Integrand size = 14, antiderivative size = 140 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx=\frac {279 i \log \left (i-3 \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 i \log \left (3 i-\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {995 i \cosh (c+d x)}{24576 d (3+5 i \sinh (c+d x))} \] Output:
279/32768*I*ln(I-3*tanh(1/2*d*x+1/2*c))/d-279/32768*I*ln(3*I-tanh(1/2*d*x+ 1/2*c))/d+5/48*I*cosh(d*x+c)/d/(3+5*I*sinh(d*x+c))^3-25/512*I*cosh(d*x+c)/ d/(3+5*I*sinh(d*x+c))^2+995/24576*I*cosh(d*x+c)/d/(3+5*I*sinh(d*x+c))
Time = 0.53 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.89 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx=\frac {-5022 \arctan \left (3 \coth \left (\frac {1}{2} (c+d x)\right )\right )-5022 \arctan \left (3 \tanh \left (\frac {1}{2} (c+d x)\right )\right )+2511 i \log (4-5 \cosh (c+d x))-2511 i \log (4+5 \cosh (c+d x))+\frac {4640 i}{\left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {1440 i}{\left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}+40 \left (\frac {80}{\left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {199}{3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )}+\frac {240}{\left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {597}{\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )}\right ) \sinh \left (\frac {1}{2} (c+d x)\right )}{589824 d} \] Input:
Integrate[(3 + (5*I)*Sinh[c + d*x])^(-4),x]
Output:
(-5022*ArcTan[3*Coth[(c + d*x)/2]] - 5022*ArcTan[3*Tanh[(c + d*x)/2]] + (2 511*I)*Log[4 - 5*Cosh[c + d*x]] - (2511*I)*Log[4 + 5*Cosh[c + d*x]] + (464 0*I)/(3*Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2 - (1440*I)/(Cosh[(c + d *x)/2] + (3*I)*Sinh[(c + d*x)/2])^2 + 40*(80/(3*Cosh[(c + d*x)/2] + I*Sinh [(c + d*x)/2])^3 + 199/(3*Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]) + 240/( Cosh[(c + d*x)/2] + (3*I)*Sinh[(c + d*x)/2])^3 + 597/(Cosh[(c + d*x)/2] + (3*I)*Sinh[(c + d*x)/2]))*Sinh[(c + d*x)/2])/(589824*d)
Time = 0.59 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3042, 3143, 25, 3042, 3233, 25, 3042, 3233, 27, 3042, 3139, 1081, 2009}
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.
| \(\displaystyle \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(3+5 \sin (i c+i d x))^4}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle \frac {1}{48} \int -\frac {9-10 i \sinh (c+d x)}{(5 i \sinh (c+d x)+3)^3}dx+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {1}{48} \int \frac {9-10 i \sinh (c+d x)}{(5 i \sinh (c+d x)+3)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {1}{48} \int \frac {9-10 \sin (i c+i d x)}{(5 \sin (i c+i d x)+3)^3}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{48} \left (-\frac {1}{32} \int -\frac {154-75 i \sinh (c+d x)}{(5 i \sinh (c+d x)+3)^2}dx-\frac {75 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}\right )+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {154-75 i \sinh (c+d x)}{(5 i \sinh (c+d x)+3)^2}dx-\frac {75 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}\right )+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {154-75 \sin (i c+i d x)}{(5 \sin (i c+i d x)+3)^2}dx-\frac {75 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}\right )+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {1}{16} \int -\frac {837}{5 i \sinh (c+d x)+3}dx+\frac {995 i \cosh (c+d x)}{16 d (3+5 i \sinh (c+d x))}\right )-\frac {75 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}\right )+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 i \cosh (c+d x)}{16 d (3+5 i \sinh (c+d x))}-\frac {837}{16} \int \frac {1}{5 i \sinh (c+d x)+3}dx\right )-\frac {75 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}\right )+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 i \cosh (c+d x)}{16 d (3+5 i \sinh (c+d x))}-\frac {837}{16} \int \frac {1}{5 \sin (i c+i d x)+3}dx\right )-\frac {75 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}\right )+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {837 i \int \frac {1}{-3 \tanh ^2\left (\frac {1}{2} (c+d x)\right )+10 i \tanh \left (\frac {1}{2} (c+d x)\right )+3}d\left (i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {995 i \cosh (c+d x)}{16 d (3+5 i \sinh (c+d x))}\right )-\frac {75 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}\right )+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 1081 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {2511 i \int \left (\frac {1}{8 \left (3 i \tanh \left (\frac {1}{2} (c+d x)\right )+1\right )}-\frac {1}{24 \left (i \tanh \left (\frac {1}{2} (c+d x)\right )+3\right )}\right )d\left (i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {995 i \cosh (c+d x)}{16 d (3+5 i \sinh (c+d x))}\right )-\frac {75 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}\right )+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}+\frac {1}{48} \left (\frac {1}{32} \left (\frac {2511 i \left (\frac {1}{24} \log \left (1+3 i \tanh \left (\frac {1}{2} (c+d x)\right )\right )-\frac {1}{24} \log \left (3+i \tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{8 d}+\frac {995 i \cosh (c+d x)}{16 d (3+5 i \sinh (c+d x))}\right )-\frac {75 i \cosh (c+d x)}{32 d (3+5 i \sinh (c+d x))^2}\right )\) |
Input:
Int[(3 + (5*I)*Sinh[c + d*x])^(-4),x]
Output:
(((((2511*I)/8)*(-1/24*Log[3 + I*Tanh[(c + d*x)/2]] + Log[1 + (3*I)*Tanh[( c + d*x)/2]]/24))/d + (((995*I)/16)*Cosh[c + d*x])/(d*(3 + (5*I)*Sinh[c + d*x])))/32 - (((75*I)/32)*Cosh[c + d*x])/(d*(3 + (5*I)*Sinh[c + d*x])^2))/ 48 + (((5*I)/48)*Cosh[c + d*x])/(d*(3 + (5*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_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c Int[ExpandIntegrand[1/((b/2 - q/2 + c*x)*(b/2 + q/2 + c*x)), x], x], x]] /; FreeQ[{a, b, c}, x] && NiceSqrtQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
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 = 0.68 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \(\frac {i \left (-62775 i {\mathrm e}^{4 d x +4 c}+20925 \,{\mathrm e}^{5 d x +5 c}+119310 i {\mathrm e}^{2 d x +2 c}-111042 \,{\mathrm e}^{3 d x +3 c}-24875 i+68625 \,{\mathrm e}^{d x +c}\right )}{12288 d \left (5 \,{\mathrm e}^{2 d x +2 c}-5-6 i {\mathrm e}^{d x +c}\right )^{3}}-\frac {279 i \ln \left ({\mathrm e}^{d x +c}+\frac {4}{5}-\frac {3 i}{5}\right )}{32768 d}+\frac {279 i \ln \left ({\mathrm e}^{d x +c}-\frac {4}{5}-\frac {3 i}{5}\right )}{32768 d}\) | \(123\) |
| derivativedivides | \(\frac {-\frac {279 i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}{32768}+\frac {75 i}{1024 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{2}}-\frac {125}{768 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{3}}+\frac {345}{8192 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}+\frac {275 i}{27648 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}+\frac {279 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}{32768}-\frac {125}{20736 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {3505}{221184 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}}{d}\) | \(144\) |
| default | \(\frac {-\frac {279 i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}{32768}+\frac {75 i}{1024 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{2}}-\frac {125}{768 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{3}}+\frac {345}{8192 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}+\frac {275 i}{27648 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}+\frac {279 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}{32768}-\frac {125}{20736 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {3505}{221184 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}}{d}\) | \(144\) |
| parallelrisch | \(\frac {28968900 i \sinh \left (d x +c \right )-2824875 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right ) \sinh \left (3 d x +3 c \right )-3957500 i \sinh \left (3 d x +3 c \right )+5151600 i \sinh \left (2 d x +2 c \right )+2824875 i \sinh \left (3 d x +3 c \right ) \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-9 i\right )+20678085 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right ) \sinh \left (d x +c \right )-20678085 i \sinh \left (d x +c \right ) \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-9 i\right )-10169550 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {i}{3}\right ) \cosh \left (2 d x +2 c \right )+10169550 \cosh \left (2 d x +2 c \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )+6105780 \cosh \left (d x +c \right )-2686500 \cosh \left (3 d x +3 c \right )-14247000 \cosh \left (2 d x +2 c \right )+2440692 \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )-2440692 \ln \left (3\right )+10169550 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {i}{3}\right )-12610242 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )+17666280}{2654208 d \left (-558 i+915 \sinh \left (d x +c \right )+450 i \cosh \left (2 d x +2 c \right )-125 \sinh \left (3 d x +3 c \right )\right )}\) | \(308\) |
Input:
int(1/(3+5*I*sinh(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/12288*I*(-62775*I*exp(4*d*x+4*c)+20925*exp(5*d*x+5*c)+119310*I*exp(2*d*x +2*c)-111042*exp(3*d*x+3*c)-24875*I+68625*exp(d*x+c))/d/(5*exp(2*d*x+2*c)- 5-6*I*exp(d*x+c))^3-279/32768*I/d*ln(exp(d*x+c)+4/5-3/5*I)+279/32768*I/d*l n(exp(d*x+c)-4/5-3/5*I)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (108) = 216\).
Time = 0.10 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.02 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx=-\frac {837 \, {\left (125 i \, e^{\left (6 \, d x + 6 \, c\right )} + 450 \, e^{\left (5 \, d x + 5 \, c\right )} - 915 i \, e^{\left (4 \, d x + 4 \, c\right )} - 1116 \, e^{\left (3 \, d x + 3 \, c\right )} + 915 i \, e^{\left (2 \, d x + 2 \, c\right )} + 450 \, e^{\left (d x + c\right )} - 125 i\right )} \log \left (e^{\left (d x + c\right )} - \frac {3}{5} i + \frac {4}{5}\right ) + 837 \, {\left (-125 i \, e^{\left (6 \, d x + 6 \, c\right )} - 450 \, e^{\left (5 \, d x + 5 \, c\right )} + 915 i \, e^{\left (4 \, d x + 4 \, c\right )} + 1116 \, e^{\left (3 \, d x + 3 \, c\right )} - 915 i \, e^{\left (2 \, d x + 2 \, c\right )} - 450 \, e^{\left (d x + c\right )} + 125 i\right )} \log \left (e^{\left (d x + c\right )} - \frac {3}{5} i - \frac {4}{5}\right ) - 167400 i \, e^{\left (5 \, d x + 5 \, c\right )} - 502200 \, e^{\left (4 \, d x + 4 \, c\right )} + 888336 i \, e^{\left (3 \, d x + 3 \, c\right )} + 954480 \, e^{\left (2 \, d x + 2 \, c\right )} - 549000 i \, e^{\left (d x + c\right )} - 199000}{98304 \, {\left (125 \, d e^{\left (6 \, d x + 6 \, c\right )} - 450 i \, d e^{\left (5 \, d x + 5 \, c\right )} - 915 \, d e^{\left (4 \, d x + 4 \, c\right )} + 1116 i \, d e^{\left (3 \, d x + 3 \, c\right )} + 915 \, d e^{\left (2 \, d x + 2 \, c\right )} - 450 i \, d e^{\left (d x + c\right )} - 125 \, d\right )}} \] Input:
integrate(1/(3+5*I*sinh(d*x+c))^4,x, algorithm="fricas")
Output:
-1/98304*(837*(125*I*e^(6*d*x + 6*c) + 450*e^(5*d*x + 5*c) - 915*I*e^(4*d* x + 4*c) - 1116*e^(3*d*x + 3*c) + 915*I*e^(2*d*x + 2*c) + 450*e^(d*x + c) - 125*I)*log(e^(d*x + c) - 3/5*I + 4/5) + 837*(-125*I*e^(6*d*x + 6*c) - 45 0*e^(5*d*x + 5*c) + 915*I*e^(4*d*x + 4*c) + 1116*e^(3*d*x + 3*c) - 915*I*e ^(2*d*x + 2*c) - 450*e^(d*x + c) + 125*I)*log(e^(d*x + c) - 3/5*I - 4/5) - 167400*I*e^(5*d*x + 5*c) - 502200*e^(4*d*x + 4*c) + 888336*I*e^(3*d*x + 3 *c) + 954480*e^(2*d*x + 2*c) - 549000*I*e^(d*x + c) - 199000)/(125*d*e^(6* d*x + 6*c) - 450*I*d*e^(5*d*x + 5*c) - 915*d*e^(4*d*x + 4*c) + 1116*I*d*e^ (3*d*x + 3*c) + 915*d*e^(2*d*x + 2*c) - 450*I*d*e^(d*x + c) - 125*d)
Time = 0.41 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx=\frac {20925 i e^{5 c} e^{5 d x} + 62775 e^{4 c} e^{4 d x} - 111042 i e^{3 c} e^{3 d x} - 119310 e^{2 c} e^{2 d x} + 68625 i e^{c} e^{d x} + 24875}{1536000 d e^{6 c} e^{6 d x} - 5529600 i d e^{5 c} e^{5 d x} - 11243520 d e^{4 c} e^{4 d x} + 13713408 i d e^{3 c} e^{3 d x} + 11243520 d e^{2 c} e^{2 d x} - 5529600 i d e^{c} e^{d x} - 1536000 d} + \frac {\operatorname {RootSum} {\left (1073741824 z^{2} + 77841, \left ( i \mapsto i \log {\left (\frac {\left (131072 i i - 837 i\right ) e^{- c}}{1395} + e^{d x} \right )} \right )\right )}}{d} \] Input:
integrate(1/(3+5*I*sinh(d*x+c))**4,x)
Output:
(20925*I*exp(5*c)*exp(5*d*x) + 62775*exp(4*c)*exp(4*d*x) - 111042*I*exp(3* c)*exp(3*d*x) - 119310*exp(2*c)*exp(2*d*x) + 68625*I*exp(c)*exp(d*x) + 248 75)/(1536000*d*exp(6*c)*exp(6*d*x) - 5529600*I*d*exp(5*c)*exp(5*d*x) - 112 43520*d*exp(4*c)*exp(4*d*x) + 13713408*I*d*exp(3*c)*exp(3*d*x) + 11243520* d*exp(2*c)*exp(2*d*x) - 5529600*I*d*exp(c)*exp(d*x) - 1536000*d) + RootSum (1073741824*_z**2 + 77841, Lambda(_i, _i*log((131072*_i*I - 837*I)*exp(-c) /1395 + exp(d*x))))/d
Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx=\frac {279 i \, \log \left (\frac {5 \, e^{\left (-d x - c\right )} + 3 i - 4}{5 \, e^{\left (-d x - c\right )} + 3 i + 4}\right )}{32768 \, d} + \frac {68625 i \, e^{\left (-d x - c\right )} + 119310 \, e^{\left (-2 \, d x - 2 \, c\right )} - 111042 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 62775 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20925 i \, e^{\left (-5 \, d x - 5 \, c\right )} - 24875}{-12288 \, d {\left (-450 i \, e^{\left (-d x - c\right )} - 915 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1116 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 915 \, e^{\left (-4 \, d x - 4 \, c\right )} - 450 i \, e^{\left (-5 \, d x - 5 \, c\right )} - 125 \, e^{\left (-6 \, d x - 6 \, c\right )} + 125\right )}} \] Input:
integrate(1/(3+5*I*sinh(d*x+c))^4,x, algorithm="maxima")
Output:
279/32768*I*log((5*e^(-d*x - c) + 3*I - 4)/(5*e^(-d*x - c) + 3*I + 4))/d + (68625*I*e^(-d*x - c) + 119310*e^(-2*d*x - 2*c) - 111042*I*e^(-3*d*x - 3* c) - 62775*e^(-4*d*x - 4*c) + 20925*I*e^(-5*d*x - 5*c) - 24875)/(d*(552960 0*I*e^(-d*x - c) + 11243520*e^(-2*d*x - 2*c) - 13713408*I*e^(-3*d*x - 3*c) - 11243520*e^(-4*d*x - 4*c) + 5529600*I*e^(-5*d*x - 5*c) + 1536000*e^(-6* d*x - 6*c) - 1536000))
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx=\frac {\frac {8 \, {\left (20925 i \, e^{\left (5 \, d x + 5 \, c\right )} + 62775 \, e^{\left (4 \, d x + 4 \, c\right )} - 111042 i \, e^{\left (3 \, d x + 3 \, c\right )} - 119310 \, e^{\left (2 \, d x + 2 \, c\right )} + 68625 i \, e^{\left (d x + c\right )} + 24875\right )}}{{\left (5 \, e^{\left (2 \, d x + 2 \, c\right )} - 6 i \, e^{\left (d x + c\right )} - 5\right )}^{3}} - 837 i \, \log \left (-\left (i - 2\right ) \, e^{\left (d x + c\right )} - 2 i + 1\right ) + 837 i \, \log \left (-\left (2 i - 1\right ) \, e^{\left (d x + c\right )} + i - 2\right )}{98304 \, d} \] Input:
integrate(1/(3+5*I*sinh(d*x+c))^4,x, algorithm="giac")
Output:
1/98304*(8*(20925*I*e^(5*d*x + 5*c) + 62775*e^(4*d*x + 4*c) - 111042*I*e^( 3*d*x + 3*c) - 119310*e^(2*d*x + 2*c) + 68625*I*e^(d*x + c) + 24875)/(5*e^ (2*d*x + 2*c) - 6*I*e^(d*x + c) - 5)^3 - 837*I*log(-(I - 2)*e^(d*x + c) - 2*I + 1) + 837*I*log(-(2*I - 1)*e^(d*x + c) + I - 2))/d
Time = 2.47 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.69 \[ \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx=-\frac {\frac {837}{102400\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,279{}\mathrm {i}}{20480\,d}}{1-{\mathrm {e}}^{2\,c+2\,d\,x}+\frac {{\mathrm {e}}^{c+d\,x}\,6{}\mathrm {i}}{5}}+\frac {\frac {7}{3750\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,39{}\mathrm {i}}{6250\,d}}{\frac {183\,{\mathrm {e}}^{4\,c+4\,d\,x}}{25}-\frac {183\,{\mathrm {e}}^{2\,c+2\,d\,x}}{25}-{\mathrm {e}}^{6\,c+6\,d\,x}+1+\frac {{\mathrm {e}}^{c+d\,x}\,18{}\mathrm {i}}{5}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,1116{}\mathrm {i}}{125}+\frac {{\mathrm {e}}^{5\,c+5\,d\,x}\,18{}\mathrm {i}}{5}}-\frac {\ln \left (-\frac {1395}{4}+{\mathrm {e}}^{c+d\,x}\,\left (-279-\frac {837}{4}{}\mathrm {i}\right )\right )\,279{}\mathrm {i}}{32768\,d}+\frac {\ln \left (\frac {1395}{4}+{\mathrm {e}}^{c+d\,x}\,\left (-279+\frac {837}{4}{}\mathrm {i}\right )\right )\,279{}\mathrm {i}}{32768\,d}-\frac {\frac {791}{80000\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,93{}\mathrm {i}}{16000\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-\frac {86\,{\mathrm {e}}^{2\,c+2\,d\,x}}{25}+1+\frac {{\mathrm {e}}^{c+d\,x}\,12{}\mathrm {i}}{5}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,12{}\mathrm {i}}{5}} \] Input:
int(1/(sinh(c + d*x)*5i + 3)^4,x)
Output:
((exp(c + d*x)*39i)/(6250*d) + 7/(3750*d))/((exp(c + d*x)*18i)/5 - (183*ex p(2*c + 2*d*x))/25 - (exp(3*c + 3*d*x)*1116i)/125 + (183*exp(4*c + 4*d*x)) /25 + (exp(5*c + 5*d*x)*18i)/5 - exp(6*c + 6*d*x) + 1) - ((exp(c + d*x)*27 9i)/(20480*d) + 837/(102400*d))/((exp(c + d*x)*6i)/5 - exp(2*c + 2*d*x) + 1) - (log(- exp(c + d*x)*(279 + 837i/4) - 1395/4)*279i)/(32768*d) + (log(1 395/4 - exp(c + d*x)*(279 - 837i/4))*279i)/(32768*d) - ((exp(c + d*x)*93i) /(16000*d) + 791/(80000*d))/((exp(c + d*x)*12i)/5 - (86*exp(2*c + 2*d*x))/ 25 - (exp(3*c + 3*d*x)*12i)/5 + exp(4*c + 4*d*x) + 1)
\[ \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx=\int \frac {1}{625 \sinh \left (d x +c \right )^{4}-1500 \sinh \left (d x +c \right )^{3} i -1350 \sinh \left (d x +c \right )^{2}+540 \sinh \left (d x +c \right ) i +81}d x \] Input:
int(1/(3+5*I*sinh(d*x+c))^4,x)
Output:
int(1/(625*sinh(c + d*x)**4 - 1500*sinh(c + d*x)**3*i - 1350*sinh(c + d*x) **2 + 540*sinh(c + d*x)*i + 81),x)