\(\int \frac {\coth ^5(x)}{(i+\sinh (x))^2} \, dx\) [226]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int \frac {\coth ^5(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{2} \text {csch}^2(x)+\frac {2}{3} i \text {csch}^3(x)+\frac {\text {csch}^4(x)}{4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2786, 45} \[ \int \frac {\coth ^5(x)}{(i+\sinh (x))^2} \, dx=\frac {\text {csch}^4(x)}{4}+\frac {2}{3} i \text {csch}^3(x)-\frac {\text {csch}^2(x)}{2} \]

[In]

Int[Coth[x]^5/(I + Sinh[x])^2,x]

[Out]

-1/2*Csch[x]^2 + ((2*I)/3)*Csch[x]^3 + Csch[x]^4/4

Rule 45

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])

Rule 2786

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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {(i-x)^2}{x^5} \, dx,x,\sinh (x)\right ) \\ & = \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^5(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{2} \text {csch}^2(x)+\frac {2}{3} i \text {csch}^3(x)+\frac {\text {csch}^4(x)}{4} \]

[In]

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

[Out]

-1/2*Csch[x]^2 + ((2*I)/3)*Csch[x]^3 + Csch[x]^4/4

Maple [A] (verified)

Time = 43.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52

method	result	size
risch	\(-\frac {2 \,{\mathrm e}^{2 x} \left (-8 i {\mathrm e}^{3 x}+3 \,{\mathrm e}^{4 x}+8 i {\mathrm e}^{x}-12 \,{\mathrm e}^{2 x}+3\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{4}}\)	\(41\)
default	\(\frac {i \tanh \left (\frac {x}{2}\right )}{4}+\frac {\tanh \left (\frac {x}{2}\right )^{4}}{64}-\frac {i \tanh \left (\frac {x}{2}\right )^{3}}{12}-\frac {3 \tanh \left (\frac {x}{2}\right )^{2}}{16}+\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}-\frac {i}{4 \tanh \left (\frac {x}{2}\right )}-\frac {3}{16 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {i}{12 \tanh \left (\frac {x}{2}\right )^{3}}\)	\(68\)

[In]

int(coth(x)^5/(I+sinh(x))^2,x,method=_RETURNVERBOSE)

[Out]

-2/3*exp(x)^2*(-8*I*exp(x)^3+3*exp(x)^4+8*I*exp(x)-12*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. 59 vs. \(2 (19) = 38\).

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.19 \[ \int \frac {\coth ^5(x)}{(i+\sinh (x))^2} \, dx=-\frac {2 \, {\left (3 \, e^{\left (6 \, x\right )} - 8 i \, e^{\left (5 \, x\right )} - 12 \, e^{\left (4 \, x\right )} + 8 i \, e^{\left (3 \, x\right )} + 3 \, e^{\left (2 \, x\right )}\right )}}{3 \, {\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}} \]

[In]

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

[Out]

-2/3*(3*e^(6*x) - 8*I*e^(5*x) - 12*e^(4*x) + 8*I*e^(3*x) + 3*e^(2*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. 65 vs. \(2 (22) = 44\).

Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \frac {\coth ^5(x)}{(i+\sinh (x))^2} \, dx=\frac {- 6 e^{6 x} + 16 i e^{5 x} + 24 e^{4 x} - 16 i e^{3 x} - 6 e^{2 x}}{3 e^{8 x} - 12 e^{6 x} + 18 e^{4 x} - 12 e^{2 x} + 3} \]

[In]

integrate(coth(x)**5/(I+sinh(x))**2,x)

[Out]

(-6*exp(6*x) + 16*I*exp(5*x) + 24*exp(4*x) - 16*I*exp(3*x) - 6*exp(2*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. 171 vs. \(2 (19) = 38\).

Time = 0.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 6.33 \[ \int \frac {\coth ^5(x)}{(i+\sinh (x))^2} \, dx=\frac {2 \, e^{\left (-2 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} - \frac {16 i \, e^{\left (-3 \, x\right )}}{3 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} - \frac {8 \, e^{\left (-4 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} + \frac {16 i \, e^{\left (-5 \, x\right )}}{3 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac {2 \, e^{\left (-6 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {\coth ^5(x)}{(i+\sinh (x))^2} \, dx=-\frac {2 \, {\left (3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 8 i \, e^{\left (-x\right )} - 8 i \, e^{x} - 6\right )}}{3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.34 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {\coth ^5(x)}{(i+\sinh (x))^2} \, dx=-\frac {2\,{\mathrm {e}}^{2\,x}\,\left (3\,{\mathrm {e}}^{4\,x}-12\,{\mathrm {e}}^{2\,x}+3-{\mathrm {e}}^{3\,x}\,8{}\mathrm {i}+{\mathrm {e}}^x\,8{}\mathrm {i}\right )}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^4} \]

[In]

int(coth(x)^5/(sinh(x) + 1i)^2,x)

[Out]

-(2*exp(2*x)*(3*exp(4*x) - exp(3*x)*8i - 12*exp(2*x) + exp(x)*8i + 3))/(3*(exp(2*x) - 1)^4)