3.81 \(\int e^{a x} \sin (b x) \, dx\)

Optimal. Leaf size=42 \[ \frac{a e^{a x} \sin (b x)}{a^2+b^2}-\frac{b e^{a x} \cos (b x)}{a^2+b^2} \]

[Out]

-((b*E^(a*x)*Cos[b*x])/(a^2 + b^2)) + (a*E^(a*x)*Sin[b*x])/(a^2 + b^2)

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Rubi [A]  time = 0.0259106, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a e^{a x} \sin (b x)}{a^2+b^2}-\frac{b e^{a x} \cos (b x)}{a^2+b^2} \]

Antiderivative was successfully verified.

[In]  Int[E^(a*x)*Sin[b*x],x]

[Out]

-((b*E^(a*x)*Cos[b*x])/(a^2 + b^2)) + (a*E^(a*x)*Sin[b*x])/(a^2 + b^2)

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Rubi in Sympy [A]  time = 1.8836, size = 36, normalized size = 0.86 \[ \frac{a e^{a x} \sin{\left (b x \right )}}{a^{2} + b^{2}} - \frac{b e^{a x} \cos{\left (b x \right )}}{a^{2} + b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(a*x)*sin(b*x),x)

[Out]

a*exp(a*x)*sin(b*x)/(a**2 + b**2) - b*exp(a*x)*cos(b*x)/(a**2 + b**2)

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Mathematica [A]  time = 0.0292896, size = 29, normalized size = 0.69 \[ \frac{e^{a x} (a \sin (b x)-b \cos (b x))}{a^2+b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(a*x)*Sin[b*x],x]

[Out]

(E^(a*x)*(-(b*Cos[b*x]) + a*Sin[b*x]))/(a^2 + b^2)

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Maple [A]  time = 0.006, size = 41, normalized size = 1. \[ -{\frac{{{\rm e}^{ax}}b\cos \left ( bx \right ) }{{a}^{2}+{b}^{2}}}+{\frac{a{{\rm e}^{ax}}\sin \left ( bx \right ) }{{a}^{2}+{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(a*x)*sin(b*x),x)

[Out]

-b*exp(a*x)*cos(b*x)/(a^2+b^2)+a*exp(a*x)*sin(b*x)/(a^2+b^2)

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Maxima [A]  time = 1.38498, size = 39, normalized size = 0.93 \[ -\frac{{\left (b \cos \left (b x\right ) - a \sin \left (b x\right )\right )} e^{\left (a x\right )}}{a^{2} + b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(a*x)*sin(b*x),x, algorithm="maxima")

[Out]

-(b*cos(b*x) - a*sin(b*x))*e^(a*x)/(a^2 + b^2)

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Fricas [A]  time = 0.223701, size = 45, normalized size = 1.07 \[ -\frac{b \cos \left (b x\right ) e^{\left (a x\right )} - a e^{\left (a x\right )} \sin \left (b x\right )}{a^{2} + b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(a*x)*sin(b*x),x, algorithm="fricas")

[Out]

-(b*cos(b*x)*e^(a*x) - a*e^(a*x)*sin(b*x))/(a^2 + b^2)

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Sympy [A]  time = 2.13496, size = 136, normalized size = 3.24 \[ \begin{cases} 0 & \text{for}\: a = 0 \wedge b = 0 \\\frac{x e^{- i b x} \sin{\left (b x \right )}}{2} - \frac{i x e^{- i b x} \cos{\left (b x \right )}}{2} - \frac{e^{- i b x} \cos{\left (b x \right )}}{2 b} & \text{for}\: a = - i b \\\frac{x e^{i b x} \sin{\left (b x \right )}}{2} + \frac{i x e^{i b x} \cos{\left (b x \right )}}{2} - \frac{e^{i b x} \cos{\left (b x \right )}}{2 b} & \text{for}\: a = i b \\\frac{a e^{a x} \sin{\left (b x \right )}}{a^{2} + b^{2}} - \frac{b e^{a x} \cos{\left (b x \right )}}{a^{2} + b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(a*x)*sin(b*x),x)

[Out]

Piecewise((0, Eq(a, 0) & Eq(b, 0)), (x*exp(-I*b*x)*sin(b*x)/2 - I*x*exp(-I*b*x)*
cos(b*x)/2 - exp(-I*b*x)*cos(b*x)/(2*b), Eq(a, -I*b)), (x*exp(I*b*x)*sin(b*x)/2
+ I*x*exp(I*b*x)*cos(b*x)/2 - exp(I*b*x)*cos(b*x)/(2*b), Eq(a, I*b)), (a*exp(a*x
)*sin(b*x)/(a**2 + b**2) - b*exp(a*x)*cos(b*x)/(a**2 + b**2), True))

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GIAC/XCAS [A]  time = 0.235554, size = 51, normalized size = 1.21 \[ -{\left (\frac{b \cos \left (b x\right )}{a^{2} + b^{2}} - \frac{a \sin \left (b x\right )}{a^{2} + b^{2}}\right )} e^{\left (a x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(a*x)*sin(b*x),x, algorithm="giac")

[Out]

-(b*cos(b*x)/(a^2 + b^2) - a*sin(b*x)/(a^2 + b^2))*e^(a*x)