3.212 \(\int \frac{\coth (x)}{i+\sinh (x)} \, dx\)

Optimal. Leaf size=19 \[ i \log (\sinh (x)+i)-i \log (\sinh (x)) \]

[Out]

(-I)*Log[Sinh[x]] + I*Log[I + Sinh[x]]

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Rubi [A]  time = 0.0324123, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2707, 36, 29, 31} \[ i \log (\sinh (x)+i)-i \log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(I + Sinh[x]),x]

[Out]

(-I)*Log[Sinh[x]] + I*Log[I + Sinh[x]]

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0082583, size = 19, normalized size = 1. \[ i \log (\sinh (x)+i)-i \log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(I + Sinh[x]),x]

[Out]

(-I)*Log[Sinh[x]] + I*Log[I + Sinh[x]]

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Maple [A]  time = 0.022, size = 17, normalized size = 0.9 \begin{align*} -i\ln \left ( \sinh \left ( x \right ) \right ) +i\ln \left ( i+\sinh \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.03739, size = 38, normalized size = 2. \begin{align*} -i \, \log \left (e^{\left (-x\right )} + 1\right ) + 2 i \, \log \left (e^{\left (-x\right )} - i\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.04954, size = 54, normalized size = 2.84 \begin{align*} -i \, \log \left (e^{\left (2 \, x\right )} - 1\right ) + 2 i \, \log \left (e^{x} + i\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: PolynomialError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: PolynomialError

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Giac [A]  time = 1.11244, size = 31, normalized size = 1.63 \begin{align*} -i \, \log \left (e^{x} + 1\right ) + 2 i \, \log \left (e^{x} + i\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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