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 = 0.0388776, antiderivative size = 19, normalized size of antiderivative = 1., 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 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 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 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*}

Mathematica [A]  time = 0.0193429, 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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Maple [A]  time = 0.028, size = 21, normalized size = 1.1 \begin{align*} 2\, \left ( \tanh \left ( x/2 \right ) +i \right ) ^{-1}-i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.18967, size = 39, normalized size = 2.05 \begin{align*} -\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 ) \end{align*}

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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Fricas [B]  time = 2.11965, size = 97, normalized size = 5.11 \begin{align*} \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} \end{align*}

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

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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Giac [A]  time = 1.41403, size = 32, normalized size = 1.68 \begin{align*} -\frac{2 i}{e^{x} + i} + i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

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))