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

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

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \log (-\sinh (x)+i)-i \sinh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/(I + Csch[x]),x]

[Out]

Log[I - Sinh[x]] - I*Sinh[x]

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fricas [B]  time = 0.60, size = 28, normalized size = 1.75 \[ -\frac {1}{2} \, {\left (2 \, x e^{x} - 4 \, e^{x} \log \left (e^{x} - i\right ) + i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\right )} \]

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)

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giac [B]  time = 0.12, size = 25, normalized size = 1.56 \[ \frac {1}{2} i \, e^{\left (-x\right )} - \frac {1}{2} i \, e^{x} - \log \left (-i \, e^{x}\right ) + 2 \, \log \left (e^{x} - i\right ) \]

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)

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maple [A]  time = 0.12, size = 20, normalized size = 1.25 \[ \ln \left (i+\mathrm {csch}\relax (x )\right )-\ln \left (\mathrm {csch}\relax (x )\right )-\frac {i}{\mathrm {csch}\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.30, size = 21, normalized size = 1.31 \[ x + \frac {1}{2} i \, e^{\left (-x\right )} - \frac {1}{2} i \, e^{x} + 2 \, \log \left (e^{\left (-x\right )} + i\right ) \]

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)

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mupad [B]  time = 0.09, size = 12, normalized size = 0.75 \[ \ln \left (\mathrm {sinh}\relax (x)-\mathrm {i}\right )-\mathrm {sinh}\relax (x)\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/(1/sinh(x) + 1i),x)

[Out]

log(sinh(x) - 1i) - sinh(x)*1i

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]

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)

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