3.3.12 \(\int \frac {\coth (x)}{i+\sinh (x)} \, dx\) [212]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {2786, 36, 29, 31} \begin {gather*} i \log (\sinh (x)+i)-i \log (\sinh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(I + Sinh[x]),x]

[Out]

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

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]

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 2786

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} -i \log (\sinh (x))+i \log (i+\sinh (x)) \end {gather*}

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 = 0.54, size = 17, normalized size = 0.89

method	result	size
derivativedivides	\(-i \ln \left (\sinh \left (x \right )\right )+i \ln \left (i+\sinh \left (x \right )\right )\)	\(17\)
default	\(-i \ln \left (\sinh \left (x \right )\right )+i \ln \left (i+\sinh \left (x \right )\right )\)	\(17\)
risch	\(2 i \ln \left ({\mathrm e}^{x}+i\right )-i \ln \left ({\mathrm e}^{2 x}-1\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(I+sinh(x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (13) = 26\).
time = 0.27, size = 28, normalized size = 1.47 \begin {gather*} -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 {gather*}

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 = 0.37, size = 17, normalized size = 0.89 \begin {gather*} -i \, \log \left (e^{\left (2 \, x\right )} - 1\right ) + 2 i \, \log \left (e^{x} + i\right ) \end {gather*}

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 [A]
time = 0.09, size = 19, normalized size = 1.00 \begin {gather*} 2 i \log {\left (e^{x} + i \right )} - i \log {\left (e^{2 x} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I*log(exp(x) + I) - I*log(exp(2*x) - 1)

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Giac [A]
time = 0.41, size = 23, normalized size = 1.21 \begin {gather*} -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 {gather*}

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))

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Mupad [B]
time = 0.15, size = 24, normalized size = 1.26 \begin {gather*} \ln \left (-36\,{\mathrm {e}}^x-36{}\mathrm {i}\right )\,2{}\mathrm {i}-\ln \left (3-3\,{\mathrm {e}}^{2\,x}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(sinh(x) + 1i),x)

[Out]

log(- 36*exp(x) - 36i)*2i - log(3 - 3*exp(2*x))*1i

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