3.751 \(\int e^{\cos ^2(x)+\sin ^2(x)} \, dx\)

Optimal. Leaf size=3 \[ e x \]

[Out]

E*x

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Rubi [A]  time = 0.0087911, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {12, 203} \[ e x \]

Antiderivative was successfully verified.

[In]

Int[E^(Cos[x]^2 + Sin[x]^2),x]

[Out]

E*x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int e^{\cos ^2(x)+\sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{e}{1+x^2} \, dx,x,\tan (x)\right )\\ &=e \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (x)\right )\\ &=e x\\ \end{align*}

Mathematica [A]  time = 0.0004321, size = 3, normalized size = 1. \[ e x \]

Antiderivative was successfully verified.

[In]

Integrate[E^(Cos[x]^2 + Sin[x]^2),x]

[Out]

E*x

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Maple [C]  time = 0.012, size = 5, normalized size = 1.7 \begin{align*}{\rm e}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(cos(x)^2+sin(x)^2),x)

[Out]

exp(1)*x

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Maxima [C]  time = 1.45693, size = 5, normalized size = 1.67 \begin{align*} x e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(x)^2+sin(x)^2),x, algorithm="maxima")

[Out]

x*e

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Fricas [C]  time = 1.9375, size = 7, normalized size = 2.33 \begin{align*} x e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(x)^2+sin(x)^2),x, algorithm="fricas")

[Out]

x*e

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Sympy [B]  time = 0.14168, size = 14, normalized size = 4.67 \begin{align*} x e^{\sin ^{2}{\left (x \right )}} e^{\cos ^{2}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(x)**2+sin(x)**2),x)

[Out]

x*exp(sin(x)**2)*exp(cos(x)**2)

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Giac [C]  time = 1.09118, size = 5, normalized size = 1.67 \begin{align*} x e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(x)^2+sin(x)^2),x, algorithm="giac")

[Out]

x*e