3.1.10 \(\int \sin (e^x) \, dx\) [10]

Optimal. Leaf size=4 \[ \text {Si}\left (e^x\right ) \]

[Out]

Si(exp(x))

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2320, 3380} \begin {gather*} \text {Si}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[E^x],x]

[Out]

SinIntegral[E^x]

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 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 4, normalized size = 1.00 \begin {gather*} \text {Si}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[E^x],x]

[Out]

SinIntegral[E^x]

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 4, normalized size = 1.00

method	result	size
derivativedivides	\(\sinIntegral \left ({\mathrm e}^{x}\right )\)	\(4\)
default	\(\sinIntegral \left ({\mathrm e}^{x}\right )\)	\(4\)
risch	\(-\frac {\pi \,\mathrm {csgn}\left ({\mathrm e}^{x}\right )}{2}+\sinIntegral \left ({\mathrm e}^{x}\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(exp(x)),x,method=_RETURNVERBOSE)

[Out]

Si(exp(x))

________________________________________________________________________________________

Maxima [C] Result contains complex when optimal does not.
time = 1.23, size = 15, normalized size = 3.75 \begin {gather*} -\frac {1}{2} i \, {\rm Ei}\left (i \, e^{x}\right ) + \frac {1}{2} i \, {\rm Ei}\left (-i \, e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(exp(x)),x, algorithm="maxima")

[Out]

-1/2*I*Ei(I*e^x) + 1/2*I*Ei(-I*e^x)

________________________________________________________________________________________

Fricas [A]
time = 0.65, size = 3, normalized size = 0.75 \begin {gather*} \operatorname {Si}\left (e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(exp(x)),x, algorithm="fricas")

[Out]

sin_integral(e^x)

________________________________________________________________________________________

Sympy [A]
time = 0.32, size = 3, normalized size = 0.75 \begin {gather*} \operatorname {Si}{\left (e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(exp(x)),x)

[Out]

Si(exp(x))

________________________________________________________________________________________

Giac [A]
time = 0.48, size = 3, normalized size = 0.75 \begin {gather*} \operatorname {Si}\left (e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(exp(x)),x, algorithm="giac")

[Out]

sin_integral(e^x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.25 \begin {gather*} \mathrm {sinint}\left ({\mathrm {e}}^x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(exp(x)),x)

[Out]

sinint(exp(x))

________________________________________________________________________________________