3.216 \(\int \frac{\coth ^5(x)}{i+\sinh (x)} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{4} i \coth ^4(x)-\frac{\text{csch}^3(x)}{3}-\text{csch}(x) \]

[Out]

(I/4)*Coth[x]^4 - Csch[x] - Csch[x]^3/3

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Rubi [A]  time = 0.0768823, antiderivative size = 23, 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, 2607, 30, 2606} \[ \frac{1}{4} i \coth ^4(x)-\frac{\text{csch}^3(x)}{3}-\text{csch}(x) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^5/(I + Sinh[x]),x]

[Out]

(I/4)*Coth[x]^4 - Csch[x] - Csch[x]^3/3

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 2607

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

Rule 30

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

Rubi steps

\begin{align*} \int \frac{\coth ^5(x)}{i+\sinh (x)} \, dx &=-\left (i \int \coth ^3(x) \text{csch}^2(x) \, dx\right )+\int \coth ^3(x) \text{csch}(x) \, dx\\ &=i \operatorname{Subst}\left (\int x^3 \, dx,x,i \coth (x)\right )+i \operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,-i \text{csch}(x)\right )\\ &=\frac{1}{4} i \coth ^4(x)-\text{csch}(x)-\frac{\text{csch}^3(x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.0096195, size = 33, normalized size = 1.43 \[ \frac{1}{4} i \text{csch}^4(x)-\frac{\text{csch}^3(x)}{3}+\frac{1}{2} i \text{csch}^2(x)-\text{csch}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^5/(I + Sinh[x]),x]

[Out]

-Csch[x] + (I/2)*Csch[x]^2 - Csch[x]^3/3 + (I/4)*Csch[x]^4

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Maple [B]  time = 0.066, size = 68, normalized size = 3. \begin{align*}{\frac{3}{8}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{i}{64}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}+{\frac{1}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{i}{16}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+{{\frac{i}{16}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{3}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{{\frac{i}{64}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^5/(I+sinh(x)),x)

[Out]

3/8*tanh(1/2*x)+1/64*I*tanh(1/2*x)^4+1/24*tanh(1/2*x)^3+1/16*I*tanh(1/2*x)^2+1/16*I/tanh(1/2*x)^2-3/8/tanh(1/2
*x)-1/24/tanh(1/2*x)^3+1/64*I/tanh(1/2*x)^4

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Maxima [B]  time = 1.06873, size = 277, normalized size = 12.04 \begin{align*} \frac{2 \, e^{\left (-x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} - \frac{2 i \, e^{\left (-2 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} - \frac{10 \, e^{\left (-3 \, x\right )}}{3 \,{\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac{10 \, e^{\left (-5 \, x\right )}}{3 \,{\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} - \frac{2 i \, e^{\left (-6 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} - \frac{2 \, e^{\left (-7 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^5/(I+sinh(x)),x, algorithm="maxima")

[Out]

2*e^(-x)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) - 2*I*e^(-2*x)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(
-6*x) - e^(-8*x) - 1) - 10/3*e^(-3*x)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) + 10/3*e^(-5*x)/(4
*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) - 2*I*e^(-6*x)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(
-8*x) - 1) - 2*e^(-7*x)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1)

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Fricas [B]  time = 1.97264, size = 176, normalized size = 7.65 \begin{align*} -\frac{6 \, e^{\left (7 \, x\right )} - 6 i \, e^{\left (6 \, x\right )} - 10 \, e^{\left (5 \, x\right )} + 10 \, e^{\left (3 \, x\right )} - 6 i \, e^{\left (2 \, x\right )} - 6 \, e^{x}}{3 \,{\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^5/(I+sinh(x)),x, algorithm="fricas")

[Out]

-1/3*(6*e^(7*x) - 6*I*e^(6*x) - 10*e^(5*x) + 10*e^(3*x) - 6*I*e^(2*x) - 6*e^x)/(e^(8*x) - 4*e^(6*x) + 6*e^(4*x
) - 4*e^(2*x) + 1)

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Sympy [B]  time = 0.754139, size = 71, normalized size = 3.09 \begin{align*} \frac{- 2 e^{7 x} + 2 i e^{6 x} + \frac{10 e^{5 x}}{3} - \frac{10 e^{3 x}}{3} + 2 i e^{2 x} + 2 e^{x}}{e^{8 x} - 4 e^{6 x} + 6 e^{4 x} - 4 e^{2 x} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**5/(I+sinh(x)),x)

[Out]

(-2*exp(7*x) + 2*I*exp(6*x) + 10*exp(5*x)/3 - 10*exp(3*x)/3 + 2*I*exp(2*x) + 2*exp(x))/(exp(8*x) - 4*exp(6*x)
+ 6*exp(4*x) - 4*exp(2*x) + 1)

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Giac [B]  time = 1.13416, size = 69, normalized size = 3. \begin{align*} \frac{6 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 6 i \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 8 \, e^{\left (-x\right )} - 8 \, e^{x} + 12 i}{3 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^5/(I+sinh(x)),x, algorithm="giac")

[Out]

1/3*(6*(e^(-x) - e^x)^3 + 6*I*(e^(-x) - e^x)^2 + 8*e^(-x) - 8*e^x + 12*I)/(e^(-x) - e^x)^4