3.107 \(\int \frac {\coth (x)}{i+\text {csch}(x)} \, dx\)

Optimal. Leaf size=13 \[ -i \log (-\sinh (x)+i) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3879, 31} \[ -i \log (-\sinh (x)+i) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(I + Csch[x]),x]

[Out]

(-I)*Log[I - Sinh[x]]

Rule 31

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

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \[ -i \log (-\sinh (x)+i) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(I + Csch[x]),x]

[Out]

(-I)*Log[I - Sinh[x]]

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fricas [A]  time = 1.31, size = 11, normalized size = 0.85 \[ i \, x - 2 i \, \log \left (e^{x} - i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*x - 2*I*log(e^x - I)

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giac [A]  time = 0.14, size = 13, normalized size = 1.00 \[ i \, x - 2 i \, \log \left (i \, e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*x - 2*I*log(I*e^x + 1)

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maple [A]  time = 0.12, size = 17, normalized size = 1.31 \[ -i \ln \left (i+\mathrm {csch}\relax (x )\right )+i \ln \left (\mathrm {csch}\relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.30, size = 15, normalized size = 1.15 \[ -i \, x - 2 i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*x - 2*I*log(I*e^(-x) - 1)

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mupad [B]  time = 1.46, size = 14, normalized size = 1.08 \[ x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,2{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(1/sinh(x) + 1i),x)

[Out]

x*1i - log(exp(x) - 1i)*2i

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(x)/(csch(x) + I), x)

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