3.87 \(\int \frac{\cosh (x)}{i+\text{csch}(x)} \, dx\)

Optimal. Leaf size=16 \[ \log (-\sinh (x)+i)-i \sinh (x) \]

[Out]

Log[I - Sinh[x]] - I*Sinh[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0584481, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3872, 2833, 43} \[ \log (-\sinh (x)+i)-i \sinh (x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(I + Csch[x]),x]

[Out]

Log[I - Sinh[x]] - I*Sinh[x]

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{\cosh (x)}{i+\text{csch}(x)} \, dx &=i \int \frac{\cosh (x) \sinh (x)}{i-\sinh (x)} \, dx\\ &=i \operatorname{Subst}\left (\int \frac{x}{i+x} \, dx,x,-\sinh (x)\right )\\ &=i \operatorname{Subst}\left (\int \left (1-\frac{i}{i+x}\right ) \, dx,x,-\sinh (x)\right )\\ &=\log (i-\sinh (x))-i \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0093764, size = 16, normalized size = 1. \[ \log (-\sinh (x)+i)-i \sinh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/(I + Csch[x]),x]

[Out]

Log[I - Sinh[x]] - I*Sinh[x]

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 20, normalized size = 1.3 \begin{align*} -\ln \left ({\rm csch} \left (x\right ) \right ) -{\frac{i}{{\rm csch} \left (x\right )}}+\ln \left ( i+{\rm csch} \left (x\right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/(I+csch(x)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02466, size = 28, normalized size = 1.75 \begin{align*} x + \frac{1}{2} i \, e^{\left (-x\right )} - \frac{1}{2} i \, e^{x} + 2 \, \log \left (e^{\left (-x\right )} + i\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.74491, size = 81, normalized size = 5.06 \begin{align*} -\frac{1}{2} \,{\left (2 \, x e^{x} - 4 \, e^{x} \log \left (e^{x} - i\right ) + i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)/(csch(x) + I), x)

________________________________________________________________________________________

Giac [B]  time = 1.1312, size = 34, normalized size = 2.12 \begin{align*} \frac{1}{2} i \, e^{\left (-x\right )} - \frac{1}{2} i \, e^{x} - \log \left (-i \, e^{x}\right ) + 2 \, \log \left (e^{x} - i\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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