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]

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2637

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

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.42, size = 18, normalized size = 0.82 \[ -\frac {1}{5} \, {\left (3 \, x + 2\right )} \cos \left (5 \, x\right ) + \frac {3}{25} \, \sin \left (5 \, x\right ) \]

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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giac [A]  time = 0.01, size = 18, normalized size = 0.82 \[ -\frac {1}{5} \, {\left (3 \, x + 2\right )} \cos \left (5 \, x\right ) + \frac {3}{25} \, \sin \left (5 \, x\right ) \]

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)

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maple [A]  time = 0.02, size = 21, normalized size = 0.95 \[ -\frac {3 x \cos \left (5 x \right )}{5}-\frac {2 \cos \left (5 x \right )}{5}+\frac {3 \sin \left (5 x \right )}{25} \]

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 = 0.54, size = 20, normalized size = 0.91 \[ -\frac {3}{5} \, x \cos \left (5 \, x\right ) - \frac {2}{5} \, \cos \left (5 \, x\right ) + \frac {3}{25} \, \sin \left (5 \, x\right ) \]

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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mupad [B]  time = 0.10, size = 20, normalized size = 0.91 \[ \frac {3\,\sin \left (5\,x\right )}{25}-\frac {2\,\cos \left (5\,x\right )}{5}-\frac {3\,x\,\cos \left (5\,x\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(5*x)*(3*x + 2),x)

[Out]

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

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sympy [A]  time = 0.20, size = 26, normalized size = 1.18 \[ - \frac {3 x \cos {\left (5 x \right )}}{5} + \frac {3 \sin {\left (5 x \right )}}{25} - \frac {2 \cos {\left (5 x \right )}}{5} \]

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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