3.1.72 \(\int e^x \cos (x) \, dx\) [72]

Optimal. Leaf size=19 \[ \frac {1}{2} e^x \cos (x)+\frac {1}{2} e^x \sin (x) \]

[Out]

1/2*exp(x)*cos(x)+1/2*exp(x)*sin(x)

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4518} \begin {gather*} \frac {1}{2} e^x \sin (x)+\frac {1}{2} e^x \cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*Cos[x],x]

[Out]

(E^x*Cos[x])/2 + (E^x*Sin[x])/2

Rule 4518

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(C
os[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] + Simp[e*F^(c*(a + b*x))*(Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 12, normalized size = 0.63 \begin {gather*} \frac {1}{2} e^x (\cos (x)+\sin (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*Cos[x],x]

[Out]

(E^x*(Cos[x] + Sin[x]))/2

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Maple [A]
time = 0.02, size = 14, normalized size = 0.74

method	result	size
default	\(\frac {{\mathrm e}^{x} \cos \left (x \right )}{2}+\frac {{\mathrm e}^{x} \sin \left (x \right )}{2}\)	\(14\)
norman	\(\frac {{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )-\frac {{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}+\frac {{\mathrm e}^{x}}{2}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(34\)
risch	\(\frac {{\mathrm e}^{\left (1+i\right ) x}}{4}-\frac {i {\mathrm e}^{\left (1+i\right ) x}}{4}+\frac {{\mathrm e}^{\left (1-i\right ) x}}{4}+\frac {i {\mathrm e}^{\left (1-i\right ) x}}{4}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*cos(x),x,method=_RETURNVERBOSE)

[Out]

1/2*exp(x)*cos(x)+1/2*exp(x)*sin(x)

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Maxima [A]
time = 4.39, size = 9, normalized size = 0.47 \begin {gather*} \frac {1}{2} \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(x),x, algorithm="maxima")

[Out]

1/2*(cos(x) + sin(x))*e^x

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Fricas [A]
time = 1.06, size = 13, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, \cos \left (x\right ) e^{x} + \frac {1}{2} \, e^{x} \sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(x),x, algorithm="fricas")

[Out]

1/2*cos(x)*e^x + 1/2*e^x*sin(x)

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Sympy [A]
time = 0.08, size = 15, normalized size = 0.79 \begin {gather*} \frac {e^{x} \sin {\left (x \right )}}{2} + \frac {e^{x} \cos {\left (x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(x),x)

[Out]

exp(x)*sin(x)/2 + exp(x)*cos(x)/2

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Giac [A]
time = 0.50, size = 9, normalized size = 0.47 \begin {gather*} \frac {1}{2} \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(x),x, algorithm="giac")

[Out]

1/2*(cos(x) + sin(x))*e^x

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Mupad [B]
time = 0.02, size = 9, normalized size = 0.47 \begin {gather*} \frac {{\mathrm {e}}^x\,\left (\cos \left (x\right )+\sin \left (x\right )\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*cos(x),x)

[Out]

(exp(x)*(cos(x) + sin(x)))/2

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