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*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]))

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Rubi [A]  time = 0.126119, antiderivative size = 160, normalized size of antiderivative = 1., 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 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]

Rule 12

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

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 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 31

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

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*}

Mathematica [A]  time = 0.579572, 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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Maple [A]  time = 0.053, size = 164, normalized size = 1. \begin{align*}{\frac{{\frac{275\,i}{27648}}}{d} \left ( 3\,\tanh \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-2}}+{\frac{{\frac{279\,i}{32768}}}{d}\ln \left ( 3\,\tanh \left ( 1/2\,dx+c/2 \right ) -i \right ) }-{\frac{125}{20736\,d} \left ( 3\,\tanh \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-3}}+{\frac{3505}{221184\,d} \left ( 3\,\tanh \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-1}}-{\frac{{\frac{279\,i}{32768}}}{d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -3\,i \right ) }+{\frac{{\frac{75\,i}{1024}}}{d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -3\,i \right ) ^{-2}}-{\frac{125}{768\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -3\,i \right ) ^{-3}}+{\frac{345}{8192\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -3\,i \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

275/27648*I/d/(3*tanh(1/2*d*x+1/2*c)-I)^2+279/32768*I/d*ln(3*tanh(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)-279/32768*I/d*ln(tanh(1/2*d*x+1/2*c)-3*I)+75/1024*I/d/(t
anh(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)

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Maxima [A]  time = 1.68885, size = 225, normalized size = 1.41 \begin{align*} \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 )}} \end{align*}

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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Fricas [B]  time = 2.21196, size = 972, normalized size = 6.08 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 5.19372, size = 223, normalized size = 1.39 \begin{align*} \frac{\frac{279 i e^{- c} e^{5 d x}}{20480 d} + \frac{837 e^{- 2 c} e^{4 d x}}{20480 d} - \frac{18507 i e^{- 3 c} e^{3 d x}}{256000 d} - \frac{3977 e^{- 4 c} e^{2 d x}}{51200 d} + \frac{183 i e^{- 5 c} e^{d x}}{4096 d} + \frac{199 e^{- 6 c}}{12288 d}}{e^{6 d x} - \frac{18 i e^{- c} e^{5 d x}}{5} - \frac{183 e^{- 2 c} e^{4 d x}}{25} + \frac{1116 i e^{- 3 c} e^{3 d x}}{125} + \frac{183 e^{- 4 c} e^{2 d x}}{25} - \frac{18 i e^{- 5 c} e^{d x}}{5} - e^{- 6 c}} + \frac{\operatorname{RootSum}{\left (1073741824 z^{2} + 77841, \left ( i \mapsto i \log{\left (\frac{131072 i i e^{- c}}{1395} + e^{d x} - \frac{3 i e^{- c}}{5} \right )} \right )\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(279*I*exp(-c)*exp(5*d*x)/(20480*d) + 837*exp(-2*c)*exp(4*d*x)/(20480*d) - 18507*I*exp(-3*c)*exp(3*d*x)/(25600
0*d) - 3977*exp(-4*c)*exp(2*d*x)/(51200*d) + 183*I*exp(-5*c)*exp(d*x)/(4096*d) + 199*exp(-6*c)/(12288*d))/(exp
(6*d*x) - 18*I*exp(-c)*exp(5*d*x)/5 - 183*exp(-2*c)*exp(4*d*x)/25 + 1116*I*exp(-3*c)*exp(3*d*x)/125 + 183*exp(
-4*c)*exp(2*d*x)/25 - 18*I*exp(-5*c)*exp(d*x)/5 - exp(-6*c)) + RootSum(1073741824*_z**2 + 77841, Lambda(_i, _i
*log(131072*_i*I*exp(-c)/1395 + exp(d*x) - 3*I*exp(-c)/5)))/d

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Giac [A]  time = 1.29235, size = 155, normalized size = 0.97 \begin{align*} -\frac{279 i \, \log \left (-\left (i - 2\right ) \, e^{\left (d x + c\right )} - 2 i + 1\right )}{32768 \, d} + \frac{279 i \, \log \left (-\left (2 i - 1\right ) \, e^{\left (d x + c\right )} + i - 2\right )}{32768 \, d} + \frac{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}{12288 \, d{\left (5 \, e^{\left (2 \, d x + 2 \, c\right )} - 6 i \, e^{\left (d x + c\right )} - 5\right )}^{3}} \end{align*}

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]

-279/32768*I*log(-(I - 2)*e^(d*x + c) - 2*I + 1)/d + 279/32768*I*log(-(2*I - 1)*e^(d*x + c) + I - 2)/d + 1/122
88*(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) + 6862
5*I*e^(d*x + c) + 24875)/(d*(5*e^(2*d*x + 2*c) - 6*I*e^(d*x + c) - 5)^3)