3.2.94 \(\int \frac {2 e^{\frac {x^2}{\log (2)}} x^2-4 \log (2)}{x \log (2)} \, dx\)

Optimal. Leaf size=18 \[ 1+e^{\frac {x^2}{\log (2)}}-\log \left (x^4\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {12, 14, 2209} \begin {gather*} e^{\frac {x^2}{\log (2)}}-4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(x^2/Log[2]) - 4*Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.83 \begin {gather*} e^{\frac {x^2}{\log (2)}}-4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(x^2/Log[2]) - 4*Log[x]

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fricas [A]  time = 1.24, size = 14, normalized size = 0.78 \begin {gather*} e^{\left (\frac {x^{2}}{\log \relax (2)}\right )} - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^2/log(2)) - 4*log(x)

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giac [B]  time = 0.30, size = 35, normalized size = 1.94 \begin {gather*} \frac {e^{\left (\frac {x^{2}}{\log \relax (2)}\right )} \log \relax (2)^{2} - 2 \, \log \relax (2)^{2} \log \left (\frac {x^{2}}{\log \relax (2)}\right )}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 15, normalized size = 0.83

method	result	size
norman	\({\mathrm e}^{\frac {x^{2}}{\ln \relax (2)}}-4 \ln \relax (x )\)	\(15\)
risch	\({\mathrm e}^{\frac {x^{2}}{\ln \relax (2)}}-4 \ln \relax (x )\)	\(15\)
default	\(\frac {-4 \ln \relax (2) \ln \relax (x )+{\mathrm e}^{\frac {x^{2}}{\ln \relax (2)}} \ln \relax (2)}{\ln \relax (2)}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x^2/ln(2))-4*ln(x)

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maxima [A]  time = 0.65, size = 24, normalized size = 1.33 \begin {gather*} \frac {e^{\left (\frac {x^{2}}{\log \relax (2)}\right )} \log \relax (2) - 4 \, \log \relax (2) \log \relax (x)}{\log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e^(x^2/log(2))*log(2) - 4*log(2)*log(x))/log(2)

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mupad [B]  time = 0.08, size = 14, normalized size = 0.78 \begin {gather*} {\mathrm {e}}^{\frac {x^2}{\ln \relax (2)}}-4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x^2/log(2)) - 4*log(x)

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sympy [A]  time = 0.12, size = 12, normalized size = 0.67 \begin {gather*} e^{\frac {x^{2}}{\log {\relax (2 )}}} - 4 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x**2/log(2)) - 4*log(x)

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