3.1.65 \(\int e^{2 x} \cos (e^{2 x}) \, dx\) [65]

Optimal. Leaf size=10 \[ \frac {1}{2} \sin \left (e^{2 x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2320, 2717} \begin {gather*} \frac {1}{2} \sin \left (e^{2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*x)*Cos[E^(2*x)],x]

[Out]

Sin[E^(2*x)]/2

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sin \left (e^{2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)*Cos[E^(2*x)],x]

[Out]

Sin[E^(2*x)]/2

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Maple [A]
time = 0.06, size = 8, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {\sin \left ({\mathrm e}^{2 x}\right )}{2}\)	\(8\)
default	\(\frac {\sin \left ({\mathrm e}^{2 x}\right )}{2}\)	\(8\)
risch	\(\frac {\sin \left ({\mathrm e}^{2 x}\right )}{2}\)	\(8\)
norman	\(\frac {\tan \left (\frac {{\mathrm e}^{2 x}}{2}\right )}{1+\tan ^{2}\left (\frac {{\mathrm e}^{2 x}}{2}\right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)*cos(exp(2*x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 7, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, \sin \left (e^{\left (2 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*cos(exp(2*x)),x, algorithm="maxima")

[Out]

1/2*sin(e^(2*x))

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Fricas [A]
time = 1.37, size = 7, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, \sin \left (e^{\left (2 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*cos(exp(2*x)),x, algorithm="fricas")

[Out]

1/2*sin(e^(2*x))

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Sympy [A]
time = 0.10, size = 7, normalized size = 0.70 \begin {gather*} \frac {\sin {\left (e^{2 x} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(exp(2*x))/2

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Giac [A]
time = 0.40, size = 7, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, \sin \left (e^{\left (2 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*cos(exp(2*x)),x, algorithm="giac")

[Out]

1/2*sin(e^(2*x))

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Mupad [B]
time = 0.05, size = 7, normalized size = 0.70 \begin {gather*} \frac {\sin \left ({\mathrm {e}}^{2\,x}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)*cos(exp(2*x)),x)

[Out]

sin(exp(2*x))/2

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