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]

((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]))

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Rubi [A]  time = 0.0596243, antiderivative size = 117, normalized size of antiderivative = 1., 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 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]

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]

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 = 0.164024, 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)

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Maple [A]  time = 0.049, size = 121, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( 2\, \left ( -i+\tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1}+{6\,i \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-12\, \left ( -i+\tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{-3}-{16\,i \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{8\,i \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}-{\frac{16}{7} \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}+{\frac{72}{5} \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.18628, size = 502, normalized size = 4.29 \begin{align*} \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 )}} \end{align*}

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))

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Fricas [A]  time = 2.07162, size = 338, normalized size = 2.89 \begin{align*} -\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} \end{align*}

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)

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Sympy [A]  time = 6.2866, size = 156, normalized size = 1.33 \begin{align*} \frac{- \frac{4 e^{- 4 c} e^{3 d x}}{d} + \frac{12 i e^{- 5 c} e^{2 d x}}{5 d} + \frac{4 e^{- 6 c} e^{d x}}{5 d} - \frac{4 i e^{- 7 c}}{35 d}}{e^{7 d x} - 7 i e^{- c} e^{6 d x} - 21 e^{- 2 c} e^{5 d x} + 35 i e^{- 3 c} e^{4 d x} + 35 e^{- 4 c} e^{3 d x} - 21 i e^{- 5 c} e^{2 d x} - 7 e^{- 6 c} e^{d x} + i e^{- 7 c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-4*exp(-4*c)*exp(3*d*x)/d + 12*I*exp(-5*c)*exp(2*d*x)/(5*d) + 4*exp(-6*c)*exp(d*x)/(5*d) - 4*I*exp(-7*c)/(35*
d))/(exp(7*d*x) - 7*I*exp(-c)*exp(6*d*x) - 21*exp(-2*c)*exp(5*d*x) + 35*I*exp(-3*c)*exp(4*d*x) + 35*exp(-4*c)*
exp(3*d*x) - 21*I*exp(-5*c)*exp(2*d*x) - 7*exp(-6*c)*exp(d*x) + I*exp(-7*c))

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Giac [A]  time = 1.43353, size = 63, normalized size = 0.54 \begin{align*} -\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}} \end{align*}

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)