3.33.38 $\int \frac{1+2x+2x^2-4e^{2x^2}{x}^2-2\log (x)}{x}dx$

Optimal. Leaf size=25 $$-5-e^{2x^2}+2x+x^2+\log (x)-{\log }^2(x)$$

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Rubi [A]  time = 0.04, antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 3, integrand size = 30, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.100, Rules used = {14, 2209, 2301} $$\begin{array}{c}{x}^2-e^{2x^2}+2x-{\log }^2(x)+\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x + 2*x^2 - 4*E^(2*x^2)*x^2 - 2*Log[x])/x,x]

[Out]

-E^(2*x^2) + 2*x + x^2 + Log[x] - Log[x]^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-4e^{2x^2}x+\frac{1+2x+2x^2-2\log (x)}{x})dx\\ & =-(4\int e^{2x^2}xdx)+\int \frac{1+2x+2x^2-2\log (x)}{x}dx\\ & =-e^{2x^2}+\int (\frac{1+2x+2x^2}{x}-\frac{2\log (x)}{x})dx\\ & =-e^{2x^2}-2\int \frac{\log (x)}{x}dx+\int \frac{1+2x+2x^2}{x}dx\\ & =-e^{2x^2}-{\log }^2(x)+\int (2+\frac{1}{x}+2x)dx\\ & =-e^{2x^2}+2x+x^2+\log (x)-{\log }^2(x)\end{array}\end{array}$$

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Mathematica [A]  time = 0.03, size = 24, normalized size = 0.96 $$\begin{array}{c}-e^{2x^2}+2x+x^2+\log (x)-{\log }^2(x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x + 2*x^2 - 4*E^(2*x^2)*x^2 - 2*Log[x])/x,x]

[Out]

-E^(2*x^2) + 2*x + x^2 + Log[x] - Log[x]^2

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fricas [A]  time = 0.62, size = 23, normalized size = 0.92 $$\begin{array}{c}{x}^2-\log (x{)}^2+2x-e^{\left(2x^2\right)}+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)-4*x^2*exp(x^2)^2+2*x^2+2*x+1)/x,x, algorithm="fricas")

[Out]

x^2 - log(x)^2 + 2*x - e^(2*x^2) + log(x)

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giac [A]  time = 0.27, size = 23, normalized size = 0.92 $$\begin{array}{c}{x}^2-\log (x{)}^2+2x-e^{\left(2x^2\right)}+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)-4*x^2*exp(x^2)^2+2*x^2+2*x+1)/x,x, algorithm="giac")

[Out]

x^2 - log(x)^2 + 2*x - e^(2*x^2) + log(x)

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maple [A]  time = 0.03, size = 24, normalized size = 0.96

method	result	size
default	$2x+\ln (x)+x^2-{\mathrm{e}}^{2x^2}-\ln (x{)}^2$	$24$
norman	$2x+\ln (x)+x^2-{\mathrm{e}}^{2x^2}-\ln (x{)}^2$	$24$
risch	$2x+\ln (x)+x^2-{\mathrm{e}}^{2x^2}-\ln (x{)}^2$	$24$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(x)-4*x^2*exp(x^2)^2+2*x^2+2*x+1)/x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.34, size = 23, normalized size = 0.92 $$\begin{array}{c}{x}^2-\log (x{)}^2+2x-e^{\left(2x^2\right)}+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)-4*x^2*exp(x^2)^2+2*x^2+2*x+1)/x,x, algorithm="maxima")

[Out]

x^2 - log(x)^2 + 2*x - e^(2*x^2) + log(x)

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mupad [B]  time = 1.98, size = 23, normalized size = 0.92 $$\begin{array}{c}2x-{\mathrm{e}}^{2x^2}+\ln (x)-{\ln (x)}^2+x^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 2*log(x) - 4*x^2*exp(2*x^2) + 2*x^2 + 1)/x,x)

[Out]

2*x - exp(2*x^2) + log(x) - log(x)^2 + x^2

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sympy [A]  time = 0.32, size = 20, normalized size = 0.80 $$\begin{array}{c}{x}^2+2x-e^{2x^2}-\log {(x)}^2+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(x)-4*x**2*exp(x**2)**2+2*x**2+2*x+1)/x,x)

[Out]

x**2 + 2*x - exp(2*x**2) - log(x)**2 + log(x)

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