\(\int (1+x) \sin (1+x) \, dx\) [786]

Optimal result

Integrand size = 8, antiderivative size = 14 \[ \int (1+x) \sin (1+x) \, dx=-((1+x) \cos (1+x))+\sin (1+x) \]

[Out]

-(1+x)*cos(1+x)+sin(1+x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3377, 2717} \[ \int (1+x) \sin (1+x) \, dx=\sin (x+1)-(x+1) \cos (x+1) \]

[In]

Int[(1 + x)*Sin[1 + x],x]

[Out]

-((1 + x)*Cos[1 + x]) + Sin[1 + 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*} \text {integral}& = -((1+x) \cos (1+x))+\int \cos (1+x) \, dx \\ & = -((1+x) \cos (1+x))+\sin (1+x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (1+x) \sin (1+x) \, dx=-((1+x) \cos (1+x))+\sin (1+x) \]

[In]

Integrate[(1 + x)*Sin[1 + x],x]

[Out]

-((1 + x)*Cos[1 + x]) + Sin[1 + x]

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(-\left (x +1\right ) \cos \left (x +1\right )+\sin \left (x +1\right )\)	\(15\)
default	\(-\left (x +1\right ) \cos \left (x +1\right )+\sin \left (x +1\right )\)	\(15\)
risch	\(\left (-x -1\right ) \cos \left (x +1\right )+\sin \left (x +1\right )\)	\(16\)
parts	\(-\cos \left (x +1\right ) x +\sin \left (x +1\right )-\cos \left (x +1\right )\)	\(19\)
norman	\(\frac {x \tan \left (\frac {x}{2}+\frac {1}{2}\right )^{2}-x +2 \tan \left (\frac {x}{2}+\frac {1}{2}\right )-2}{1+\tan \left (\frac {x}{2}+\frac {1}{2}\right )^{2}}\)	\(37\)
parallelrisch	\(\frac {x \tan \left (\frac {x}{2}+\frac {1}{2}\right )^{2}-x +2 \tan \left (\frac {x}{2}+\frac {1}{2}\right )-2}{1+\tan \left (\frac {x}{2}+\frac {1}{2}\right )^{2}}\)	\(37\)
meijerg	\(2 \sin \left (1\right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (x \right )}{2 \sqrt {\pi }}+\frac {x \sin \left (x \right )}{2 \sqrt {\pi }}\right )+2 \cos \left (1\right ) \sqrt {\pi }\, \left (-\frac {x \cos \left (x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (x \right )}{2 \sqrt {\pi }}\right )+\sin \left (1\right ) \sin \left (x \right )+\cos \left (1\right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (x \right )}{\sqrt {\pi }}\right )\)	\(75\)

[In]

int((x+1)*sin(x+1),x,method=_RETURNVERBOSE)

[Out]

-(x+1)*cos(x+1)+sin(x+1)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (1+x) \sin (1+x) \, dx=-{\left (x + 1\right )} \cos \left (x + 1\right ) + \sin \left (x + 1\right ) \]

[In]

integrate((1+x)*sin(1+x),x, algorithm="fricas")

[Out]

-(x + 1)*cos(x + 1) + sin(x + 1)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int (1+x) \sin (1+x) \, dx=- x \cos {\left (x + 1 \right )} + \sin {\left (x + 1 \right )} - \cos {\left (x + 1 \right )} \]

[In]

integrate((1+x)*sin(1+x),x)

[Out]

-x*cos(x + 1) + sin(x + 1) - cos(x + 1)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (1+x) \sin (1+x) \, dx=-{\left (x + 1\right )} \cos \left (x + 1\right ) + \sin \left (x + 1\right ) \]

[In]

integrate((1+x)*sin(1+x),x, algorithm="maxima")

[Out]

-(x + 1)*cos(x + 1) + sin(x + 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (1+x) \sin (1+x) \, dx=-{\left (x + 1\right )} \cos \left (x + 1\right ) + \sin \left (x + 1\right ) \]

[In]

integrate((1+x)*sin(1+x),x, algorithm="giac")

[Out]

-(x + 1)*cos(x + 1) + sin(x + 1)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (1+x) \sin (1+x) \, dx=\sin \left (x+1\right )-\cos \left (x+1\right )\,\left (x+1\right ) \]

[In]

int(sin(x + 1)*(x + 1),x)

[Out]

sin(x + 1) - cos(x + 1)*(x + 1)