3.46 \(\int (2+3 x) \sin (5 x) \, dx\)

Optimal. Leaf size=22 \[ \frac{3}{25} \sin (5 x)-\frac{1}{5} (3 x+2) \cos (5 x) \]

[Out]

-((2 + 3*x)*Cos[5*x])/5 + (3*Sin[5*x])/25

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Rubi [A]  time = 0.0112121, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3296, 2637} \[ \frac{3}{25} \sin (5 x)-\frac{1}{5} (3 x+2) \cos (5 x) \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)*Sin[5*x],x]

[Out]

-((2 + 3*x)*Cos[5*x])/5 + (3*Sin[5*x])/25

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (2+3 x) \sin (5 x) \, dx &=-\frac{1}{5} (2+3 x) \cos (5 x)+\frac{3}{5} \int \cos (5 x) \, dx\\ &=-\frac{1}{5} (2+3 x) \cos (5 x)+\frac{3}{25} \sin (5 x)\\ \end{align*}

Mathematica [A]  time = 0.0105456, size = 26, normalized size = 1.18 \[ \frac{3}{25} \sin (5 x)-\frac{3}{5} x \cos (5 x)-\frac{2}{5} \cos (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)*Sin[5*x],x]

[Out]

(-2*Cos[5*x])/5 - (3*x*Cos[5*x])/5 + (3*Sin[5*x])/25

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Maple [A]  time = 0.006, size = 21, normalized size = 1. \begin{align*} -{\frac{2\,\cos \left ( 5\,x \right ) }{5}}+{\frac{3\,\sin \left ( 5\,x \right ) }{25}}-{\frac{3\,\cos \left ( 5\,x \right ) x}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)*sin(5*x),x)

[Out]

-2/5*cos(5*x)+3/25*sin(5*x)-3/5*cos(5*x)*x

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Maxima [A]  time = 0.946682, size = 27, normalized size = 1.23 \begin{align*} -\frac{3}{5} \, x \cos \left (5 \, x\right ) - \frac{2}{5} \, \cos \left (5 \, x\right ) + \frac{3}{25} \, \sin \left (5 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)*sin(5*x),x, algorithm="maxima")

[Out]

-3/5*x*cos(5*x) - 2/5*cos(5*x) + 3/25*sin(5*x)

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Fricas [A]  time = 0.447508, size = 55, normalized size = 2.5 \begin{align*} -\frac{1}{5} \,{\left (3 \, x + 2\right )} \cos \left (5 \, x\right ) + \frac{3}{25} \, \sin \left (5 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)*sin(5*x),x, algorithm="fricas")

[Out]

-1/5*(3*x + 2)*cos(5*x) + 3/25*sin(5*x)

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Sympy [A]  time = 0.171743, size = 26, normalized size = 1.18 \begin{align*} - \frac{3 x \cos{\left (5 x \right )}}{5} + \frac{3 \sin{\left (5 x \right )}}{25} - \frac{2 \cos{\left (5 x \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)*sin(5*x),x)

[Out]

-3*x*cos(5*x)/5 + 3*sin(5*x)/25 - 2*cos(5*x)/5

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Giac [A]  time = 1.08561, size = 24, normalized size = 1.09 \begin{align*} -\frac{1}{5} \,{\left (3 \, x + 2\right )} \cos \left (5 \, x\right ) + \frac{3}{25} \, \sin \left (5 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)*sin(5*x),x, algorithm="giac")

[Out]

-1/5*(3*x + 2)*cos(5*x) + 3/25*sin(5*x)