3.91 \(\int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx\)

Optimal. Leaf size=160 \[ \frac {995 i \cosh (c+d x)}{24576 d (3+5 i \sinh (c+d x))}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {279 i \log \left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \]

[Out]

-279/32768*I*ln(3*cosh(1/2*d*x+1/2*c)+I*sinh(1/2*d*x+1/2*c))/d+279/32768*I*ln(cosh(1/2*d*x+1/2*c)+3*I*sinh(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+995/245
76*I*cosh(d*x+c)/d/(3+5*I*sinh(d*x+c))

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Rubi [A]  time = 0.13, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2664, 2754, 12, 2660, 616, 31} \[ \frac {995 i \cosh (c+d x)}{24576 d (3+5 i \sinh (c+d x))}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {279 i \log \left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \]

Antiderivative was successfully verified.

[In]

Int[(3 + (5*I)*Sinh[c + d*x])^(-4),x]

[Out]

(((-279*I)/32768)*Log[3*Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]])/d + (((279*I)/32768)*Log[Cosh[(c + d*x)/2] +
(3*I)*Sinh[(c + d*x)/2]])/d + (((5*I)/48)*Cosh[c + d*x])/(d*(3 + (5*I)*Sinh[c + d*x])^3) - (((25*I)/512)*Cosh
[c + d*x])/(d*(3 + (5*I)*Sinh[c + d*x])^2) + (((995*I)/24576)*Cosh[c + d*x])/(d*(3 + (5*I)*Sinh[c + d*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(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]

Rule 2664

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] + Dist[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] && Integer
Q[2*n]

Rule 2754

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] + Dist[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]

Rubi steps

\begin {align*} \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx &=\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)}{(3+5 i \sinh (c+d x))^3} \, dx\\ &=\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 {\int \frac {154-75 i \sinh (c+d x)}{(3+5 i \sinh (c+d x))^2} \, dx}{1536}\\ &=\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))}+\frac {\int -\frac {837}{3+5 i \sinh (c+d x)} \, dx}{24576}\\ &=\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))}-\frac {279 \int \frac {1}{3+5 i \sinh (c+d x)} \, dx}{8192}\\ &=\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))}+\frac {(279 i) \operatorname {Subst}\left (\int \frac {1}{3+10 x+3 x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{4096 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))}+\frac {(837 i) \operatorname {Subst}\left (\int \frac {1}{1+3 x} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{32768 d}-\frac {(837 i) \operatorname {Subst}\left (\int \frac {1}{9+3 x} \, dx,x,\tan \left (\frac {1}{2} (i c+i 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 {279 i \log \left (1+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))}\\ \end {align*}

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Mathematica [A]  time = 0.63, size = 265, normalized size = 1.66 \[ \frac {-5022 \tan ^{-1}\left (3 \tanh \left (\frac {1}{2} (c+d x)\right )\right )+2511 i \log (4-5 \cosh (c+d x))-2511 i \log (5 \cosh (c+d x)+4)+40 \sinh \left (\frac {1}{2} (c+d x)\right ) \left (\frac {597}{\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )}+\frac {240}{\left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {199}{3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )}+\frac {80}{\left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^3}\right )+\frac {4640 i}{\left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {1440 i}{\left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}-5022 \tan ^{-1}\left (3 \coth \left (\frac {1}{2} (c+d x)\right )\right )}{589824 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + (5*I)*Sinh[c + d*x])^(-4),x]

[Out]

(-5022*ArcTan[3*Coth[(c + d*x)/2]] - 5022*ArcTan[3*Tanh[(c + d*x)/2]] + (2511*I)*Log[4 - 5*Cosh[c + d*x]] - (2
511*I)*Log[4 + 5*Cosh[c + d*x]] + (4640*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)

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fricas [B]  time = 0.95, size = 280, normalized size = 1.75 \[ \frac {{\left (-104625 i \, e^{\left (6 \, d x + 6 \, c\right )} - 376650 \, e^{\left (5 \, d x + 5 \, c\right )} + 765855 i \, e^{\left (4 \, d x + 4 \, c\right )} + 934092 \, e^{\left (3 \, d x + 3 \, c\right )} - 765855 i \, e^{\left (2 \, d x + 2 \, c\right )} - 376650 \, e^{\left (d x + c\right )} + 104625 i\right )} \log \left (e^{\left (d x + c\right )} - \frac {3}{5} i + \frac {4}{5}\right ) + {\left (104625 i \, e^{\left (6 \, d x + 6 \, c\right )} + 376650 \, e^{\left (5 \, d x + 5 \, c\right )} - 765855 i \, e^{\left (4 \, d x + 4 \, c\right )} - 934092 \, e^{\left (3 \, d x + 3 \, c\right )} + 765855 i \, e^{\left (2 \, d x + 2 \, c\right )} + 376650 \, e^{\left (d x + c\right )} - 104625 i\right )} \log \left (e^{\left (d x + c\right )} - \frac {3}{5} i - \frac {4}{5}\right ) + 167400 i \, e^{\left (5 \, d x + 5 \, c\right )} + 502200 \, e^{\left (4 \, d x + 4 \, c\right )} - 888336 i \, e^{\left (3 \, d x + 3 \, c\right )} - 954480 \, e^{\left (2 \, d x + 2 \, c\right )} + 549000 i \, e^{\left (d x + c\right )} + 199000}{12288000 \, d e^{\left (6 \, d x + 6 \, c\right )} - 44236800 i \, d e^{\left (5 \, d x + 5 \, c\right )} - 89948160 \, d e^{\left (4 \, d x + 4 \, c\right )} + 109707264 i \, d e^{\left (3 \, d x + 3 \, c\right )} + 89948160 \, d e^{\left (2 \, d x + 2 \, c\right )} - 44236800 i \, d e^{\left (d x + c\right )} - 12288000 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*I*sinh(d*x+c))^4,x, algorithm="fricas")

[Out]

((-104625*I*e^(6*d*x + 6*c) - 376650*e^(5*d*x + 5*c) + 765855*I*e^(4*d*x + 4*c) + 934092*e^(3*d*x + 3*c) - 765
855*I*e^(2*d*x + 2*c) - 376650*e^(d*x + c) + 104625*I)*log(e^(d*x + c) - 3/5*I + 4/5) + (104625*I*e^(6*d*x + 6
*c) + 376650*e^(5*d*x + 5*c) - 765855*I*e^(4*d*x + 4*c) - 934092*e^(3*d*x + 3*c) + 765855*I*e^(2*d*x + 2*c) +
376650*e^(d*x + c) - 104625*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)/(12288000*d*e^(6*d*x
+ 6*c) - 44236800*I*d*e^(5*d*x + 5*c) - 89948160*d*e^(4*d*x + 4*c) + 109707264*I*d*e^(3*d*x + 3*c) + 89948160*
d*e^(2*d*x + 2*c) - 44236800*I*d*e^(d*x + c) - 12288000*d)

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giac [A]  time = 0.16, size = 111, normalized size = 0.69 \[ \frac {\frac {8 \, {\left (20925 i \, e^{\left (5 \, d x + 5 \, c\right )} + 62775 \, e^{\left (4 \, d x + 4 \, c\right )} - 111042 i \, e^{\left (3 \, d x + 3 \, c\right )} - 119310 \, e^{\left (2 \, d x + 2 \, c\right )} + 68625 i \, e^{\left (d x + c\right )} + 24875\right )}}{{\left (5 \, e^{\left (2 \, d x + 2 \, c\right )} - 6 i \, e^{\left (d x + c\right )} - 5\right )}^{3}} - 837 i \, \log \left (-\left (i - 2\right ) \, e^{\left (d x + c\right )} - 2 i + 1\right ) + 837 i \, \log \left (-\left (2 i - 1\right ) \, e^{\left (d x + c\right )} + i - 2\right )}{98304 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*I*sinh(d*x+c))^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.09, size = 164, normalized size = 1.02 \[ -\frac {279 i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}{32768 d}+\frac {75 i}{1024 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{2}}-\frac {125}{768 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{3}}+\frac {345}{8192 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}+\frac {275 i}{27648 d \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}+\frac {279 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}{32768 d}-\frac {125}{20736 d \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {3505}{221184 d \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3+5*I*sinh(d*x+c))^4,x)

[Out]

-279/32768*I/d*ln(tanh(1/2*d*x+1/2*c)-3*I)+75/1024*I/d/(tanh(1/2*d*x+1/2*c)-3*I)^2-125/768/d/(tanh(1/2*d*x+1/2
*c)-3*I)^3+345/8192/d/(tanh(1/2*d*x+1/2*c)-3*I)+275/27648*I/d/(3*tanh(1/2*d*x+1/2*c)-I)^2+279/32768*I/d*ln(3*t
anh(1/2*d*x+1/2*c)-I)-125/20736/d/(3*tanh(1/2*d*x+1/2*c)-I)^3+3505/221184/d/(3*tanh(1/2*d*x+1/2*c)-I)

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maxima [A]  time = 0.43, size = 167, normalized size = 1.04 \[ \frac {279 i \, \log \left (\frac {10 \, e^{\left (-d x - c\right )} + 6 i - 8}{10 \, e^{\left (-d x - c\right )} + 6 i + 8}\right )}{32768 \, d} + \frac {68625 i \, e^{\left (-d x - c\right )} + 119310 \, e^{\left (-2 \, d x - 2 \, c\right )} - 111042 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 62775 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20925 i \, e^{\left (-5 \, d x - 5 \, c\right )} - 24875}{d {\left (5529600 i \, e^{\left (-d x - c\right )} + 11243520 \, e^{\left (-2 \, d x - 2 \, c\right )} - 13713408 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 11243520 \, e^{\left (-4 \, d x - 4 \, c\right )} + 5529600 i \, e^{\left (-5 \, d x - 5 \, c\right )} + 1536000 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1536000\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*I*sinh(d*x+c))^4,x, algorithm="maxima")

[Out]

279/32768*I*log((10*e^(-d*x - c) + 6*I - 8)/(10*e^(-d*x - c) + 6*I + 8))/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*(55
29600*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) + 5
529600*I*e^(-5*d*x - 5*c) + 1536000*e^(-6*d*x - 6*c) - 1536000))

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mupad [B]  time = 1.26, size = 237, normalized size = 1.48 \[ -\frac {\frac {837}{102400\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,279{}\mathrm {i}}{20480\,d}}{1-{\mathrm {e}}^{2\,c+2\,d\,x}+\frac {{\mathrm {e}}^{c+d\,x}\,6{}\mathrm {i}}{5}}+\frac {\frac {7}{3750\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,39{}\mathrm {i}}{6250\,d}}{\frac {183\,{\mathrm {e}}^{4\,c+4\,d\,x}}{25}-\frac {183\,{\mathrm {e}}^{2\,c+2\,d\,x}}{25}-{\mathrm {e}}^{6\,c+6\,d\,x}+1+\frac {{\mathrm {e}}^{c+d\,x}\,18{}\mathrm {i}}{5}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,1116{}\mathrm {i}}{125}+\frac {{\mathrm {e}}^{5\,c+5\,d\,x}\,18{}\mathrm {i}}{5}}-\frac {\ln \left (-\frac {1395}{4}+{\mathrm {e}}^{c+d\,x}\,\left (-279-\frac {837}{4}{}\mathrm {i}\right )\right )\,279{}\mathrm {i}}{32768\,d}+\frac {\ln \left (\frac {1395}{4}+{\mathrm {e}}^{c+d\,x}\,\left (-279+\frac {837}{4}{}\mathrm {i}\right )\right )\,279{}\mathrm {i}}{32768\,d}-\frac {\frac {791}{80000\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,93{}\mathrm {i}}{16000\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-\frac {86\,{\mathrm {e}}^{2\,c+2\,d\,x}}{25}+1+\frac {{\mathrm {e}}^{c+d\,x}\,12{}\mathrm {i}}{5}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,12{}\mathrm {i}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*5i + 3)^4,x)

[Out]

((exp(c + d*x)*39i)/(6250*d) + 7/(3750*d))/((exp(c + d*x)*18i)/5 - (183*exp(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)*
279i)/(20480*d) + 837/(102400*d))/((exp(c + d*x)*6i)/5 - exp(2*c + 2*d*x) + 1) - (log(- exp(c + d*x)*(279 + 83
7i/4) - 1395/4)*279i)/(32768*d) + (log(1395/4 - 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)

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sympy [A]  time = 0.73, size = 209, normalized size = 1.31 \[ \frac {24875 e^{6 c} - 68625 i e^{5 c} e^{- d x} - 119310 e^{4 c} e^{- 2 d x} + 111042 i e^{3 c} e^{- 3 d x} + 62775 e^{2 c} e^{- 4 d x} - 20925 i e^{c} e^{- 5 d x}}{1536000 d e^{6 c} - 5529600 i d e^{5 c} e^{- d x} - 11243520 d e^{4 c} e^{- 2 d x} + 13713408 i d e^{3 c} e^{- 3 d x} + 11243520 d e^{2 c} e^{- 4 d x} - 5529600 i d e^{c} e^{- 5 d x} - 1536000 d e^{- 6 d x}} + \frac {\operatorname {RootSum} {\left (419430400000000 z^{2} + 77841, \left (i \mapsto i \log {\left (- \frac {16384000 i i e^{c}}{279} + \frac {3 i e^{c}}{5} + e^{- d x} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*I*sinh(d*x+c))**4,x)

[Out]

(24875*exp(6*c) - 68625*I*exp(5*c)*exp(-d*x) - 119310*exp(4*c)*exp(-2*d*x) + 111042*I*exp(3*c)*exp(-3*d*x) + 6
2775*exp(2*c)*exp(-4*d*x) - 20925*I*exp(c)*exp(-5*d*x))/(1536000*d*exp(6*c) - 5529600*I*d*exp(5*c)*exp(-d*x) -
11243520*d*exp(4*c)*exp(-2*d*x) + 13713408*I*d*exp(3*c)*exp(-3*d*x) + 11243520*d*exp(2*c)*exp(-4*d*x) - 55296
00*I*d*exp(c)*exp(-5*d*x) - 1536000*d*exp(-6*d*x)) + RootSum(419430400000000*_z**2 + 77841, Lambda(_i, _i*log(
-16384000*_i*I*exp(c)/279 + 3*I*exp(c)/5 + exp(-d*x))))/d

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