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

Optimal. Leaf size=7 \[ \log (\sinh (x)+i) \]

[Out]

ln(I+sinh(x))

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Rubi [A]  time = 0.02, antiderivative size = 7, 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 = {2667, 31} \[ \log (\sinh (x)+i) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(I + Sinh[x]),x]

[Out]

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 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(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]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 7, normalized size = 1.00 \[ \log (\sinh (x)+i) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/(I + Sinh[x]),x]

[Out]

Log[I + Sinh[x]]

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fricas [B]  time = 0.50, size = 11, normalized size = 1.57 \[ -x + 2 \, \log \left (e^{x} + i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 2*log(e^x + I)

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giac [B]  time = 0.30, size = 11, normalized size = 1.57 \[ -x + 2 \, \log \left (e^{x} + i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 2*log(e^x + I)

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maple [A]  time = 0.02, size = 7, normalized size = 1.00 \[ \ln \left (i+\sinh \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(I+sinh(x))

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maxima [A]  time = 0.32, size = 5, normalized size = 0.71 \[ \log \left (\sinh \relax (x) + i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sinh(x) + I)

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mupad [B]  time = 0.47, size = 10, normalized size = 1.43 \[ \ln \left (\mathrm {cosh}\relax (x)\right )-\mathrm {atan}\left (\mathrm {sinh}\relax (x)\right )\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/(sinh(x) + 1i),x)

[Out]

log(cosh(x)) - atan(sinh(x))*1i

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sympy [B]  time = 0.13, size = 15, normalized size = 2.14 \[ x \left (1 + 2 i\right ) - 2 i \log {\left (e^{x} + i \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(1 + 2*I) - 2*I*log(exp(x) + I)

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