3.182 \(\int \frac{\cosh ^2(x)}{(1+i \sinh (x))^3} \, dx\)

Optimal. Leaf size=20 \[ \frac{i \cosh ^3(x)}{3 (1+i \sinh (x))^3} \]

[Out]

((I/3)*Cosh[x]^3)/(1 + I*Sinh[x])^3

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Rubi [A]  time = 0.0330749, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2671} \[ \frac{i \cosh ^3(x)}{3 (1+i \sinh (x))^3} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^2/(1 + I*Sinh[x])^3,x]

[Out]

((I/3)*Cosh[x]^3)/(1 + I*Sinh[x])^3

Rule 2671

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

Rubi steps

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

Mathematica [A]  time = 0.0641376, size = 40, normalized size = 2. \[ -\frac{i \left (\cosh \left (\frac{3 x}{2}\right )-3 \cosh \left (\frac{x}{2}\right )\right )}{3 \left (\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^2/(1 + I*Sinh[x])^3,x]

[Out]

((-I/3)*(-3*Cosh[x/2] + Cosh[(3*x)/2]))/(Cosh[x/2] + I*Sinh[x/2])^3

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Maple [B]  time = 0.047, size = 36, normalized size = 1.8 \begin{align*}{4\,i \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}+2\, \left ( \tanh \left ( x/2 \right ) -i \right ) ^{-1}-{\frac{8}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^2/(1+I*sinh(x))^3,x)

[Out]

4*I/(tanh(1/2*x)-I)^2+2/(tanh(1/2*x)-I)-8/3/(tanh(1/2*x)-I)^3

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Maxima [B]  time = 1.11053, size = 72, normalized size = 3.6 \begin{align*} \frac{6 \, e^{\left (-2 \, x\right )}}{-9 i \, e^{\left (-x\right )} - 9 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} + 3} - \frac{2}{-9 i \, e^{\left (-x\right )} - 9 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2/(1+I*sinh(x))^3,x, algorithm="maxima")

[Out]

6*e^(-2*x)/(-9*I*e^(-x) - 9*e^(-2*x) + 3*I*e^(-3*x) + 3) - 2/(-9*I*e^(-x) - 9*e^(-2*x) + 3*I*e^(-3*x) + 3)

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Fricas [B]  time = 1.70145, size = 84, normalized size = 4.2 \begin{align*} \frac{-6 i \, e^{\left (2 \, x\right )} + 2 i}{3 \, e^{\left (3 \, x\right )} - 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} + 3 i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2/(1+I*sinh(x))^3,x, algorithm="fricas")

[Out]

(-6*I*e^(2*x) + 2*I)/(3*e^(3*x) - 9*I*e^(2*x) - 9*e^x + 3*I)

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Sympy [B]  time = 0.330437, size = 32, normalized size = 1.6 \begin{align*} \frac{- 2 i e^{2 x} + \frac{2 i}{3}}{e^{3 x} - 3 i e^{2 x} - 3 e^{x} + i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**2/(1+I*sinh(x))**3,x)

[Out]

(-2*I*exp(2*x) + 2*I/3)/(exp(3*x) - 3*I*exp(2*x) - 3*exp(x) + I)

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Giac [A]  time = 1.25264, size = 22, normalized size = 1.1 \begin{align*} -\frac{6 i \, e^{\left (2 \, x\right )} - 2 i}{3 \,{\left (e^{x} - i\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2/(1+I*sinh(x))^3,x, algorithm="giac")

[Out]

-1/3*(6*I*e^(2*x) - 2*I)/(e^x - I)^3