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

Optimal. Leaf size=124 \[ -\frac{311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}-\frac{25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac{i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac{385 i \tan ^{-1}\left (\frac{\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{16384 d}+\frac{385 x}{32768} \]

[Out]

(385*x)/32768 - (((385*I)/16384)*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/d - ((I/16)*Cosh[c + d*x])/(d*(5
+ (3*I)*Sinh[c + d*x])^3) - (((25*I)/512)*Cosh[c + d*x])/(d*(5 + (3*I)*Sinh[c + d*x])^2) - (((311*I)/8192)*Co
sh[c + d*x])/(d*(5 + (3*I)*Sinh[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.101751, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2664, 2754, 12, 2657} \[ -\frac{311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}-\frac{25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac{i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac{385 i \tan ^{-1}\left (\frac{\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{16384 d}+\frac{385 x}{32768} \]

Antiderivative was successfully verified.

[In]

Int[(5 + (3*I)*Sinh[c + d*x])^(-4),x]

[Out]

(385*x)/32768 - (((385*I)/16384)*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/d - ((I/16)*Cosh[c + d*x])/(d*(5
+ (3*I)*Sinh[c + d*x])^3) - (((25*I)/512)*Cosh[c + d*x])/(d*(5 + (3*I)*Sinh[c + d*x])^2) - (((311*I)/8192)*Co
sh[c + d*x])/(d*(5 + (3*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 2657

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp
[(2*ArcTan[(b*Cos[c + d*x])/(a + q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && PosQ[a]

Rubi steps

\begin{align*} \int \frac{1}{(5+3 i \sinh (c+d x))^4} \, dx &=-\frac{i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac{1}{48} \int \frac{-15+6 i \sinh (c+d x)}{(5+3 i \sinh (c+d x))^3} \, dx\\ &=-\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{\int \frac{186-75 i \sinh (c+d x)}{(5+3 i \sinh (c+d x))^2} \, dx}{1536}\\ &=-\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))}-\frac{\int -\frac{1155}{5+3 i \sinh (c+d x)} \, dx}{24576}\\ &=-\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))}+\frac{385 \int \frac{1}{5+3 i \sinh (c+d x)} \, dx}{8192}\\ &=\frac{385 x}{32768}-\frac{385 i \tan ^{-1}\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))}\\ \end{align*}

Mathematica [B]  time = 1.76811, size = 308, normalized size = 2.48 \[ \frac{\frac{2 (-298563 i \sinh (c+d x)+89364 i \sinh (2 (c+d x))+8397 i \sinh (3 (c+d x))+166615 \cosh (c+d x)+82530 \cosh (2 (c+d x))-13995 \cosh (3 (c+d x))-235150)}{(3 \sinh (c+d x)-5 i)^3}+\frac{2656-192 i}{\left ((1+2 i) \cosh \left (\frac{1}{2} (c+d x)\right )-(2+i) \sinh \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{2656+192 i}{\left ((1+2 i) \sinh \left (\frac{1}{2} (c+d x)\right )+(2+i) \cosh \left (\frac{1}{2} (c+d x)\right )\right )^2}-1925 \log (5 \cosh (c+d x)-4 \sinh (c+d x))+1925 \log (4 \sinh (c+d x)+5 \cosh (c+d x))-3850 i \tan ^{-1}\left (\frac{2 \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )}{\cosh \left (\frac{1}{2} (c+d x)\right )-2 \sinh \left (\frac{1}{2} (c+d x)\right )}\right )+3850 i \tan ^{-1}\left (\frac{2 \sinh \left (\frac{1}{2} (c+d x)\right )+\cosh \left (\frac{1}{2} (c+d x)\right )}{\sinh \left (\frac{1}{2} (c+d x)\right )+2 \cosh \left (\frac{1}{2} (c+d x)\right )}\right )}{327680 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-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*C
osh[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)] - 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)/(32
7680*d)

________________________________________________________________________________________

Maple [B]  time = 0.052, size = 314, normalized size = 2.5 \begin{align*}{\frac{1053}{32000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-3}}-{\frac{{\frac{99\,i}{8000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-3}}+{\frac{783}{128000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-2}}-{\frac{{\frac{3753\,i}{64000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-2}}-{\frac{39933}{1024000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-1}}-{\frac{{\frac{8361\,i}{256000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) ^{-1}}+{\frac{385}{32768\,d}\ln \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) }+{\frac{1053}{32000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-3}}+{\frac{{\frac{99\,i}{8000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-3}}-{\frac{783}{128000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-2}}-{\frac{{\frac{3753\,i}{64000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-2}}-{\frac{39933}{1024000\,d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-1}}+{\frac{{\frac{8361\,i}{256000}}}{d} \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) ^{-1}}-{\frac{385}{32768\,d}\ln \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1053/32000/d/(5*tanh(1/2*d*x+1/2*c)+4-3*I)^3-99/8000*I/d/(5*tanh(1/2*d*x+1/2*c)+4-3*I)^3+783/128000/d/(5*tanh(
1/2*d*x+1/2*c)+4-3*I)^2-3753/64000*I/d/(5*tanh(1/2*d*x+1/2*c)+4-3*I)^2-39933/1024000/d/(5*tanh(1/2*d*x+1/2*c)+
4-3*I)-8361/256000*I/d/(5*tanh(1/2*d*x+1/2*c)+4-3*I)+385/32768/d*ln(5*tanh(1/2*d*x+1/2*c)+4-3*I)+1053/32000/d/
(5*tanh(1/2*d*x+1/2*c)-4-3*I)^3+99/8000*I/d/(5*tanh(1/2*d*x+1/2*c)-4-3*I)^3-783/128000/d/(5*tanh(1/2*d*x+1/2*c
)-4-3*I)^2-3753/64000*I/d/(5*tanh(1/2*d*x+1/2*c)-4-3*I)^2-39933/1024000/d/(5*tanh(1/2*d*x+1/2*c)-4-3*I)+8361/2
56000*I/d/(5*tanh(1/2*d*x+1/2*c)-4-3*I)-385/32768/d*ln(5*tanh(1/2*d*x+1/2*c)-4-3*I)

________________________________________________________________________________________

Maxima [A]  time = 1.71295, size = 205, normalized size = 1.65 \begin{align*} -\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}{d{\left (3317760 i \, e^{\left (-d x - c\right )} + 12054528 \, e^{\left (-2 \, d x - 2 \, c\right )} - 18923520 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 12054528 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3317760 i \, e^{\left (-5 \, d x - 5 \, c\right )} + 331776 \, e^{\left (-6 \, d x - 6 \, c\right )} - 331776\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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) + 12054
528*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))

________________________________________________________________________________________

Fricas [B]  time = 2.17341, size = 946, normalized size = 7.63 \begin{align*} \frac{{\left (31185 \, e^{\left (6 \, d x + 6 \, c\right )} - 311850 i \, e^{\left (5 \, d x + 5 \, c\right )} - 1133055 \, e^{\left (4 \, d x + 4 \, c\right )} + 1778700 i \, e^{\left (3 \, d x + 3 \, c\right )} + 1133055 \, e^{\left (2 \, d x + 2 \, c\right )} - 311850 i \, e^{\left (d x + c\right )} - 31185\right )} \log \left (e^{\left (d x + c\right )} - \frac{1}{3} i\right ) -{\left (31185 \, e^{\left (6 \, d x + 6 \, c\right )} - 311850 i \, e^{\left (5 \, d x + 5 \, c\right )} - 1133055 \, e^{\left (4 \, d x + 4 \, c\right )} + 1778700 i \, e^{\left (3 \, d x + 3 \, c\right )} + 1133055 \, e^{\left (2 \, d x + 2 \, c\right )} - 311850 i \, e^{\left (d x + c\right )} - 31185\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}{2654208 \, d e^{\left (6 \, d x + 6 \, c\right )} - 26542080 i \, d e^{\left (5 \, d x + 5 \, c\right )} - 96436224 \, d e^{\left (4 \, d x + 4 \, c\right )} + 151388160 i \, d e^{\left (3 \, d x + 3 \, c\right )} + 96436224 \, d e^{\left (2 \, d x + 2 \, c\right )} - 26542080 i \, d e^{\left (d x + c\right )} - 2654208 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((31185*e^(6*d*x + 6*c) - 311850*I*e^(5*d*x + 5*c) - 1133055*e^(4*d*x + 4*c) + 1778700*I*e^(3*d*x + 3*c) + 113
3055*e^(2*d*x + 2*c) - 311850*I*e^(d*x + c) - 31185)*log(e^(d*x + c) - 1/3*I) - (31185*e^(6*d*x + 6*c) - 31185
0*I*e^(5*d*x + 5*c) - 1133055*e^(4*d*x + 4*c) + 1778700*I*e^(3*d*x + 3*c) + 1133055*e^(2*d*x + 2*c) - 311850*I
*e^(d*x + c) - 31185)*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)/(2654208*d*e^(6*d*x + 6*c) - 26542080*
I*d*e^(5*d*x + 5*c) - 96436224*d*e^(4*d*x + 4*c) + 151388160*I*d*e^(3*d*x + 3*c) + 96436224*d*e^(2*d*x + 2*c)
- 26542080*I*d*e^(d*x + c) - 2654208*d)

________________________________________________________________________________________

Sympy [B]  time = 5.23034, size = 219, normalized size = 1.77 \begin{align*} \frac{- \frac{385 i e^{- c} e^{5 d x}}{12288 d} - \frac{9625 e^{- 2 c} e^{4 d x}}{36864 d} + \frac{119735 i e^{- 3 c} e^{3 d x}}{165888 d} + \frac{12137 e^{- 4 c} e^{2 d x}}{18432 d} - \frac{2725 i e^{- 5 c} e^{d x}}{12288 d} - \frac{311 e^{- 6 c}}{12288 d}}{e^{6 d x} - 10 i e^{- c} e^{5 d x} - \frac{109 e^{- 2 c} e^{4 d x}}{3} + \frac{1540 i e^{- 3 c} e^{3 d x}}{27} + \frac{109 e^{- 4 c} e^{2 d x}}{3} - 10 i e^{- 5 c} e^{d x} - e^{- 6 c}} + \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-385*I*exp(-c)*exp(5*d*x)/(12288*d) - 9625*exp(-2*c)*exp(4*d*x)/(36864*d) + 119735*I*exp(-3*c)*exp(3*d*x)/(16
5888*d) + 12137*exp(-4*c)*exp(2*d*x)/(18432*d) - 2725*I*exp(-5*c)*exp(d*x)/(12288*d) - 311*exp(-6*c)/(12288*d)
)/(exp(6*d*x) - 10*I*exp(-c)*exp(5*d*x) - 109*exp(-2*c)*exp(4*d*x)/3 + 1540*I*exp(-3*c)*exp(3*d*x)/27 + 109*ex
p(-4*c)*exp(2*d*x)/3 - 10*I*exp(-5*c)*exp(d*x) - exp(-6*c)) + (-385*log(exp(d*x) - 3*I*exp(-c))/32768 + 385*lo
g(exp(d*x) - I*exp(-c)/3)/32768)/d

________________________________________________________________________________________

Giac [A]  time = 1.29281, size = 153, normalized size = 1.23 \begin{align*} \frac{385 \, \log \left (3 \, e^{\left (d x + c\right )} - i\right )}{32768 \, d} - \frac{385 \, \log \left (e^{\left (d x + c\right )} - 3 i\right )}{32768 \, d} - \frac{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}{12288 \, d{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} - 10 i \, e^{\left (d x + c\right )} - 3\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

385/32768*log(3*e^(d*x + c) - I)/d - 385/32768*log(e^(d*x + c) - 3*I)/d - 1/12288*(10395*I*e^(5*d*x + 5*c) + 8
6625*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)/(d*(3*e
^(2*d*x + 2*c) - 10*I*e^(d*x + c) - 3)^3)