3.214 \(\int \frac{\coth ^3(x)}{i+\sinh (x)} \, dx\)

Optimal. Leaf size=15 \[ -\text{csch}(x)+\frac{1}{2} i \text{csch}^2(x) \]

[Out]

-Csch[x] + (I/2)*Csch[x]^2

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Rubi [A]  time = 0.0585608, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 2606, 30, 8} \[ -\text{csch}(x)+\frac{1}{2} i \text{csch}^2(x) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^3/(I + Sinh[x]),x]

[Out]

-Csch[x] + (I/2)*Csch[x]^2

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 2606

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

Rule 30

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

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0114535, size = 15, normalized size = 1. \[ -\text{csch}(x)+\frac{1}{2} i \text{csch}^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^3/(I + Sinh[x]),x]

[Out]

-Csch[x] + (I/2)*Csch[x]^2

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Maple [B]  time = 0.038, size = 34, normalized size = 2.3 \begin{align*}{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+{{\frac{i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.04582, size = 90, normalized size = 6. \begin{align*} \frac{2 \, e^{\left (-x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac{2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac{2 \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.98198, size = 84, normalized size = 5.6 \begin{align*} -\frac{2 \, e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} - 2 \, e^{x}}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.256396, size = 32, normalized size = 2.13 \begin{align*} \frac{- 2 e^{3 x} + 2 i e^{2 x} + 2 e^{x}}{e^{4 x} - 2 e^{2 x} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*exp(3*x) + 2*I*exp(2*x) + 2*exp(x))/(exp(4*x) - 2*exp(2*x) + 1)

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Giac [B]  time = 1.14897, size = 32, normalized size = 2.13 \begin{align*} \frac{2 \, e^{\left (-x\right )} - 2 \, e^{x} + 2 i}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*e^(-x) - 2*e^x + 2*I)/(e^(-x) - e^x)^2