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

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 = 0.0599475, 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 = 0.150469, 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 = 0.043, size = 121, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( 2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-1}+{\frac{72}{5} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +i \right ) ^{-5}}-{6\,i \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +i \right ) ^{-2}}-{\frac{16}{7} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +i \right ) ^{-7}}-12\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-3}+{16\,i \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +i \right ) ^{-4}}-{8\,i \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +i \right ) ^{-6}} \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/(tanh(1/2*d*x+1/2*c)+I)+72/5/(tanh(1/2*d*x+1/2*c)+I)^5-6*I/(tanh(1/2*d*x+1/2*c)+I)^2-16/7/(tanh(1/2*d*x
+1/2*c)+I)^7-12/(tanh(1/2*d*x+1/2*c)+I)^3+16*I/(tanh(1/2*d*x+1/2*c)+I)^4-8*I/(tanh(1/2*d*x+1/2*c)+I)^6)

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Maxima [B]  time = 1.16414, 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 = 1.9639, 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 = 6.2558, 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 = 1.38993, 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)