3.44 \(\int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx\)

Optimal. Leaf size=19 \[ \frac {\cosh (x)}{\sinh (x)+i}+i \tanh ^{-1}(\cosh (x)) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2747, 2648, 3770} \[ \frac {\cosh (x)}{\sinh (x)+i}+i \tanh ^{-1}(\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]/(I + Sinh[x]),x]

[Out]

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

Rule 2648

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 2747

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 3770

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 30, normalized size = 1.58 \[ \text {sech}(x) \left (\sinh (x)+i \sqrt {\cosh ^2(x)} \tanh ^{-1}\left (\sqrt {\cosh ^2(x)}\right )-i\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]/(I + Sinh[x]),x]

[Out]

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

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fricas [B]  time = 0.53, size = 33, normalized size = 1.74 \[ \frac {{\left (i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) + {\left (-i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 2 i}{e^{x} + i} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.21, size = 24, normalized size = 1.26 \[ -\frac {2 i}{e^{x} + i} + i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.04, size = 21, normalized size = 1.11 \[ -i \ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {2}{\tanh \left (\frac {x}{2}\right )+i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 29, normalized size = 1.53 \[ -\frac {2 i}{e^{\left (-x\right )} - i} + i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.49, size = 35, normalized size = 1.84 \[ -\ln \left ({\mathrm {e}}^x\,2{}\mathrm {i}-2{}\mathrm {i}\right )\,1{}\mathrm {i}+\ln \left ({\mathrm {e}}^x\,2{}\mathrm {i}+2{}\mathrm {i}\right )\,1{}\mathrm {i}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(x)/(sinh(x) + I), x)

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