\(\int \frac {\coth ^5(x)}{i+\sinh (x)} \, dx\) [216]

Optimal result

Integrand size = 13, antiderivative size = 23 \[ \int \frac {\coth ^5(x)}{i+\sinh (x)} \, dx=\frac {1}{4} i \coth ^4(x)-\text {csch}(x)-\frac {\text {csch}^3(x)}{3} \]

[Out]

1/4*I*coth(x)^4-csch(x)-1/3*csch(x)^3

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2785, 2687, 30, 2686} \[ \int \frac {\coth ^5(x)}{i+\sinh (x)} \, dx=\frac {1}{4} i \coth ^4(x)-\frac {\text {csch}^3(x)}{3}-\text {csch}(x) \]

[In]

Int[Coth[x]^5/(I + Sinh[x]),x]

[Out]

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

Rule 30

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

Rule 2686

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 2687

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 2785

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]

Rubi steps \begin{align*} \text {integral}& = -\left (i \int \coth ^3(x) \text {csch}^2(x) \, dx\right )+\int \coth ^3(x) \text {csch}(x) \, dx \\ & = i \text {Subst}\left (\int x^3 \, dx,x,i \coth (x)\right )+i \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {\coth ^5(x)}{i+\sinh (x)} \, dx=-\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x)-\frac {\text {csch}^3(x)}{3}+\frac {1}{4} i \text {csch}^4(x) \]

[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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (18 ) = 36\).

Time = 17.70 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96

method	result	size
risch	\(-\frac {2 \,{\mathrm e}^{x} \left (-3 i {\mathrm e}^{5 x}+3 \,{\mathrm e}^{6 x}-5 \,{\mathrm e}^{4 x}-3 i {\mathrm e}^{x}+5 \,{\mathrm e}^{2 x}-3\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{4}}\)	\(45\)
default	\(\frac {3 \tanh \left (\frac {x}{2}\right )}{8}+\frac {i \tanh \left (\frac {x}{2}\right )^{4}}{64}+\frac {\tanh \left (\frac {x}{2}\right )^{3}}{24}+\frac {i \tanh \left (\frac {x}{2}\right )^{2}}{16}+\frac {i}{64 \tanh \left (\frac {x}{2}\right )^{4}}-\frac {3}{8 \tanh \left (\frac {x}{2}\right )}+\frac {i}{16 \tanh \left (\frac {x}{2}\right )^{2}}-\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}\)	\(68\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (17) = 34\).

Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {\coth ^5(x)}{i+\sinh (x)} \, dx=-\frac {2 \, {\left (3 \, e^{\left (7 \, x\right )} - 3 i \, e^{\left (6 \, x\right )} - 5 \, e^{\left (5 \, x\right )} + 5 \, e^{\left (3 \, x\right )} - 3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x}\right )}}{3 \, {\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (17) = 34\).

Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int \frac {\coth ^5(x)}{i+\sinh (x)} \, dx=\frac {- 6 e^{7 x} + 6 i e^{6 x} + 10 e^{5 x} - 10 e^{3 x} + 6 i e^{2 x} + 6 e^{x}}{3 e^{8 x} - 12 e^{6 x} + 18 e^{4 x} - 12 e^{2 x} + 3} \]

[In]

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

[Out]

(-6*exp(7*x) + 6*I*exp(6*x) + 10*exp(5*x) - 10*exp(3*x) + 6*I*exp(2*x) + 6*exp(x))/(3*exp(8*x) - 12*exp(6*x) +
18*exp(4*x) - 12*exp(2*x) + 3)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (17) = 34\).

Time = 0.20 (sec) , antiderivative size = 205, normalized size of antiderivative = 8.91 \[ \int \frac {\coth ^5(x)}{i+\sinh (x)} \, dx=\frac {2 \, e^{\left (-x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} - \frac {2 i \, e^{\left (-2 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} - \frac {10 \, e^{\left (-3 \, x\right )}}{3 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac {10 \, e^{\left (-5 \, x\right )}}{3 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} - \frac {2 i \, e^{\left (-6 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} - \frac {2 \, e^{\left (-7 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} \]

[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)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (17) = 34\).

Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {\coth ^5(x)}{i+\sinh (x)} \, dx=\frac {2 \, {\left (3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 3 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, e^{\left (-x\right )} - 4 \, e^{x} + 6 i\right )}}{3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {\coth ^5(x)}{i+\sinh (x)} \, dx=\frac {2\,{\mathrm {e}}^x\,\left (5\,{\mathrm {e}}^{4\,x}-5\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{6\,x}+3+{\mathrm {e}}^{5\,x}\,3{}\mathrm {i}+{\mathrm {e}}^x\,3{}\mathrm {i}\right )}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^4} \]

[In]

int(coth(x)^5/(sinh(x) + 1i),x)

[Out]

(2*exp(x)*(5*exp(4*x) - 5*exp(2*x) + exp(5*x)*3i - 3*exp(6*x) + exp(x)*3i + 3))/(3*(exp(2*x) - 1)^4)