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

Optimal result

Integrand size = 3, antiderivative size = 11 \[ \int x^n \, dx=\frac {x^{1+n}}{1+n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {30} \[ \int x^n \, dx=\frac {x^{n+1}}{n+1} \]

[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*} \text {integral}& = \frac {x^{1+n}}{1+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int x^n \, dx=\frac {x^{1+n}}{1+n} \]

[In]

Integrate[x^n,x]

[Out]

x^(1 + n)/(1 + n)

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

x/(1+n)*x^n

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int x^n \, dx=\frac {x x^{n}}{n + 1} \]

[In]

integrate(x^n,x, algorithm="fricas")

[Out]

x*x^n/(n + 1)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int x^n \, dx=\begin {cases} \frac {x^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**n,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int x^n \, dx=\frac {x^{n + 1}}{n + 1} \]

[In]

integrate(x^n,x, algorithm="maxima")

[Out]

x^(n + 1)/(n + 1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int x^n \, dx=\frac {x^{n + 1}}{n + 1} \]

[In]

integrate(x^n,x, algorithm="giac")

[Out]

x^(n + 1)/(n + 1)

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.82 \[ \int x^n \, dx=\left \{\begin {array}{cl} \ln \left (x\right ) & \text {\ if\ \ }n=-1\\ \frac {x^{n+1}}{n+1} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]

[In]

int(x^n,x)

[Out]

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