3.35.25 \(\int \frac {-1+(7 x-2 x^2-2 e^{x^2} x^2) \log (4 x)}{x \log (4 x)} \, dx\)

Optimal. Leaf size=24 \[ 4-e^{x^2}-(-3+x)^2+x-\log (\log (4 x)) \]

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Rubi [A]  time = 0.24, antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {6688, 14, 2209, 2302, 29} \begin {gather*} -x^2-e^{x^2}+7 x-\log (\log (4 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + (7*x - 2*x^2 - 2*E^x^2*x^2)*Log[4*x])/(x*Log[4*x]),x]

[Out]

-E^x^2 + 7*x - x^2 - Log[Log[4*x]]

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (7-2 \left (1+e^{x^2}\right ) x-\frac {1}{x \log (4 x)}\right ) \, dx\\ &=7 x-2 \int \left (1+e^{x^2}\right ) x \, dx-\int \frac {1}{x \log (4 x)} \, dx\\ &=7 x-2 \int \left (x+e^{x^2} x\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (4 x)\right )\\ &=7 x-x^2-\log (\log (4 x))-2 \int e^{x^2} x \, dx\\ &=-e^{x^2}+7 x-x^2-\log (\log (4 x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 23, normalized size = 0.96 \begin {gather*} -e^{x^2}+7 x-x^2-\log (\log (4 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + (7*x - 2*x^2 - 2*E^x^2*x^2)*Log[4*x])/(x*Log[4*x]),x]

[Out]

-E^x^2 + 7*x - x^2 - Log[Log[4*x]]

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fricas [A]  time = 0.52, size = 22, normalized size = 0.92 \begin {gather*} -x^{2} + 7 \, x - e^{\left (x^{2}\right )} - \log \left (\log \left (4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(x^2)-2*x^2+7*x)*log(4*x)-1)/x/log(4*x),x, algorithm="fricas")

[Out]

-x^2 + 7*x - e^(x^2) - log(log(4*x))

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giac [A]  time = 0.21, size = 22, normalized size = 0.92 \begin {gather*} -x^{2} + 7 \, x - e^{\left (x^{2}\right )} - \log \left (\log \left (4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(x^2)-2*x^2+7*x)*log(4*x)-1)/x/log(4*x),x, algorithm="giac")

[Out]

-x^2 + 7*x - e^(x^2) - log(log(4*x))

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maple [A]  time = 0.02, size = 23, normalized size = 0.96

method	result	size
default	\(-x^{2}+7 x -\ln \left (\ln \left (4 x \right )\right )-{\mathrm e}^{x^{2}}\)	\(23\)
norman	\(-x^{2}+7 x -\ln \left (\ln \left (4 x \right )\right )-{\mathrm e}^{x^{2}}\)	\(23\)
risch	\(-x^{2}+7 x -\ln \left (\ln \left (4 x \right )\right )-{\mathrm e}^{x^{2}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2*exp(x^2)-2*x^2+7*x)*ln(4*x)-1)/x/ln(4*x),x,method=_RETURNVERBOSE)

[Out]

-x^2+7*x-ln(ln(4*x))-exp(x^2)

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maxima [A]  time = 0.45, size = 22, normalized size = 0.92 \begin {gather*} -x^{2} + 7 \, x - e^{\left (x^{2}\right )} - \log \left (\log \left (4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(x^2)-2*x^2+7*x)*log(4*x)-1)/x/log(4*x),x, algorithm="maxima")

[Out]

-x^2 + 7*x - e^(x^2) - log(log(4*x))

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mupad [B]  time = 2.09, size = 22, normalized size = 0.92 \begin {gather*} 7\,x-{\mathrm {e}}^{x^2}-\ln \left (\ln \left (4\,x\right )\right )-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(4*x)*(2*x^2*exp(x^2) - 7*x + 2*x^2) + 1)/(x*log(4*x)),x)

[Out]

7*x - exp(x^2) - log(log(4*x)) - x^2

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sympy [A]  time = 0.32, size = 17, normalized size = 0.71 \begin {gather*} - x^{2} + 7 x - e^{x^{2}} - \log {\left (\log {\left (4 x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2*exp(x**2)-2*x**2+7*x)*ln(4*x)-1)/x/ln(4*x),x)

[Out]

-x**2 + 7*x - exp(x**2) - log(log(4*x))

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