\(\int e^x \sin (x) \, dx\) [71]

Optimal result

Integrand size = 6, antiderivative size = 19 \[ \int e^x \sin (x) \, dx=-\frac {1}{2} e^x \cos (x)+\frac {1}{2} e^x \sin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4517} \[ \int e^x \sin (x) \, dx=\frac {1}{2} e^x \sin (x)-\frac {1}{2} e^x \cos (x) \]

[In]

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

[Out]

-1/2*(E^x*Cos[x]) + (E^x*Sin[x])/2

Rule 4517

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(S
in[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] - Simp[e*F^(c*(a + b*x))*(Cos[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*} \text {integral}& = -\frac {1}{2} e^x \cos (x)+\frac {1}{2} e^x \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int e^x \sin (x) \, dx=\frac {1}{2} e^x (-\cos (x)+\sin (x)) \]

[In]

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

[Out]

(E^x*(-Cos[x] + Sin[x]))/2

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{x} \left (-\cos \left (x \right )+\sin \left (x \right )\right )}{2}\)	\(12\)
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	\(-\frac {{\mathrm e}^{\left (1+i\right ) x}}{4}-\frac {i {\mathrm e}^{\left (1+i\right ) x}}{4}-\frac {{\mathrm e}^{\left (1-i\right ) x}}{4}+\frac {i {\mathrm e}^{\left (1-i\right ) x}}{4}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sin (x) \, dx=-\frac {1}{2} \, \cos \left (x\right ) e^{x} + \frac {1}{2} \, e^{x} \sin \left (x\right ) \]

[In]

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

[Out]

-1/2*cos(x)*e^x + 1/2*e^x*sin(x)

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int e^x \sin (x) \, dx=\frac {e^{x} \sin {\left (x \right )}}{2} - \frac {e^{x} \cos {\left (x \right )}}{2} \]

[In]

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

[Out]

exp(x)*sin(x)/2 - exp(x)*cos(x)/2

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int e^x \sin (x) \, dx=-\frac {1}{2} \, {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} e^{x} \]

[In]

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

[Out]

-1/2*(cos(x) - sin(x))*e^x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int e^x \sin (x) \, dx=-\frac {1}{2} \, {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} e^{x} \]

[In]

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

[Out]

-1/2*(cos(x) - sin(x))*e^x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int e^x \sin (x) \, dx=-\frac {{\mathrm {e}}^x\,\left (\cos \left (x\right )-\sin \left (x\right )\right )}{2} \]

[In]

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

[Out]

-(exp(x)*(cos(x) - sin(x)))/2