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

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

[Out]

Log[I + Sinh[x]]

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Rubi [A]  time = 0.0184888, antiderivative size = 7, normalized size of antiderivative = 1., 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 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])

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{\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*}

Mathematica [A]  time = 0.0053324, size = 7, normalized size = 1. \[ \log (\sinh (x)+i) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/(I + Sinh[x]),x]

[Out]

Log[I + Sinh[x]]

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Maple [A]  time = 0.008, size = 7, normalized size = 1. \begin{align*} \ln \left ( i+\sinh \left ( x \right ) \right ) \end{align*}

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 = 1.18454, size = 7, normalized size = 1. \begin{align*} \log \left (\sinh \left (x\right ) + i\right ) \end{align*}

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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Fricas [B]  time = 1.82319, size = 28, normalized size = 4. \begin{align*} -x + 2 \, \log \left (e^{x} + i\right ) \end{align*}

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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Sympy [A]  time = 0.147357, size = 8, normalized size = 1.14 \begin{align*} - x + 2 \log{\left (e^{x} + i \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 2*log(exp(x) + I)

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Giac [B]  time = 1.3252, size = 15, normalized size = 2.14 \begin{align*} -x + 2 \, \log \left (e^{x} + i\right ) \end{align*}

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)