3.32.91 \(\int \frac {4 \log (5 e^4 x^2)}{x} \, dx\)

Optimal. Leaf size=13 \[ 25+\log ^2\left (5 e^4 x^2\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2301} \begin {gather*} \log ^2\left (5 e^4 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*Log[5*E^4*x^2])/x,x]

[Out]

Log[5*E^4*x^2]^2

Rule 12

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

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 {gather*} \begin {aligned} \text {integral} &=4 \int \frac {\log \left (5 e^4 x^2\right )}{x} \, dx\\ &=\log ^2\left (5 e^4 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.31 \begin {gather*} 16 \log (x)+4 \log (5) \log (x)+\log ^2\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*Log[5*E^4*x^2])/x,x]

[Out]

16*Log[x] + 4*Log[5]*Log[x] + Log[x^2]^2

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fricas [A]  time = 0.52, size = 10, normalized size = 0.77 \begin {gather*} \log \left (5 \, x^{2} e^{4}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log(5*x^2*exp(2)^2)/x,x, algorithm="fricas")

[Out]

log(5*x^2*e^4)^2

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giac [A]  time = 0.14, size = 10, normalized size = 0.77 \begin {gather*} \log \left (5 \, x^{2} e^{4}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log(5*x^2*exp(2)^2)/x,x, algorithm="giac")

[Out]

log(5*x^2*e^4)^2

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maple [A]  time = 0.02, size = 11, normalized size = 0.85

method	result	size
risch	\(\ln \left (5 x^{2} {\mathrm e}^{4}\right )^{2}\)	\(11\)
derivativedivides	\(\ln \left (5 x^{2} {\mathrm e}^{4}\right )^{2}\)	\(13\)
default	\(\ln \left (5 x^{2} {\mathrm e}^{4}\right )^{2}\)	\(13\)
norman	\(\ln \left (5 x^{2} {\mathrm e}^{4}\right )^{2}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*ln(5*x^2*exp(2)^2)/x,x,method=_RETURNVERBOSE)

[Out]

ln(5*x^2*exp(4))^2

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maxima [A]  time = 0.34, size = 10, normalized size = 0.77 \begin {gather*} \log \left (5 \, x^{2} e^{4}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log(5*x^2*exp(2)^2)/x,x, algorithm="maxima")

[Out]

log(5*x^2*e^4)^2

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mupad [B]  time = 2.68, size = 10, normalized size = 0.77 \begin {gather*} {\left (\ln \left (5\,x^2\right )+4\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(5*x^2*exp(4)))/x,x)

[Out]

(log(5*x^2) + 4)^2

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sympy [A]  time = 0.09, size = 10, normalized size = 0.77 \begin {gather*} \log {\left (5 x^{2} e^{4} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*ln(5*x**2*exp(2)**2)/x,x)

[Out]

log(5*x**2*exp(4))**2

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