3.1.46 \(\int (2+3 x) \sin (5 x) \, dx\) [46]

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

[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 = {3377, 2717} \begin {gather*} \frac {3}{25} \sin (5 x)-\frac {1}{5} (3 x+2) \cos (5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)*Sin[5*x],x]

[Out]

-1/5*((2 + 3*x)*Cos[5*x]) + (3*Sin[5*x])/25

Rule 2717

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

Rule 3377

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 \begin {gather*} -\frac {2}{5} \cos (5 x)-\frac {3}{5} x \cos (5 x)+\frac {3}{25} \sin (5 x) \end {gather*}

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 = 0.02, size = 21, normalized size = 0.95

method	result	size
risch	\(\left (-\frac {2}{5}-\frac {3 x}{5}\right ) \cos \left (5 x \right )+\frac {3 \sin \left (5 x \right )}{25}\)	\(18\)
derivativedivides	\(-\frac {2 \cos \left (5 x \right )}{5}+\frac {3 \sin \left (5 x \right )}{25}-\frac {3 \cos \left (5 x \right ) x}{5}\)	\(21\)
default	\(-\frac {2 \cos \left (5 x \right )}{5}+\frac {3 \sin \left (5 x \right )}{25}-\frac {3 \cos \left (5 x \right ) x}{5}\)	\(21\)
norman	\(\frac {-\frac {3 x}{5}+\frac {3 x \left (\tan ^{2}\left (\frac {5 x}{2}\right )\right )}{5}+\frac {6 \tan \left (\frac {5 x}{2}\right )}{25}-\frac {4}{5}}{1+\tan ^{2}\left (\frac {5 x}{2}\right )}\)	\(32\)
meijerg	\(\frac {2 \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (5 x \right )}{\sqrt {\pi }}\right )}{5}+\frac {6 \sqrt {\pi }\, \left (-\frac {5 x \cos \left (5 x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (5 x \right )}{2 \sqrt {\pi }}\right )}{25}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)*sin(5*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.06, size = 20, normalized size = 0.91 \begin {gather*} -\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 {gather*}

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 = 2.31, size = 18, normalized size = 0.82 \begin {gather*} -\frac {1}{5} \, {\left (3 \, x + 2\right )} \cos \left (5 \, x\right ) + \frac {3}{25} \, \sin \left (5 \, x\right ) \end {gather*}

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 = 0.06, size = 26, normalized size = 1.18 \begin {gather*} - \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 {gather*}

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 = 0.51, size = 18, normalized size = 0.82 \begin {gather*} -\frac {1}{5} \, {\left (3 \, x + 2\right )} \cos \left (5 \, x\right ) + \frac {3}{25} \, \sin \left (5 \, x\right ) \end {gather*}

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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Mupad [B]
time = 0.10, size = 20, normalized size = 0.91 \begin {gather*} \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} \end {gather*}

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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