3.732 \(\int \frac {\cosh (x)-i \sinh (x)}{\cosh (x)+i \sinh (x)} \, dx\)

Optimal. Leaf size=14 \[ -i \log (\cosh (x)+i \sinh (x)) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3133} \[ -i \log (\cosh (x)+i \sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[x] - I*Sinh[x])/(Cosh[x] + I*Sinh[x]),x]

[Out]

(-I)*Log[Cosh[x] + I*Sinh[x]]

Rule 3133

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[((b*B + c*C)*x)/(b^2 + c^2), x] + Simp[((c*B - b*C)*L
og[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2
+ c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]

Rubi steps

\begin {align*} \int \frac {\cosh (x)-i \sinh (x)}{\cosh (x)+i \sinh (x)} \, dx &=-i \log (\cosh (x)+i \sinh (x))\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 15, normalized size = 1.07 \[ \tan ^{-1}(\tanh (x))-\frac {1}{2} i \log (\cosh (2 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[x] - I*Sinh[x])/(Cosh[x] + I*Sinh[x]),x]

[Out]

ArcTan[Tanh[x]] - (I/2)*Log[Cosh[2*x]]

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fricas [A]  time = 0.49, size = 13, normalized size = 0.93 \[ i \, x - i \, \log \left (e^{\left (2 \, x\right )} - i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)-I*sinh(x))/(cosh(x)+I*sinh(x)),x, algorithm="fricas")

[Out]

I*x - I*log(e^(2*x) - I)

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giac [A]  time = 0.12, size = 13, normalized size = 0.93 \[ i \, x - i \, \log \left (e^{\left (2 \, x\right )} - i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)-I*sinh(x))/(cosh(x)+I*sinh(x)),x, algorithm="giac")

[Out]

I*x - I*log(e^(2*x) - I)

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maple [A]  time = 0.17, size = 13, normalized size = 0.93 \[ -i \ln \left (\cosh \relax (x )+i \sinh \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x)-I*sinh(x))/(cosh(x)+I*sinh(x)),x)

[Out]

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

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maxima [A]  time = 0.56, size = 10, normalized size = 0.71 \[ -i \, \log \left (\cosh \relax (x) + i \, \sinh \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)-I*sinh(x))/(cosh(x)+I*sinh(x)),x, algorithm="maxima")

[Out]

-I*log(cosh(x) + I*sinh(x))

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mupad [B]  time = 0.08, size = 16, normalized size = 1.14 \[ x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^{2\,x}-\mathrm {i}\right )\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x) - sinh(x)*1i)/(cosh(x) + sinh(x)*1i),x)

[Out]

x*1i - log(exp(2*x) - 1i)*1i

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sympy [A]  time = 0.12, size = 14, normalized size = 1.00 \[ x \left (-2 - i\right ) + \log {\left (e^{2 x} - i \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)-I*sinh(x))/(cosh(x)+I*sinh(x)),x)

[Out]

x*(-2 - I) + log(exp(2*x) - I)

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