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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]