### 3.1 $$\int x^n \, dx$$

Optimal. Leaf size=11 $\frac{x^{n+1}}{n+1}$

[Out]

x^(1 + n)/(1 + n)

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Rubi [A]  time = 0.0023506, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 3, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {30} $\frac{x^{n+1}}{n+1}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^n,x]

[Out]

x^(1 + n)/(1 + n)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^n \, dx &=\frac{x^{1+n}}{1+n}\\ \end{align*}

Mathematica [A]  time = 0.0014014, size = 11, normalized size = 1. $\frac{x^{n+1}}{n+1}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^n,x]

[Out]

x^(1 + n)/(1 + n)

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Maple [A]  time = 0.003, size = 12, normalized size = 1.1 \begin{align*}{\frac{{x}^{1+n}}{1+n}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^n,x)

[Out]

x^(1+n)/(1+n)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89629, size = 20, normalized size = 1.82 \begin{align*} \frac{x x^{n}}{n + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n,x, algorithm="fricas")

[Out]

x*x^n/(n + 1)

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Sympy [A]  time = 0.053293, size = 12, normalized size = 1.09 \begin{align*} \begin{cases} \frac{x^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**n,x)

[Out]

Piecewise((x**(n + 1)/(n + 1), Ne(n, -1)), (log(x), True))

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Giac [A]  time = 1.0705, size = 15, normalized size = 1.36 \begin{align*} \frac{x^{n + 1}}{n + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n,x, algorithm="giac")

[Out]

x^(n + 1)/(n + 1)