3.786 \(\int (1+x) \sin (1+x) \, dx\)

Optimal. Leaf size=14 \[ \sin (x+1)-(x+1) \cos (x+1) \]

[Out]

-((1 + x)*Cos[1 + x]) + Sin[1 + x]

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Rubi [A]  time = 0.0111327, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3296, 2637} \[ \sin (x+1)-(x+1) \cos (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)*Sin[1 + x],x]

[Out]

-((1 + x)*Cos[1 + x]) + Sin[1 + 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]

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 (1+x) \sin (1+x) \, dx &=-(1+x) \cos (1+x)+\int \cos (1+x) \, dx\\ &=-(1+x) \cos (1+x)+\sin (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0300698, size = 14, normalized size = 1. \[ \sin (x+1)-(x+1) \cos (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)*Sin[1 + x],x]

[Out]

-((1 + x)*Cos[1 + x]) + Sin[1 + x]

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Maple [A]  time = 0.006, size = 15, normalized size = 1.1 \begin{align*} - \left ( 1+x \right ) \cos \left ( 1+x \right ) +\sin \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.965903, size = 19, normalized size = 1.36 \begin{align*} -{\left (x + 1\right )} \cos \left (x + 1\right ) + \sin \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*cos(x + 1) + sin(x + 1)

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Fricas [A]  time = 1.95472, size = 46, normalized size = 3.29 \begin{align*} -{\left (x + 1\right )} \cos \left (x + 1\right ) + \sin \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*cos(x + 1) + sin(x + 1)

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Sympy [A]  time = 0.170035, size = 15, normalized size = 1.07 \begin{align*} - x \cos{\left (x + 1 \right )} + \sin{\left (x + 1 \right )} - \cos{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08431, size = 19, normalized size = 1.36 \begin{align*} -{\left (x + 1\right )} \cos \left (x + 1\right ) + \sin \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*cos(x + 1) + sin(x + 1)