\(\int (3+\log (2 x^2)) \, dx\) [6500]

Optimal result

Integrand size = 8, antiderivative size = 15 \[ \int \left (3+\log \left (2 x^2\right )\right ) \, dx=8+x+\log ^2(15)+x \log \left (2 x^2\right ) \]

[Out]

x*ln(2*x^2)+x+ln(15)^2+8

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2332} \[ \int \left (3+\log \left (2 x^2\right )\right ) \, dx=x \log \left (2 x^2\right )+x \]

[In]

Int[3 + Log[2*x^2],x]

[Out]

x + x*Log[2*x^2]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps \begin{align*} \text {integral}& = 3 x+\int \log \left (2 x^2\right ) \, dx \\ & = x+x \log \left (2 x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \left (3+\log \left (2 x^2\right )\right ) \, dx=x+x \log \left (2 x^2\right ) \]

[In]

Integrate[3 + Log[2*x^2],x]

[Out]

x + x*Log[2*x^2]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73

method	result	size
norman	\(x +x \ln \left (2 x^{2}\right )\)	\(11\)
risch	\(x +x \ln \left (2 x^{2}\right )\)	\(11\)
parallelrisch	\(x +x \ln \left (2 x^{2}\right )\)	\(11\)
default	\(x +x \ln \left (2\right )+x \ln \left (x^{2}\right )\)	\(13\)
parts	\(x +x \ln \left (2\right )+x \ln \left (x^{2}\right )\)	\(13\)

[In]

int(ln(2*x^2)+3,x,method=_RETURNVERBOSE)

[Out]

x+x*ln(2*x^2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \left (3+\log \left (2 x^2\right )\right ) \, dx=x \log \left (2 \, x^{2}\right ) + x \]

[In]

integrate(log(2*x^2)+3,x, algorithm="fricas")

[Out]

x*log(2*x^2) + x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.53 \[ \int \left (3+\log \left (2 x^2\right )\right ) \, dx=x \log {\left (2 x^{2} \right )} + x \]

[In]

integrate(ln(2*x**2)+3,x)

[Out]

x*log(2*x**2) + x

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \left (3+\log \left (2 x^2\right )\right ) \, dx=x \log \left (2 \, x^{2}\right ) + x \]

[In]

integrate(log(2*x^2)+3,x, algorithm="maxima")

[Out]

x*log(2*x^2) + x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \left (3+\log \left (2 x^2\right )\right ) \, dx=x \log \left (2 \, x^{2}\right ) + x \]

[In]

integrate(log(2*x^2)+3,x, algorithm="giac")

[Out]

x*log(2*x^2) + x

Mupad [B] (verification not implemented)

Time = 12.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \left (3+\log \left (2 x^2\right )\right ) \, dx=x\,\left (\ln \left (2\,x^2\right )+1\right ) \]

[In]

int(log(2*x^2) + 3,x)

[Out]

x*(log(2*x^2) + 1)