3.5 \(\int \sin (x) \, dx\)

Optimal. Leaf size=4 \[ -\cos (x) \]

[Out]

-Cos[x]

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Rubi [A]  time = 0.0022196, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2638} \[ -\cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x],x]

[Out]

-Cos[x]

Rule 2638

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

Rubi steps

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

Mathematica [A]  time = 0.000953, size = 4, normalized size = 1. \[ -\cos (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x],x]

[Out]

-Cos[x]

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Maple [A]  time = 0., size = 5, normalized size = 1.3 \begin{align*} -\cos \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x),x)

[Out]

-cos(x)

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Maxima [A]  time = 0.926118, size = 5, normalized size = 1.25 \begin{align*} -\cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x),x, algorithm="maxima")

[Out]

-cos(x)

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Fricas [A]  time = 2.00473, size = 12, normalized size = 3. \begin{align*} -\cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x),x, algorithm="fricas")

[Out]

-cos(x)

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Sympy [A]  time = 0.053336, size = 3, normalized size = 0.75 \begin{align*} - \cos{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x),x)

[Out]

-cos(x)

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Giac [A]  time = 1.06566, size = 5, normalized size = 1.25 \begin{align*} -\cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x),x, algorithm="giac")

[Out]

-cos(x)