3.1.34 \(\int x^n \, dx\) [34]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^{n+1}}{n+1} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {x^{1+n}}{1+n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^n,x]

[Out]

x^(1 + n)/(1 + n)

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Maple [A]
time = 0.01, size = 12, normalized size = 1.09

method	result	size
risch	\(\frac {x \,x^{n}}{1+n}\)	\(11\)
gosper	\(\frac {x^{1+n}}{1+n}\)	\(12\)
default	\(\frac {x^{1+n}}{1+n}\)	\(12\)
norman	\(\frac {x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^n,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 11, normalized size = 1.00 \begin {gather*} \frac {x^{n + 1}}{n + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n,x, algorithm="maxima")

[Out]

x^(n + 1)/(n + 1)

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Fricas [A]
time = 0.99, size = 10, normalized size = 0.91 \begin {gather*} \frac {x x^{n}}{n + 1} \end {gather*}

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 = 0.01, size = 12, normalized size = 1.09 \begin {gather*} \begin {cases} \frac {x^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}

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 = 1.56, size = 11, normalized size = 1.00 \begin {gather*} \frac {x^{n + 1}}{n + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n,x, algorithm="giac")

[Out]

x^(n + 1)/(n + 1)

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Mupad [B]
time = 0.35, size = 20, normalized size = 1.82 \begin {gather*} \left \{\begin {array}{cl} \ln \left (x\right ) & \text {\ if\ \ }n=-1\\ \frac {x^{n+1}}{n+1} & \text {\ if\ \ }n\neq -1 \end {array}\right . \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^n,x)

[Out]

piecewise(n == -1, log(x), n ~= -1, x^(n + 1)/(n + 1))

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