\(\int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx\) [44]

Optimal result

Integrand size = 11, antiderivative size = 19 \[ \int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx=i \text {arctanh}(\cosh (x))+\frac {\cosh (x)}{i+\sinh (x)} \]

[Out]

I*arctanh(cosh(x))+cosh(x)/(I+sinh(x))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2826, 2727, 3855} \[ \int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx=i \text {arctanh}(\cosh (x))+\frac {\cosh (x)}{\sinh (x)+i} \]

[In]

Int[Csch[x]/(I + Sinh[x]),x]

[Out]

I*ArcTanh[Cosh[x]] + Cosh[x]/(I + Sinh[x])

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2826

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[b/(
b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; Fre
eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = -(i \int \text {csch}(x) \, dx)+i \int \frac {1}{i+\sinh (x)} \, dx \\ & = i \text {arctanh}(\cosh (x))+\frac {\cosh (x)}{i+\sinh (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx=\text {sech}(x) \left (-i+i \text {arctanh}\left (\sqrt {\cosh ^2(x)}\right ) \sqrt {\cosh ^2(x)}+\sinh (x)\right ) \]

[In]

Integrate[Csch[x]/(I + Sinh[x]),x]

[Out]

Sech[x]*(-I + I*ArcTanh[Sqrt[Cosh[x]^2]]*Sqrt[Cosh[x]^2] + Sinh[x])

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11

method	result	size
default	\(-i \ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {2}{\tanh \left (\frac {x}{2}\right )+i}\)	\(21\)
risch	\(-\frac {2 i}{{\mathrm e}^{x}+i}+i \ln \left ({\mathrm e}^{x}+1\right )-i \ln \left ({\mathrm e}^{x}-1\right )\)	\(28\)
parallelrisch	\(\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )-i \ln \left (\tanh \left (\frac {x}{2}\right )\right ) \tanh \left (\frac {x}{2}\right )+2}{\tanh \left (\frac {x}{2}\right )+i}\)	\(30\)

[In]

int(csch(x)/(I+sinh(x)),x,method=_RETURNVERBOSE)

[Out]

-I*ln(tanh(1/2*x))+2/(tanh(1/2*x)+I)

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. 33 vs. \(2 (15) = 30\).

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx=\frac {{\left (i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) + {\left (-i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 2 i}{e^{x} + i} \]

[In]

integrate(csch(x)/(I+sinh(x)),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(csch(x)/(I+sinh(x)),x)

[Out]

Integral(csch(x)/(sinh(x) + I), x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53 \[ \int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx=-\frac {2 i}{e^{\left (-x\right )} - i} + i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

integrate(csch(x)/(I+sinh(x)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx=-\frac {2 i}{e^{x} + i} + i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

[In]

integrate(csch(x)/(I+sinh(x)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx=-\ln \left ({\mathrm {e}}^x\,2{}\mathrm {i}-2{}\mathrm {i}\right )\,1{}\mathrm {i}+\ln \left ({\mathrm {e}}^x\,2{}\mathrm {i}+2{}\mathrm {i}\right )\,1{}\mathrm {i}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]

[In]

int(1/(sinh(x)*(sinh(x) + 1i)),x)

[Out]

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