3.3.17 \(\int x \sin (a+x) \, dx\) [217]

Optimal. Leaf size=12 \[ -x \cos (a+x)+\sin (a+x) \]

[Out]

-x*cos(a+x)+sin(a+x)

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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3377, 2717} \begin {gather*} \sin (a+x)-x \cos (a+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sin[a + x],x]

[Out]

-(x*Cos[a + x]) + Sin[a + x]

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 x \sin (a+x) \, dx &=-x \cos (a+x)+\int \cos (a+x) \, dx\\ &=-x \cos (a+x)+\sin (a+x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 12, normalized size = 1.00 \begin {gather*} -x \cos (a+x)+\sin (a+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sin[a + x],x]

[Out]

-(x*Cos[a + x]) + Sin[a + x]

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Maple [A]
time = 0.02, size = 21, normalized size = 1.75

method	result	size
risch	\(-x \cos \left (a +x \right )+\sin \left (a +x \right )\)	\(13\)
derivativedivides	\(a \cos \left (a +x \right )+\sin \left (a +x \right )-\left (a +x \right ) \cos \left (a +x \right )\)	\(21\)
default	\(a \cos \left (a +x \right )+\sin \left (a +x \right )-\left (a +x \right ) \cos \left (a +x \right )\)	\(21\)
norman	\(\frac {x \left (\tan ^{2}\left (\frac {a}{2}+\frac {x}{2}\right )\right )-x +2 \tan \left (\frac {a}{2}+\frac {x}{2}\right )}{1+\tan ^{2}\left (\frac {a}{2}+\frac {x}{2}\right )}\)	\(42\)
meijerg	\(2 \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (x \right )}{2 \sqrt {\pi }}+\frac {x \sin \left (x \right )}{2 \sqrt {\pi }}\right )+2 \cos \left (a \right ) \sqrt {\pi }\, \left (-\frac {x \cos \left (x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (x \right )}{2 \sqrt {\pi }}\right )\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sin(a+x),x,method=_RETURNVERBOSE)

[Out]

a*cos(a+x)+sin(a+x)-(a+x)*cos(a+x)

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Maxima [A]
time = 2.20, size = 20, normalized size = 1.67 \begin {gather*} -{\left (a + x\right )} \cos \left (a + x\right ) + a \cos \left (a + x\right ) + \sin \left (a + x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(a+x),x, algorithm="maxima")

[Out]

-(a + x)*cos(a + x) + a*cos(a + x) + sin(a + x)

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Fricas [A]
time = 0.89, size = 12, normalized size = 1.00 \begin {gather*} -x \cos \left (a + x\right ) + \sin \left (a + x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(a+x),x, algorithm="fricas")

[Out]

-x*cos(a + x) + sin(a + x)

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.83 \begin {gather*} - x \cos {\left (a + x \right )} + \sin {\left (a + x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(a+x),x)

[Out]

-x*cos(a + x) + sin(a + x)

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Giac [A]
time = 0.49, size = 12, normalized size = 1.00 \begin {gather*} -x \cos \left (a + x\right ) + \sin \left (a + x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(a+x),x, algorithm="giac")

[Out]

-x*cos(a + x) + sin(a + x)

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Mupad [B]
time = 0.08, size = 12, normalized size = 1.00 \begin {gather*} \sin \left (a+x\right )-x\,\cos \left (a+x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sin(a + x),x)

[Out]

sin(a + x) - x*cos(a + x)

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