3.158 \(\int \frac{\cosh ^8(x)}{i+\sinh (x)} \, dx\)

Optimal. Leaf size=50 \[ -\frac{5 i x}{16}+\frac{\cosh ^7(x)}{7}-\frac{1}{6} i \sinh (x) \cosh ^5(x)-\frac{5}{24} i \sinh (x) \cosh ^3(x)-\frac{5}{16} i \sinh (x) \cosh (x) \]

[Out]

((-5*I)/16)*x + Cosh[x]^7/7 - ((5*I)/16)*Cosh[x]*Sinh[x] - ((5*I)/24)*Cosh[x]^3*Sinh[x] - (I/6)*Cosh[x]^5*Sinh
[x]

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Rubi [A]  time = 0.0549461, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2682, 2635, 8} \[ -\frac{5 i x}{16}+\frac{\cosh ^7(x)}{7}-\frac{1}{6} i \sinh (x) \cosh ^5(x)-\frac{5}{24} i \sinh (x) \cosh ^3(x)-\frac{5}{16} i \sinh (x) \cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^8/(I + Sinh[x]),x]

[Out]

((-5*I)/16)*x + Cosh[x]^7/7 - ((5*I)/16)*Cosh[x]*Sinh[x] - ((5*I)/24)*Cosh[x]^3*Sinh[x] - (I/6)*Cosh[x]^5*Sinh
[x]

Rule 2682

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e
+ f*x])^(p - 1))/(b*f*(p - 1)), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\cosh ^8(x)}{i+\sinh (x)} \, dx &=\frac{\cosh ^7(x)}{7}-i \int \cosh ^6(x) \, dx\\ &=\frac{\cosh ^7(x)}{7}-\frac{1}{6} i \cosh ^5(x) \sinh (x)-\frac{5}{6} i \int \cosh ^4(x) \, dx\\ &=\frac{\cosh ^7(x)}{7}-\frac{5}{24} i \cosh ^3(x) \sinh (x)-\frac{1}{6} i \cosh ^5(x) \sinh (x)-\frac{5}{8} i \int \cosh ^2(x) \, dx\\ &=\frac{\cosh ^7(x)}{7}-\frac{5}{16} i \cosh (x) \sinh (x)-\frac{5}{24} i \cosh ^3(x) \sinh (x)-\frac{1}{6} i \cosh ^5(x) \sinh (x)-\frac{5}{16} i \int 1 \, dx\\ &=-\frac{5 i x}{16}+\frac{\cosh ^7(x)}{7}-\frac{5}{16} i \cosh (x) \sinh (x)-\frac{5}{24} i \cosh ^3(x) \sinh (x)-\frac{1}{6} i \cosh ^5(x) \sinh (x)\\ \end{align*}

Mathematica [B]  time = 0.150883, size = 219, normalized size = 4.38 \[ \frac{\cosh ^9(x) \left (48 \sqrt{1+i \sinh (x)} \sinh ^7(x)-8 i \sqrt{1+i \sinh (x)} \sinh ^6(x)+200 \sqrt{1+i \sinh (x)} \sinh ^5(x)-38 i \sqrt{1+i \sinh (x)} \sinh ^4(x)+326 \sqrt{1+i \sinh (x)} \sinh ^3(x)-87 i \sqrt{1+i \sinh (x)} \sinh ^2(x)+279 \sqrt{1+i \sinh (x)} \sinh (x)+6 i \left (8 \sqrt{1+i \sinh (x)}+35 \sqrt{1-i \sinh (x)} \sin ^{-1}\left (\frac{\sqrt{1-i \sinh (x)}}{\sqrt{2}}\right )\right )\right )}{336 \sqrt{1+i \sinh (x)} (\sinh (x)-i)^4 (\sinh (x)+i)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^8/(I + Sinh[x]),x]

[Out]

(Cosh[x]^9*((6*I)*(35*ArcSin[Sqrt[1 - I*Sinh[x]]/Sqrt[2]]*Sqrt[1 - I*Sinh[x]] + 8*Sqrt[1 + I*Sinh[x]]) + 279*S
qrt[1 + I*Sinh[x]]*Sinh[x] - (87*I)*Sqrt[1 + I*Sinh[x]]*Sinh[x]^2 + 326*Sqrt[1 + I*Sinh[x]]*Sinh[x]^3 - (38*I)
*Sqrt[1 + I*Sinh[x]]*Sinh[x]^4 + 200*Sqrt[1 + I*Sinh[x]]*Sinh[x]^5 - (8*I)*Sqrt[1 + I*Sinh[x]]*Sinh[x]^6 + 48*
Sqrt[1 + I*Sinh[x]]*Sinh[x]^7))/(336*Sqrt[1 + I*Sinh[x]]*(-I + Sinh[x])^4*(I + Sinh[x])^5)

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Maple [B]  time = 0.066, size = 292, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^8/(I+sinh(x)),x)

[Out]

-11/16/(tanh(1/2*x)+1)^2+5/16/(tanh(1/2*x)+1)+9/8/(tanh(1/2*x)+1)^3-5/16/(tanh(1/2*x)-1)-11/16/(tanh(1/2*x)-1)
^2-9/8/(tanh(1/2*x)-1)^3-11/16*I/(tanh(1/2*x)+1)-7/6*I/(tanh(1/2*x)+1)^3-11/16*I/(tanh(1/2*x)-1)-7/6*I/(tanh(1
/2*x)-1)^3+19/16*I/(tanh(1/2*x)+1)^2+1/6*I/(tanh(1/2*x)+1)^6-1/2*I/(tanh(1/2*x)+1)^5+5/16*I*ln(tanh(1/2*x)-1)-
I/(tanh(1/2*x)-1)^4-19/16*I/(tanh(1/2*x)-1)^2+I/(tanh(1/2*x)+1)^4-1/2*I/(tanh(1/2*x)-1)^5-1/6*I/(tanh(1/2*x)-1
)^6-5/16*I*ln(tanh(1/2*x)+1)-1/7/(tanh(1/2*x)-1)^7-5/4/(tanh(1/2*x)-1)^4+1/(tanh(1/2*x)+1)^5-1/2/(tanh(1/2*x)-
1)^6-5/4/(tanh(1/2*x)+1)^4-1/2/(tanh(1/2*x)+1)^6-1/(tanh(1/2*x)-1)^5+1/7/(tanh(1/2*x)+1)^7

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Maxima [B]  time = 1.35429, size = 122, normalized size = 2.44 \begin{align*} -\frac{1}{5376} \,{\left (14 i \, e^{\left (-x\right )} - 42 \, e^{\left (-2 \, x\right )} + 126 i \, e^{\left (-3 \, x\right )} - 126 \, e^{\left (-4 \, x\right )} + 630 i \, e^{\left (-5 \, x\right )} - 210 \, e^{\left (-6 \, x\right )} - 6\right )} e^{\left (7 \, x\right )} - \frac{5}{16} i \, x + \frac{5}{128} \, e^{\left (-x\right )} + \frac{15}{128} i \, e^{\left (-2 \, x\right )} + \frac{3}{128} \, e^{\left (-3 \, x\right )} + \frac{3}{128} i \, e^{\left (-4 \, x\right )} + \frac{1}{128} \, e^{\left (-5 \, x\right )} + \frac{1}{384} i \, e^{\left (-6 \, x\right )} + \frac{1}{896} \, e^{\left (-7 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^8/(I+sinh(x)),x, algorithm="maxima")

[Out]

-1/5376*(14*I*e^(-x) - 42*e^(-2*x) + 126*I*e^(-3*x) - 126*e^(-4*x) + 630*I*e^(-5*x) - 210*e^(-6*x) - 6)*e^(7*x
) - 5/16*I*x + 5/128*e^(-x) + 15/128*I*e^(-2*x) + 3/128*e^(-3*x) + 3/128*I*e^(-4*x) + 1/128*e^(-5*x) + 1/384*I
*e^(-6*x) + 1/896*e^(-7*x)

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Fricas [B]  time = 1.8912, size = 301, normalized size = 6.02 \begin{align*} \frac{1}{2688} \,{\left (-840 i \, x e^{\left (7 \, x\right )} + 3 \, e^{\left (14 \, x\right )} - 7 i \, e^{\left (13 \, x\right )} + 21 \, e^{\left (12 \, x\right )} - 63 i \, e^{\left (11 \, x\right )} + 63 \, e^{\left (10 \, x\right )} - 315 i \, e^{\left (9 \, x\right )} + 105 \, e^{\left (8 \, x\right )} + 105 \, e^{\left (6 \, x\right )} + 315 i \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 i \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 i \, e^{x} + 3\right )} e^{\left (-7 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^8/(I+sinh(x)),x, algorithm="fricas")

[Out]

1/2688*(-840*I*x*e^(7*x) + 3*e^(14*x) - 7*I*e^(13*x) + 21*e^(12*x) - 63*I*e^(11*x) + 63*e^(10*x) - 315*I*e^(9*
x) + 105*e^(8*x) + 105*e^(6*x) + 315*I*e^(5*x) + 63*e^(4*x) + 63*I*e^(3*x) + 21*e^(2*x) + 7*I*e^x + 3)*e^(-7*x
)

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Sympy [B]  time = 0.729779, size = 124, normalized size = 2.48 \begin{align*} - \frac{5 i x}{16} + \frac{e^{7 x}}{896} - \frac{i e^{6 x}}{384} + \frac{e^{5 x}}{128} - \frac{3 i e^{4 x}}{128} + \frac{3 e^{3 x}}{128} - \frac{15 i e^{2 x}}{128} + \frac{5 e^{x}}{128} + \frac{5 e^{- x}}{128} + \frac{15 i e^{- 2 x}}{128} + \frac{3 e^{- 3 x}}{128} + \frac{3 i e^{- 4 x}}{128} + \frac{e^{- 5 x}}{128} + \frac{i e^{- 6 x}}{384} + \frac{e^{- 7 x}}{896} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**8/(I+sinh(x)),x)

[Out]

-5*I*x/16 + exp(7*x)/896 - I*exp(6*x)/384 + exp(5*x)/128 - 3*I*exp(4*x)/128 + 3*exp(3*x)/128 - 15*I*exp(2*x)/1
28 + 5*exp(x)/128 + 5*exp(-x)/128 + 15*I*exp(-2*x)/128 + 3*exp(-3*x)/128 + 3*I*exp(-4*x)/128 + exp(-5*x)/128 +
I*exp(-6*x)/384 + exp(-7*x)/896

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Giac [B]  time = 1.20025, size = 116, normalized size = 2.32 \begin{align*} \frac{1}{2688} \,{\left (105 \, e^{\left (6 \, x\right )} + 315 i \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 i \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 i \, e^{x} + 3\right )} e^{\left (-7 \, x\right )} - \frac{5}{16} i \, x + \frac{1}{896} \, e^{\left (7 \, x\right )} - \frac{1}{384} i \, e^{\left (6 \, x\right )} + \frac{1}{128} \, e^{\left (5 \, x\right )} - \frac{3}{128} i \, e^{\left (4 \, x\right )} + \frac{3}{128} \, e^{\left (3 \, x\right )} - \frac{15}{128} i \, e^{\left (2 \, x\right )} + \frac{5}{128} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^8/(I+sinh(x)),x, algorithm="giac")

[Out]

1/2688*(105*e^(6*x) + 315*I*e^(5*x) + 63*e^(4*x) + 63*I*e^(3*x) + 21*e^(2*x) + 7*I*e^x + 3)*e^(-7*x) - 5/16*I*
x + 1/896*e^(7*x) - 1/384*I*e^(6*x) + 1/128*e^(5*x) - 3/128*I*e^(4*x) + 3/128*e^(3*x) - 15/128*I*e^(2*x) + 5/1
28*e^x