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

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

[Out]

x-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 = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2735, 2648} \[ x-\frac {\cosh (x)}{\sinh (x)+i} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(I + Sinh[x]),x]

[Out]

x - Cosh[x]/(I + Sinh[x])

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rubi steps

\begin {align*} \int \frac {\sinh (x)}{i+\sinh (x)} \, dx &=x-i \int \frac {1}{i+\sinh (x)} \, dx\\ &=x-\frac {\cosh (x)}{i+\sinh (x)}\\ \end {align*}

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Mathematica [B]  time = 0.06, size = 43, normalized size = 3.07 \[ i \text {sech}(x) \left (i \sinh (x)+2 \sqrt {\cosh ^2(x)} \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(I + Sinh[x]),x]

[Out]

I*Sech[x]*(1 + 2*ArcSin[Sqrt[1 - I*Sinh[x]]/Sqrt[2]]*Sqrt[Cosh[x]^2] + I*Sinh[x])

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fricas [A]  time = 1.02, size = 16, normalized size = 1.14 \[ \frac {x e^{x} + i \, x + 2 i}{e^{x} + i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e^x + I*x + 2*I)/(e^x + I)

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giac [A]  time = 0.19, size = 10, normalized size = 0.71 \[ x + \frac {2 i}{e^{x} + i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2*I/(e^x + I)

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maple [B]  time = 0.04, size = 29, normalized size = 2.07 \[ -\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {2}{\tanh \left (\frac {x}{2}\right )+i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(tanh(1/2*x)-1)+ln(tanh(1/2*x)+1)-2/(tanh(1/2*x)+I)

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maxima [A]  time = 0.59, size = 12, normalized size = 0.86 \[ x + \frac {2 i}{e^{\left (-x\right )} - i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2*I/(e^(-x) - I)

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mupad [B]  time = 0.41, size = 12, normalized size = 0.86 \[ x+\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(sinh(x) + 1i),x)

[Out]

x + 2i/(exp(x) + 1i)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.57 \[ x + \frac {2}{- i e^{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2/(-I*exp(x) + 1)

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