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 = 0.01, antiderivative size = 3, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 3, normalized size = 1.00 \[ e x \]

Antiderivative was successfully verified.

[In]

Integrate[E^(Cos[x]^2 + Sin[x]^2),x]

[Out]

E*x

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fricas [C]  time = 0.55, size = 4, normalized size = 1.33 \[ x e \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [C]  time = 0.06, size = 5, normalized size = 1.67 \[ {\mathrm e} x \]

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 = 0.43, size = 4, normalized size = 1.33 \[ x e \]

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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mupad [B]  time = 0.03, size = 4, normalized size = 1.33 \[ x\,\mathrm {e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(cos(x)^2 + sin(x)^2),x)

[Out]

x*exp(1)

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sympy [B]  time = 0.14, size = 14, normalized size = 4.67 \[ x e^{\sin ^{2}{\relax (x )}} e^{\cos ^{2}{\relax (x )}} \]

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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