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{3}}+\frac {-\frac {783}{128000}-\frac {3753 i}{64000}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{2}}+\frac {-\frac {39933}{1024000}+\frac {8361 i}{256000}}{5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i}-\frac {385 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}{32768}}{d}\) | \(168\) |
| parallelrisch | \(\frac {\left (-47210625 i \sinh \left (d x +c \right )+1299375 i \sinh \left (3 d x +3 c \right )+12993750 \cosh \left (2 d x +2 c \right )-37056250\right ) \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )+\left (47210625 i \sinh \left (d x +c \right )-1299375 i \sinh \left (3 d x +3 c \right )-12993750 \cosh \left (2 d x +2 c \right )+37056250\right ) \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )-21973500 i \cosh \left (d x +c \right )+11109960 i \cosh \left (2 d x +2 c \right )+1399500 i \cosh \left (3 d x +3 c \right )-31683960 i+40366188 \sinh \left (d x +c \right )+10530000 \sinh \left (2 d x +2 c \right )-1110996 \sinh \left (3 d x +3 c \right )}{4096000 d \left (770-27 i \sinh \left (3 d x +3 c \right )+981 i \sinh \left (d x +c \right )-270 \cosh \left (2 d x +2 c \right )\right )}\) | \(210\) |
Input:
int(1/(5+3*I*sinh(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
-1/12288*I*(-86625*I*exp(4*d*x+4*c)+10395*exp(5*d*x+5*c)+218466*I*exp(2*d* x+2*c)-239470*exp(3*d*x+3*c)-8397*I+73575*exp(d*x+c))/d/(3*exp(2*d*x+2*c)- 3-10*I*exp(d*x+c))^3-385/32768/d*ln(exp(d*x+c)-3*I)+385/32768/d*ln(-1/3*I+ exp(d*x+c))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (98) = 196\).
Time = 0.10 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.28 \[ \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx=\frac {1155 \, {\left (27 \, e^{\left (6 \, d x + 6 \, c\right )} - 270 i \, e^{\left (5 \, d x + 5 \, c\right )} - 981 \, e^{\left (4 \, d x + 4 \, c\right )} + 1540 i \, e^{\left (3 \, d x + 3 \, c\right )} + 981 \, e^{\left (2 \, d x + 2 \, c\right )} - 270 i \, e^{\left (d x + c\right )} - 27\right )} \log \left (e^{\left (d x + c\right )} - \frac {1}{3} i\right ) - 1155 \, {\left (27 \, e^{\left (6 \, d x + 6 \, c\right )} - 270 i \, e^{\left (5 \, d x + 5 \, c\right )} - 981 \, e^{\left (4 \, d x + 4 \, c\right )} + 1540 i \, e^{\left (3 \, d x + 3 \, c\right )} + 981 \, e^{\left (2 \, d x + 2 \, c\right )} - 270 i \, e^{\left (d x + c\right )} - 27\right )} \log \left (e^{\left (d x + c\right )} - 3 i\right ) - 83160 i \, e^{\left (5 \, d x + 5 \, c\right )} - 693000 \, e^{\left (4 \, d x + 4 \, c\right )} + 1915760 i \, e^{\left (3 \, d x + 3 \, c\right )} + 1747728 \, e^{\left (2 \, d x + 2 \, c\right )} - 588600 i \, e^{\left (d x + c\right )} - 67176}{98304 \, {\left (27 \, d e^{\left (6 \, d x + 6 \, c\right )} - 270 i \, d e^{\left (5 \, d x + 5 \, c\right )} - 981 \, d e^{\left (4 \, d x + 4 \, c\right )} + 1540 i \, d e^{\left (3 \, d x + 3 \, c\right )} + 981 \, d e^{\left (2 \, d x + 2 \, c\right )} - 270 i \, d e^{\left (d x + c\right )} - 27 \, d\right )}} \] Input:
integrate(1/(5+3*I*sinh(d*x+c))^4,x, algorithm="fricas")
Output:
1/98304*(1155*(27*e^(6*d*x + 6*c) - 270*I*e^(5*d*x + 5*c) - 981*e^(4*d*x + 4*c) + 1540*I*e^(3*d*x + 3*c) + 981*e^(2*d*x + 2*c) - 270*I*e^(d*x + c) - 27)*log(e^(d*x + c) - 1/3*I) - 1155*(27*e^(6*d*x + 6*c) - 270*I*e^(5*d*x + 5*c) - 981*e^(4*d*x + 4*c) + 1540*I*e^(3*d*x + 3*c) + 981*e^(2*d*x + 2*c ) - 270*I*e^(d*x + c) - 27)*log(e^(d*x + c) - 3*I) - 83160*I*e^(5*d*x + 5* c) - 693000*e^(4*d*x + 4*c) + 1915760*I*e^(3*d*x + 3*c) + 1747728*e^(2*d*x + 2*c) - 588600*I*e^(d*x + c) - 67176)/(27*d*e^(6*d*x + 6*c) - 270*I*d*e^ (5*d*x + 5*c) - 981*d*e^(4*d*x + 4*c) + 1540*I*d*e^(3*d*x + 3*c) + 981*d*e ^(2*d*x + 2*c) - 270*I*d*e^(d*x + c) - 27*d)
Time = 0.35 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx=\frac {- 10395 i e^{5 c} e^{5 d x} - 86625 e^{4 c} e^{4 d x} + 239470 i e^{3 c} e^{3 d x} + 218466 e^{2 c} e^{2 d x} - 73575 i e^{c} e^{d x} - 8397}{331776 d e^{6 c} e^{6 d x} - 3317760 i d e^{5 c} e^{5 d x} - 12054528 d e^{4 c} e^{4 d x} + 18923520 i d e^{3 c} e^{3 d x} + 12054528 d e^{2 c} e^{2 d x} - 3317760 i d e^{c} e^{d x} - 331776 d} + \frac {- \frac {385 \log {\left (e^{d x} - 3 i e^{- c} \right )}}{32768} + \frac {385 \log {\left (e^{d x} - \frac {i e^{- c}}{3} \right )}}{32768}}{d} \] Input:
integrate(1/(5+3*I*sinh(d*x+c))**4,x)
Output:
(-10395*I*exp(5*c)*exp(5*d*x) - 86625*exp(4*c)*exp(4*d*x) + 239470*I*exp(3 *c)*exp(3*d*x) + 218466*exp(2*c)*exp(2*d*x) - 73575*I*exp(c)*exp(d*x) - 83 97)/(331776*d*exp(6*c)*exp(6*d*x) - 3317760*I*d*exp(5*c)*exp(5*d*x) - 1205 4528*d*exp(4*c)*exp(4*d*x) + 18923520*I*d*exp(3*c)*exp(3*d*x) + 12054528*d *exp(2*c)*exp(2*d*x) - 3317760*I*d*exp(c)*exp(d*x) - 331776*d) + (-385*log (exp(d*x) - 3*I*exp(-c))/32768 + 385*log(exp(d*x) - I*exp(-c)/3)/32768)/d
Time = 0.12 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx=-\frac {385 i \, \arctan \left (\frac {3}{4} \, e^{\left (-d x - c\right )} + \frac {5}{4} i\right )}{16384 \, d} - \frac {73575 i \, e^{\left (-d x - c\right )} + 218466 \, e^{\left (-2 \, d x - 2 \, c\right )} - 239470 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 86625 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10395 i \, e^{\left (-5 \, d x - 5 \, c\right )} - 8397}{-12288 \, d {\left (-270 i \, e^{\left (-d x - c\right )} - 981 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1540 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 981 \, e^{\left (-4 \, d x - 4 \, c\right )} - 270 i \, e^{\left (-5 \, d x - 5 \, c\right )} - 27 \, e^{\left (-6 \, d x - 6 \, c\right )} + 27\right )}} \] Input:
integrate(1/(5+3*I*sinh(d*x+c))^4,x, algorithm="maxima")
Output:
-385/16384*I*arctan(3/4*e^(-d*x - c) + 5/4*I)/d - (73575*I*e^(-d*x - c) + 218466*e^(-2*d*x - 2*c) - 239470*I*e^(-3*d*x - 3*c) - 86625*e^(-4*d*x - 4* c) + 10395*I*e^(-5*d*x - 5*c) - 8397)/(d*(3317760*I*e^(-d*x - c) + 1205452 8*e^(-2*d*x - 2*c) - 18923520*I*e^(-3*d*x - 3*c) - 12054528*e^(-4*d*x - 4* c) + 3317760*I*e^(-5*d*x - 5*c) + 331776*e^(-6*d*x - 6*c) - 331776))
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx=-\frac {\frac {8 \, {\left (10395 i \, e^{\left (5 \, d x + 5 \, c\right )} + 86625 \, e^{\left (4 \, d x + 4 \, c\right )} - 239470 i \, e^{\left (3 \, d x + 3 \, c\right )} - 218466 \, e^{\left (2 \, d x + 2 \, c\right )} + 73575 i \, e^{\left (d x + c\right )} + 8397\right )}}{{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} - 10 i \, e^{\left (d x + c\right )} - 3\right )}^{3}} - 1155 \, \log \left (3 \, e^{\left (d x + c\right )} - i\right ) + 1155 \, \log \left (e^{\left (d x + c\right )} - 3 i\right )}{98304 \, d} \] Input:
integrate(1/(5+3*I*sinh(d*x+c))^4,x, algorithm="giac")
Output:
-1/98304*(8*(10395*I*e^(5*d*x + 5*c) + 86625*e^(4*d*x + 4*c) - 239470*I*e^ (3*d*x + 3*c) - 218466*e^(2*d*x + 2*c) + 73575*I*e^(d*x + c) + 8397)/(3*e^ (2*d*x + 2*c) - 10*I*e^(d*x + c) - 3)^3 - 1155*log(3*e^(d*x + c) - I) + 11 55*log(e^(d*x + c) - 3*I))/d
Time = 3.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.87 \[ \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx=\frac {\frac {1925}{36864\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,385{}\mathrm {i}}{12288\,d}}{1-{\mathrm {e}}^{2\,c+2\,d\,x}+\frac {{\mathrm {e}}^{c+d\,x}\,10{}\mathrm {i}}{3}}+\frac {\frac {41}{486\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,365{}\mathrm {i}}{1458\,d}}{\frac {109\,{\mathrm {e}}^{4\,c+4\,d\,x}}{3}-\frac {109\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3}-{\mathrm {e}}^{6\,c+6\,d\,x}+1+{\mathrm {e}}^{c+d\,x}\,10{}\mathrm {i}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,1540{}\mathrm {i}}{27}+{\mathrm {e}}^{5\,c+5\,d\,x}\,10{}\mathrm {i}}-\frac {385\,\ln \left (-\frac {385\,{\mathrm {e}}^{c+d\,x}}{4}+\frac {1155}{4}{}\mathrm {i}\right )}{32768\,d}+\frac {385\,\ln \left (\frac {3465\,{\mathrm {e}}^{c+d\,x}}{4}-\frac {1155}{4}{}\mathrm {i}\right )}{32768\,d}-\frac {\frac {3461}{31104\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,385{}\mathrm {i}}{10368\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-\frac {118\,{\mathrm {e}}^{2\,c+2\,d\,x}}{9}+1+\frac {{\mathrm {e}}^{c+d\,x}\,20{}\mathrm {i}}{3}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,20{}\mathrm {i}}{3}} \] Input:
int(1/(sinh(c + d*x)*3i + 5)^4,x)
Output:
((exp(c + d*x)*385i)/(12288*d) + 1925/(36864*d))/((exp(c + d*x)*10i)/3 - e xp(2*c + 2*d*x) + 1) + ((exp(c + d*x)*365i)/(1458*d) + 41/(486*d))/(exp(c + d*x)*10i - (109*exp(2*c + 2*d*x))/3 - (exp(3*c + 3*d*x)*1540i)/27 + (109 *exp(4*c + 4*d*x))/3 + exp(5*c + 5*d*x)*10i - exp(6*c + 6*d*x) + 1) - (385 *log(1155i/4 - (385*exp(c + d*x))/4))/(32768*d) + (385*log((3465*exp(c + d *x))/4 - 1155i/4))/(32768*d) - ((exp(c + d*x)*385i)/(10368*d) + 3461/(3110 4*d))/((exp(c + d*x)*20i)/3 - (118*exp(2*c + 2*d*x))/9 - (exp(3*c + 3*d*x) *20i)/3 + exp(4*c + 4*d*x) + 1)
\[ \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx=\int \frac {1}{81 \sinh \left (d x +c \right )^{4}-540 \sinh \left (d x +c \right )^{3} i -1350 \sinh \left (d x +c \right )^{2}+1500 \sinh \left (d x +c \right ) i +625}d x \] Input:
int(1/(5+3*I*sinh(d*x+c))^4,x)
Output:
int(1/(81*sinh(c + d*x)**4 - 540*sinh(c + d*x)**3*i - 1350*sinh(c + d*x)** 2 + 1500*sinh(c + d*x)*i + 625),x)