3.226 \(\int \frac{\coth ^5(x)}{(i+\sinh (x))^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{\text{csch}^4(x)}{4}+\frac{2}{3} i \text{csch}^3(x)-\frac{\text{csch}^2(x)}{2} \]

[Out]

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

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Rubi [A]  time = 0.0427058, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2707, 43} \[ \frac{\text{csch}^4(x)}{4}+\frac{2}{3} i \text{csch}^3(x)-\frac{\text{csch}^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^5/(I + Sinh[x])^2,x]

[Out]

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

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(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]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0123971, size = 27, normalized size = 1. \[ \frac{\text{csch}^4(x)}{4}+\frac{2}{3} i \text{csch}^3(x)-\frac{\text{csch}^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^5/(I + Sinh[x])^2,x]

[Out]

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

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Maple [B]  time = 0.067, size = 68, normalized size = 2.5 \begin{align*}{\frac{i}{4}}\tanh \left ({\frac{x}{2}} \right ) +{\frac{1}{64} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}}-{\frac{i}{12}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}-{\frac{3}{16} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{3}{16} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{{\frac{i}{4}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{{\frac{i}{12}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{\frac{1}{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))^2,x)

[Out]

1/4*I*tanh(1/2*x)+1/64*tanh(1/2*x)^4-1/12*I*tanh(1/2*x)^3-3/16*tanh(1/2*x)^2-3/16/tanh(1/2*x)^2-1/4*I/tanh(1/2
*x)+1/12*I/tanh(1/2*x)^3+1/64/tanh(1/2*x)^4

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Maxima [B]  time = 1.02155, size = 231, normalized size = 8.56 \begin{align*} \frac{2 \, 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{16 i \, 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{8 \, e^{\left (-4 \, 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{16 i \, 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 \, 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^5/(I+sinh(x))^2,x, algorithm="maxima")

[Out]

2*e^(-2*x)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) - 16/3*I*e^(-3*x)/(4*e^(-2*x) - 6*e^(-4*x) +
4*e^(-6*x) - e^(-8*x) - 1) - 8*e^(-4*x)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) + 16/3*I*e^(-5*x
)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) + 2*e^(-6*x)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e
^(-8*x) - 1)

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Fricas [B]  time = 1.95788, size = 166, normalized size = 6.15 \begin{align*} -\frac{6 \, e^{\left (6 \, x\right )} - 16 i \, e^{\left (5 \, x\right )} - 24 \, e^{\left (4 \, x\right )} + 16 i \, e^{\left (3 \, x\right )} + 6 \, e^{\left (2 \, x\right )}}{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))^2,x, algorithm="fricas")

[Out]

-1/3*(6*e^(6*x) - 16*I*e^(5*x) - 24*e^(4*x) + 16*I*e^(3*x) + 6*e^(2*x))/(e^(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e
^(2*x) + 1)

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Sympy [B]  time = 0.74758, size = 66, normalized size = 2.44 \begin{align*} \frac{- 2 e^{6 x} + \frac{16 i e^{5 x}}{3} + 8 e^{4 x} - \frac{16 i e^{3 x}}{3} - 2 e^{2 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))**2,x)

[Out]

(-2*exp(6*x) + 16*I*exp(5*x)/3 + 8*exp(4*x) - 16*I*exp(3*x)/3 - 2*exp(2*x))/(exp(8*x) - 4*exp(6*x) + 6*exp(4*x
) - 4*exp(2*x) + 1)

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Giac [A]  time = 1.10279, size = 51, normalized size = 1.89 \begin{align*} -\frac{6 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 16 i \, e^{\left (-x\right )} - 16 i \, e^{x} - 12}{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))^2,x, algorithm="giac")

[Out]

-1/3*(6*(e^(-x) - e^x)^2 + 16*I*e^(-x) - 16*I*e^x - 12)/(e^(-x) - e^x)^4