\(\int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx\) [109]

Optimal result

Integrand size = 13, antiderivative size = 12 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-\text {csch}(x)-i \log (\sinh (x)) \]

[Out]

-csch(x)-I*ln(sinh(x))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 12, 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, 45} \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-\text {csch}(x)-i \log (\sinh (x)) \]

[In]

Int[Coth[x]^3/(I + Csch[x]),x]

[Out]

-Csch[x] - I*Log[Sinh[x]]

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 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}{x^2} \, dx,x,i \sinh (x)\right ) \\ & = \text {Subst}\left (\int \left (\frac {i}{x^2}-\frac {i}{x}\right ) \, dx,x,i \sinh (x)\right ) \\ & = -\text {csch}(x)-i \log (\sinh (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-\text {csch}(x)-i \log (\sinh (x)) \]

[In]

Integrate[Coth[x]^3/(I + Csch[x]),x]

[Out]

-Csch[x] - I*Log[Sinh[x]]

Maple [A] (verified)

Time = 2.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(-\operatorname {csch}\left (x \right )+i \ln \left (\operatorname {csch}\left (x \right )\right )\)	\(12\)
default	\(-\operatorname {csch}\left (x \right )+i \ln \left (\operatorname {csch}\left (x \right )\right )\)	\(12\)
risch	\(i x -\frac {2 \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}-1}-i \ln \left ({\mathrm e}^{2 x}-1\right )\)	\(28\)

[In]

int(coth(x)^3/(I+csch(x)),x,method=_RETURNVERBOSE)

[Out]

-csch(x)+I*ln(csch(x))

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. 40 vs. \(2 (10) = 20\).

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 3.33 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=\frac {i \, x e^{\left (2 \, x\right )} + {\left (-i \, e^{\left (2 \, x\right )} + i\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - i \, x - 2 \, e^{x}}{e^{\left (2 \, x\right )} - 1} \]

[In]

integrate(coth(x)^3/(I+csch(x)),x, algorithm="fricas")

[Out]

(I*x*e^(2*x) + (-I*e^(2*x) + I)*log(e^(2*x) - 1) - I*x - 2*e^x)/(e^(2*x) - 1)

Sympy [F]

\[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]

[In]

integrate(coth(x)**3/(I+csch(x)),x)

[Out]

Integral(coth(x)**3/(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. 36 vs. \(2 (10) = 20\).

Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 3.00 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-i \, x + \frac {2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} - 1} - i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

integrate(coth(x)^3/(I+csch(x)),x, algorithm="maxima")

[Out]

-I*x + 2*e^(-x)/(e^(-2*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. 39 vs. \(2 (10) = 20\).

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 3.25 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {-i \, e^{\left (-x\right )} + i \, e^{x} - 2}{e^{\left (-x\right )} - e^{x}} - i \, \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right ) \]

[In]

integrate(coth(x)^3/(I+csch(x)),x, algorithm="giac")

[Out]

-(-I*e^(-x) + I*e^x - 2)/(e^(-x) - e^x) - I*log(abs(-e^(-x) + e^x))

Mupad [B] (verification not implemented)

Time = 2.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.25 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}+x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,1{}\mathrm {i} \]

[In]

int(coth(x)^3/(1/sinh(x) + 1i),x)

[Out]

x*1i - log(exp(2*x) - 1)*1i - (2*exp(x))/(exp(2*x) - 1)