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

Optimal. Leaf size=117 \[ \frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4} \]

[Out]

1/7*I*cosh(d*x+c)/d/(1+I*sinh(d*x+c))^4+3/35*I*cosh(d*x+c)/d/(1+I*sinh(d*x+c))^3+2/35*I*cosh(d*x+c)/d/(1+I*sin
h(d*x+c))^2+2/35*I*cosh(d*x+c)/d/(1+I*sinh(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2650, 2648} \[ \frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Int[(1 + I*Sinh[c + d*x])^(-4),x]

[Out]

((I/7)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x])^4) + (((3*I)/35)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x])^3) + (
((2*I)/35)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x])^2) + (((2*I)/35)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x]))

Rule 2648

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

Rule 2650

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(a*
d*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rubi steps

\begin {align*} \int \frac {1}{(1+i \sinh (c+d x))^4} \, dx &=\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3}{7} \int \frac {1}{(1+i \sinh (c+d x))^3} \, dx\\ &=\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {6}{35} \int \frac {1}{(1+i \sinh (c+d x))^2} \, dx\\ &=\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {2}{35} \int \frac {1}{1+i \sinh (c+d x)} \, dx\\ &=\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 87, normalized size = 0.74 \[ \frac {35 \sinh \left (\frac {1}{2} (c+d x)\right )-7 \sinh \left (\frac {5}{2} (c+d x)\right )+21 i \cosh \left (\frac {3}{2} (c+d x)\right )-i \cosh \left (\frac {7}{2} (c+d x)\right )}{70 d \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + I*Sinh[c + d*x])^(-4),x]

[Out]

((21*I)*Cosh[(3*(c + d*x))/2] - I*Cosh[(7*(c + d*x))/2] + 35*Sinh[(c + d*x)/2] - 7*Sinh[(5*(c + d*x))/2])/(70*
d*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^7)

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 121, normalized size = 1.03 \[ -\frac {140 \, e^{\left (3 \, d x + 3 \, c\right )} - 84 i \, e^{\left (2 \, d x + 2 \, c\right )} - 28 \, e^{\left (d x + c\right )} + 4 i}{35 \, d e^{\left (7 \, d x + 7 \, c\right )} - 245 i \, d e^{\left (6 \, d x + 6 \, c\right )} - 735 \, d e^{\left (5 \, d x + 5 \, c\right )} + 1225 i \, d e^{\left (4 \, d x + 4 \, c\right )} + 1225 \, d e^{\left (3 \, d x + 3 \, c\right )} - 735 i \, d e^{\left (2 \, d x + 2 \, c\right )} - 245 \, d e^{\left (d x + c\right )} + 35 i \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(140*e^(3*d*x + 3*c) - 84*I*e^(2*d*x + 2*c) - 28*e^(d*x + c) + 4*I)/(35*d*e^(7*d*x + 7*c) - 245*I*d*e^(6*d*x
+ 6*c) - 735*d*e^(5*d*x + 5*c) + 1225*I*d*e^(4*d*x + 4*c) + 1225*d*e^(3*d*x + 3*c) - 735*I*d*e^(2*d*x + 2*c) -
245*d*e^(d*x + c) + 35*I*d)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 47, normalized size = 0.40 \[ -\frac {140 \, e^{\left (3 \, d x + 3 \, c\right )} - 84 i \, e^{\left (2 \, d x + 2 \, c\right )} - 28 \, e^{\left (d x + c\right )} + 4 i}{35 \, d {\left (e^{\left (d x + c\right )} - i\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(140*e^(3*d*x + 3*c) - 84*I*e^(2*d*x + 2*c) - 28*e^(d*x + c) + 4*I)/(d*(e^(d*x + c) - I)^7)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 121, normalized size = 1.03 \[ \frac {\frac {8 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {72}{5 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {6 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {16}{7 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {16 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {12}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(8*I/(-I+tanh(1/2*d*x+1/2*c))^6+72/5/(-I+tanh(1/2*d*x+1/2*c))^5+6*I/(-I+tanh(1/2*d*x+1/2*c))^2-16/7/(-I+ta
nh(1/2*d*x+1/2*c))^7-16*I/(-I+tanh(1/2*d*x+1/2*c))^4-12/(-I+tanh(1/2*d*x+1/2*c))^3+2/(-I+tanh(1/2*d*x+1/2*c)))

________________________________________________________________________________________

maxima [B]  time = 0.35, size = 372, normalized size = 3.18 \[ \frac {28 \, e^{\left (-d x - c\right )}}{d {\left (245 \, e^{\left (-d x - c\right )} - 735 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 1225 \, e^{\left (-3 \, d x - 3 \, c\right )} + 1225 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 735 \, e^{\left (-5 \, d x - 5 \, c\right )} - 245 i \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 i\right )}} - \frac {84 i \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (245 \, e^{\left (-d x - c\right )} - 735 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 1225 \, e^{\left (-3 \, d x - 3 \, c\right )} + 1225 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 735 \, e^{\left (-5 \, d x - 5 \, c\right )} - 245 i \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 i\right )}} - \frac {140 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (245 \, e^{\left (-d x - c\right )} - 735 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 1225 \, e^{\left (-3 \, d x - 3 \, c\right )} + 1225 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 735 \, e^{\left (-5 \, d x - 5 \, c\right )} - 245 i \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 i\right )}} + \frac {4 i}{d {\left (245 \, e^{\left (-d x - c\right )} - 735 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 1225 \, e^{\left (-3 \, d x - 3 \, c\right )} + 1225 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 735 \, e^{\left (-5 \, d x - 5 \, c\right )} - 245 i \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 i\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

28*e^(-d*x - c)/(d*(245*e^(-d*x - c) - 735*I*e^(-2*d*x - 2*c) - 1225*e^(-3*d*x - 3*c) + 1225*I*e^(-4*d*x - 4*c
) + 735*e^(-5*d*x - 5*c) - 245*I*e^(-6*d*x - 6*c) - 35*e^(-7*d*x - 7*c) + 35*I)) - 84*I*e^(-2*d*x - 2*c)/(d*(2
45*e^(-d*x - c) - 735*I*e^(-2*d*x - 2*c) - 1225*e^(-3*d*x - 3*c) + 1225*I*e^(-4*d*x - 4*c) + 735*e^(-5*d*x - 5
*c) - 245*I*e^(-6*d*x - 6*c) - 35*e^(-7*d*x - 7*c) + 35*I)) - 140*e^(-3*d*x - 3*c)/(d*(245*e^(-d*x - c) - 735*
I*e^(-2*d*x - 2*c) - 1225*e^(-3*d*x - 3*c) + 1225*I*e^(-4*d*x - 4*c) + 735*e^(-5*d*x - 5*c) - 245*I*e^(-6*d*x
- 6*c) - 35*e^(-7*d*x - 7*c) + 35*I)) + 4*I/(d*(245*e^(-d*x - c) - 735*I*e^(-2*d*x - 2*c) - 1225*e^(-3*d*x - 3
*c) + 1225*I*e^(-4*d*x - 4*c) + 735*e^(-5*d*x - 5*c) - 245*I*e^(-6*d*x - 6*c) - 35*e^(-7*d*x - 7*c) + 35*I))

________________________________________________________________________________________

mupad [B]  time = 0.96, size = 53, normalized size = 0.45 \[ -\frac {\left (7\,{\mathrm {e}}^{c+d\,x}+{\mathrm {e}}^{2\,c+2\,d\,x}\,21{}\mathrm {i}-35\,{\mathrm {e}}^{3\,c+3\,d\,x}-\mathrm {i}\right )\,4{}\mathrm {i}}{35\,d\,{\left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*1i + 1)^4,x)

[Out]

-((7*exp(c + d*x) + exp(2*c + 2*d*x)*21i - 35*exp(3*c + 3*d*x) - 1i)*4i)/(35*d*(exp(c + d*x)*1i + 1)^7)

________________________________________________________________________________________

sympy [A]  time = 0.44, size = 162, normalized size = 1.38 \[ \frac {- 4 e^{7 c} + 28 i e^{6 c} e^{- d x} + 84 e^{5 c} e^{- 2 d x} - 140 i e^{4 c} e^{- 3 d x}}{- 35 d e^{7 c} + 245 i d e^{6 c} e^{- d x} + 735 d e^{5 c} e^{- 2 d x} - 1225 i d e^{4 c} e^{- 3 d x} - 1225 d e^{3 c} e^{- 4 d x} + 735 i d e^{2 c} e^{- 5 d x} + 245 d e^{c} e^{- 6 d x} - 35 i d e^{- 7 d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-4*exp(7*c) + 28*I*exp(6*c)*exp(-d*x) + 84*exp(5*c)*exp(-2*d*x) - 140*I*exp(4*c)*exp(-3*d*x))/(-35*d*exp(7*c)
+ 245*I*d*exp(6*c)*exp(-d*x) + 735*d*exp(5*c)*exp(-2*d*x) - 1225*I*d*exp(4*c)*exp(-3*d*x) - 1225*d*exp(3*c)*e
xp(-4*d*x) + 735*I*d*exp(2*c)*exp(-5*d*x) + 245*d*exp(c)*exp(-6*d*x) - 35*I*d*exp(-7*d*x))

________________________________________________________________________________________