\(\int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx\) [111]

Optimal result

Integrand size = 13, antiderivative size = 30 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=-\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x)-\frac {\text {csch}^3(x)}{3}-i \log (\sinh (x)) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, 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 = {3964, 76} \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=-\frac {1}{3} \text {csch}^3(x)+\frac {1}{2} i \text {csch}^2(x)-\text {csch}(x)-i \log (\sinh (x)) \]

[In]

Int[Coth[x]^5/(I + Csch[x]),x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rule 3964

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=-\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x)-\frac {\text {csch}^3(x)}{3}-i \log (\sinh (x)) \]

[In]

Integrate[Coth[x]^5/(I + Csch[x]),x]

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24 ) = 48\).

Time = 6.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80

method	result	size
risch	\(i x -\frac {2 \,{\mathrm e}^{x} \left (-3 i {\mathrm e}^{3 x}+3 \,{\mathrm e}^{4 x}+3 i {\mathrm e}^{x}-2 \,{\mathrm e}^{2 x}+3\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{3}}-i \ln \left ({\mathrm e}^{2 x}-1\right )\)	\(54\)
default	\(\frac {3 \tanh \left (\frac {x}{2}\right )}{8}+\frac {\tanh \left (\frac {x}{2}\right )^{3}}{24}+\frac {i \tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}-i \ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {i}{8 \tanh \left (\frac {x}{2}\right )^{2}}-\frac {3}{8 \tanh \left (\frac {x}{2}\right )}+i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\)	\(78\)

[In]

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

[Out]

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

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. 100 vs. \(2 (22) = 44\).

Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.33 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=\frac {3 i \, x e^{\left (6 \, x\right )} - 3 \, {\left (3 i \, x - 2 i\right )} e^{\left (4 \, x\right )} - 3 \, {\left (-3 i \, x + 2 i\right )} e^{\left (2 \, x\right )} - 3 \, {\left (i \, e^{\left (6 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 3 i \, x - 6 \, e^{\left (5 \, x\right )} + 4 \, e^{\left (3 \, x\right )} - 6 \, e^{x}}{3 \, {\left (e^{\left (6 \, x\right )} - 3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} - 1\right )}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=\int \frac {\coth ^{5}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]

[In]

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

[Out]

Integral(coth(x)**5/(csch(x) + I), x)

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. 75 vs. \(2 (22) = 44\).

Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.50 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=-i \, x + \frac {2 \, {\left (3 \, e^{\left (-x\right )} - 3 i \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} + 3 i \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

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

[Out]

-I*x + 2/3*(3*e^(-x) - 3*I*e^(-2*x) - 2*e^(-3*x) + 3*I*e^(-4*x) + 3*e^(-5*x))/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6
*x) - 1) - I*log(e^(-x) + 1) - I*log(e^(-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. 65 vs. \(2 (22) = 44\).

Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.17 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=\frac {11 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 12 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 12 i \, e^{\left (-x\right )} - 12 i \, e^{x} + 16}{6 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} - i \, \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right ) \]

[In]

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

[Out]

1/6*(11*I*(e^(-x) - e^x)^3 + 12*(e^(-x) - e^x)^2 + 12*I*e^(-x) - 12*I*e^x + 16)/(e^(-x) - e^x)^3 - I*log(abs(-
e^(-x) + e^x))

Mupad [B] (verification not implemented)

Time = 2.38 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,1{}\mathrm {i}-\frac {8\,{\mathrm {e}}^x}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {\frac {8\,{\mathrm {e}}^x}{3}-2{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {2\,{\mathrm {e}}^x-2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]

[In]

int(coth(x)^5/(1/sinh(x) + 1i),x)

[Out]

x*1i - log(exp(2*x) - 1)*1i - (8*exp(x))/(3*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - ((8*exp(x))/3 - 2i)/(e
xp(4*x) - 2*exp(2*x) + 1) - (2*exp(x) - 2i)/(exp(2*x) - 1)