\(\int x^{1+x^2} (1+2 \log (x)) \, dx\) [211]

Optimal result

Integrand size = 14, antiderivative size = 5 \[ \int x^{1+x^2} (1+2 \log (x)) \, dx=x^{x^2} \]

[Out]

x^(x^2)

Rubi [F]

\[ \int x^{1+x^2} (1+2 \log (x)) \, dx=\int x^{1+x^2} (1+2 \log (x)) \, dx \]

[In]

Int[x^(1 + x^2)*(1 + 2*Log[x]),x]

[Out]

Defer[Int][x^(1 + x^2), x] + 2*Log[x]*Defer[Int][x^(1 + x^2), x] - 2*Defer[Int][Defer[Int][x^(1 + x^2), x]/x,
x]

Rubi steps \begin{align*} \text {integral}& = \int \left (x^{1+x^2}+2 x^{1+x^2} \log (x)\right ) \, dx \\ & = 2 \int x^{1+x^2} \log (x) \, dx+\int x^{1+x^2} \, dx \\ & = -\left (2 \int \frac {\int x^{1+x^2} \, dx}{x} \, dx\right )+(2 \log (x)) \int x^{1+x^2} \, dx+\int x^{1+x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int x^{1+x^2} (1+2 \log (x)) \, dx=x^{x^2} \]

[In]

Integrate[x^(1 + x^2)*(1 + 2*Log[x]),x]

[Out]

x^x^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(11\) vs. \(2(5)=10\).

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 2.40

method	result	size
risch	\(\frac {x^{x^{2}+1}}{x}\)	\(12\)
parallelrisch	\(\frac {x^{x^{2}+1}}{x}\)	\(12\)

[In]

int(x^(x^2+1)*(1+2*ln(x)),x,method=_RETURNVERBOSE)

[Out]

x^(x^2+1)/x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (5) = 10\).

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 2.20 \[ \int x^{1+x^2} (1+2 \log (x)) \, dx=\frac {x^{x^{2} + 1}}{x} \]

[In]

integrate(x^(x^2+1)*(1+2*log(x)),x, algorithm="fricas")

[Out]

x^(x^2 + 1)/x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7 vs. \(2 (3) = 6\).

Time = 0.07 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.40 \[ \int x^{1+x^2} (1+2 \log (x)) \, dx=\frac {x^{x^{2} + 1}}{x} \]

[In]

integrate(x**(x**2+1)*(1+2*ln(x)),x)

[Out]

x**(x**2 + 1)/x

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int x^{1+x^2} (1+2 \log (x)) \, dx=x^{\left (x^{2}\right )} \]

[In]

integrate(x^(x^2+1)*(1+2*log(x)),x, algorithm="maxima")

[Out]

x^(x^2)

Giac [F]

\[ \int x^{1+x^2} (1+2 \log (x)) \, dx=\int { x^{x^{2} + 1} {\left (2 \, \log \left (x\right ) + 1\right )} \,d x } \]

[In]

integrate(x^(x^2+1)*(1+2*log(x)),x, algorithm="giac")

[Out]

integrate(x^(x^2 + 1)*(2*log(x) + 1), x)

Mupad [B] (verification not implemented)

Time = 15.46 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int x^{1+x^2} (1+2 \log (x)) \, dx=x^{x^2} \]

[In]

int(x^(x^2 + 1)*(2*log(x) + 1),x)

[Out]

x^(x^2)