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

Optimal. Leaf size=15 \[ \frac{\sinh ^2(x)}{2}-i \sinh (x) \]

[Out]

(-I)*Sinh[x] + Sinh[x]^2/2

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Rubi [A]  time = 0.0342303, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2667} \[ \frac{\sinh ^2(x)}{2}-i \sinh (x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^3/(I + Sinh[x]),x]

[Out]

(-I)*Sinh[x] + Sinh[x]^2/2

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

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

Mathematica [A]  time = 0.0102805, size = 12, normalized size = 0.8 \[ \frac{1}{2} \sinh (x) (\sinh (x)-2 i) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^3/(I + Sinh[x]),x]

[Out]

(Sinh[x]*(-2*I + Sinh[x]))/2

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Maple [A]  time = 0.014, size = 13, normalized size = 0.9 \begin{align*} -i\sinh \left ( x \right ) +{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*sinh(x)+1/2*sinh(x)^2

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Maxima [B]  time = 1.06961, size = 36, normalized size = 2.4 \begin{align*} \frac{1}{8} \,{\left (-4 i \, e^{\left (-x\right )} + 1\right )} e^{\left (2 \, x\right )} + \frac{1}{2} i \, e^{\left (-x\right )} + \frac{1}{8} \, e^{\left (-2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.7818, size = 70, normalized size = 4.67 \begin{align*} \frac{1}{8} \,{\left (e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} + 4 i \, e^{x} + 1\right )} e^{\left (-2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.207659, size = 27, normalized size = 1.8 \begin{align*} \frac{e^{2 x}}{8} - \frac{i e^{x}}{2} + \frac{i e^{- x}}{2} + \frac{e^{- 2 x}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2*x)/8 - I*exp(x)/2 + I*exp(-x)/2 + exp(-2*x)/8

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Giac [B]  time = 1.23027, size = 31, normalized size = 2.07 \begin{align*} -\frac{1}{8} \,{\left (-4 i \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} + \frac{1}{8} \, e^{\left (2 \, x\right )} - \frac{1}{2} i \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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