3.85 \(\int \frac{\cosh ^3(x)}{i+\text{csch}(x)} \, dx\)

Optimal. Leaf size=19 \[ \frac{\sinh ^2(x)}{2}-\frac{1}{3} i \sinh ^3(x) \]

[Out]

Sinh[x]^2/2 - (I/3)*Sinh[x]^3

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Rubi [A]  time = 0.105568, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3872, 2835, 2564, 30} \[ \frac{\sinh ^2(x)}{2}-\frac{1}{3} i \sinh ^3(x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^3/(I + Csch[x]),x]

[Out]

Sinh[x]^2/2 - (I/3)*Sinh[x]^3

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2835

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\cosh ^3(x)}{i+\text{csch}(x)} \, dx &=i \int \frac{\cosh ^3(x) \sinh (x)}{i-\sinh (x)} \, dx\\ &=-\left (i \int \cosh (x) \sinh ^2(x) \, dx\right )+\int \cosh (x) \sinh (x) \, dx\\ &=-\operatorname{Subst}(\int x \, dx,x,i \sinh (x))+\operatorname{Subst}\left (\int x^2 \, dx,x,i \sinh (x)\right )\\ &=\frac{\sinh ^2(x)}{2}-\frac{1}{3} i \sinh ^3(x)\\ \end{align*}

Mathematica [A]  time = 0.0097886, size = 19, normalized size = 1. \[ \frac{\sinh ^2(x)}{2}-\frac{1}{3} i \sinh ^3(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^3/(I + Csch[x]),x]

[Out]

Sinh[x]^2/2 - (I/3)*Sinh[x]^3

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Maple [A]  time = 0.022, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{2\, \left ({\rm csch} \left (x\right ) \right ) ^{2}}}-{\frac{{\frac{i}{3}}}{ \left ({\rm csch} \left (x\right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/csch(x)^2-1/3*I/csch(x)^3

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Maxima [B]  time = 1.03729, size = 53, normalized size = 2.79 \begin{align*} \frac{1}{24} \,{\left (3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - i\right )} e^{\left (3 \, x\right )} - \frac{1}{8} i \, e^{\left (-x\right )} + \frac{1}{8} \, e^{\left (-2 \, x\right )} + \frac{1}{24} i \, e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*e^(-x) + 3*I*e^(-2*x) - I)*e^(3*x) - 1/8*I*e^(-x) + 1/8*e^(-2*x) + 1/24*I*e^(-3*x)

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Fricas [B]  time = 1.85812, size = 108, normalized size = 5.68 \begin{align*} \frac{1}{24} \,{\left (-i \, e^{\left (6 \, x\right )} + 3 \, e^{\left (5 \, x\right )} + 3 i \, e^{\left (4 \, x\right )} - 3 i \, e^{\left (2 \, x\right )} + 3 \, e^{x} + i\right )} e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(-I*e^(6*x) + 3*e^(5*x) + 3*I*e^(4*x) - 3*I*e^(2*x) + 3*e^x + I)*e^(-3*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.14124, size = 47, normalized size = 2.47 \begin{align*} -\frac{1}{24} \,{\left (3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} e^{\left (-3 \, x\right )} - \frac{1}{24} i \, e^{\left (3 \, x\right )} + \frac{1}{8} \, e^{\left (2 \, x\right )} + \frac{1}{8} i \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(3*I*e^(2*x) - 3*e^x - I)*e^(-3*x) - 1/24*I*e^(3*x) + 1/8*e^(2*x) + 1/8*I*e^x