3.4.68 \(\int e^{\sin (x)} \sin (2 x) \, dx\) [368]

Optimal. Leaf size=15 \[ -2 e^{\sin (x)}+2 e^{\sin (x)} \sin (x) \]

[Out]

-2*exp(sin(x))+2*exp(sin(x))*sin(x)

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {12, 2207, 2225} \begin {gather*} 2 e^{\sin (x)} \sin (x)-2 e^{\sin (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^Sin[x]*Sin[2*x],x]

[Out]

-2*E^Sin[x] + 2*E^Sin[x]*Sin[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 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 11, normalized size = 0.73 \begin {gather*} e^{\sin (x)} (-2+2 \sin (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^Sin[x]*Sin[2*x],x]

[Out]

E^Sin[x]*(-2 + 2*Sin[x])

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Mathics [A]
time = 2.73, size = 10, normalized size = 0.67 \begin {gather*} 2 \left (-1+\text {Sin}\left [x\right ]\right ) E^{\text {Sin}\left [x\right ]} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[E^Sin[x]*Sin[2*x],x]')

[Out]

2 (-1 + Sin[x]) E ^ Sin[x]

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Maple [A]
time = 0.05, size = 14, normalized size = 0.93

method	result	size
derivativedivides	\(-2 \,{\mathrm e}^{\sin \left (x \right )}+2 \,{\mathrm e}^{\sin \left (x \right )} \sin \left (x \right )\)	\(14\)
default	\(-2 \,{\mathrm e}^{\sin \left (x \right )}+2 \,{\mathrm e}^{\sin \left (x \right )} \sin \left (x \right )\)	\(14\)
risch	\(-2 \,{\mathrm e}^{\sin \left (x \right )}+2 \,{\mathrm e}^{\sin \left (x \right )} \sin \left (x \right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*exp(sin(x))+2*exp(sin(x))*sin(x)

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Maxima [A]
time = 0.28, size = 9, normalized size = 0.60 \begin {gather*} 2 \, {\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sin(x) - 1)*e^sin(x)

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Fricas [A]
time = 0.34, size = 9, normalized size = 0.60 \begin {gather*} 2 \, {\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sin(x) - 1)*e^sin(x)

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Sympy [A]
time = 0.96, size = 15, normalized size = 1.00 \begin {gather*} 2 e^{\sin {\left (x \right )}} \sin {\left (x \right )} - 2 e^{\sin {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*exp(sin(x))*sin(x) - 2*exp(sin(x))

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Giac [A]
time = 0.00, size = 9, normalized size = 0.60 \begin {gather*} 2 \left (\sin x-1\right ) \mathrm {e}^{\sin x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^sin(x)*(sin(x) - 1)

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Mupad [B]
time = 0.27, size = 9, normalized size = 0.60 \begin {gather*} 2\,{\mathrm {e}}^{\sin \left (x\right )}\,\left (\sin \left (x\right )-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*exp(sin(x))*(sin(x) - 1)

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