3.96.10 \(\int \frac {1}{2} (-36+45 x-18 \log (4)-18 \log (x^2-e^4 x^2)) \, dx\)

Optimal. Leaf size=28 \[ 9 x^2 \left (\frac {5}{4}-\frac {\log (4)+\log \left (x \left (x-e^4 x\right )\right )}{x}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.18, 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, 2453, 2295} \begin {gather*} \frac {45 x^2}{4}-9 x \log \left (\left (1-e^4\right ) x^2\right )+18 x-9 x (2+\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-36 + 45*x - 18*Log[4] - 18*Log[x^2 - E^4*x^2])/2,x]

[Out]

18*x + (45*x^2)/4 - 9*x*(2 + Log[4]) - 9*x*Log[(1 - E^4)*x^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 2295

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

Rule 2453

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.), x_Symbol] :> Int[(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /;
FreeQ[{a, b, c, p, q}, x] && BinomialQ[v, x] &&  !BinomialMatchQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (-36+45 x-18 \log (4)-18 \log \left (x^2-e^4 x^2\right )\right ) \, dx\\ &=\frac {45 x^2}{4}-9 x (2+\log (4))-9 \int \log \left (x^2-e^4 x^2\right ) \, dx\\ &=\frac {45 x^2}{4}-9 x (2+\log (4))-9 \int \log \left (\left (1-e^4\right ) x^2\right ) \, dx\\ &=18 x+\frac {45 x^2}{4}-9 x (2+\log (4))-9 x \log \left (\left (1-e^4\right ) x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} \frac {45 x^2}{4}-9 x \log (4)-9 x \log \left (\left (1-e^4\right ) x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36 + 45*x - 18*Log[4] - 18*Log[x^2 - E^4*x^2])/2,x]

[Out]

(45*x^2)/4 - 9*x*Log[4] - 9*x*Log[(1 - E^4)*x^2]

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fricas [A]  time = 0.71, size = 26, normalized size = 0.93 \begin {gather*} \frac {45}{4} \, x^{2} - 18 \, x \log \relax (2) - 9 \, x \log \left (-x^{2} e^{4} + x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*log(-x^2*exp(4)+x^2)-18*log(2)+45/2*x-18,x, algorithm="fricas")

[Out]

45/4*x^2 - 18*x*log(2) - 9*x*log(-x^2*e^4 + x^2)

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giac [A]  time = 0.19, size = 26, normalized size = 0.93 \begin {gather*} \frac {45}{4} \, x^{2} - 18 \, x \log \relax (2) - 9 \, x \log \left (-x^{2} e^{4} + x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*log(-x^2*exp(4)+x^2)-18*log(2)+45/2*x-18,x, algorithm="giac")

[Out]

45/4*x^2 - 18*x*log(2) - 9*x*log(-x^2*e^4 + x^2)

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maple [A]  time = 0.04, size = 27, normalized size = 0.96

method	result	size
default	\(\frac {45 x^{2}}{4}-18 x \ln \relax (2)-9 x \ln \left (-x^{2} {\mathrm e}^{4}+x^{2}\right )\)	\(27\)
norman	\(\frac {45 x^{2}}{4}-18 x \ln \relax (2)-9 x \ln \left (-x^{2} {\mathrm e}^{4}+x^{2}\right )\)	\(27\)
risch	\(\frac {45 x^{2}}{4}-18 x \ln \relax (2)-9 x \ln \left (-x^{2} {\mathrm e}^{4}+x^{2}\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-9*ln(-x^2*exp(4)+x^2)-18*ln(2)+45/2*x-18,x,method=_RETURNVERBOSE)

[Out]

45/4*x^2-18*x*ln(2)-9*x*ln(-x^2*exp(4)+x^2)

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maxima [A]  time = 0.34, size = 26, normalized size = 0.93 \begin {gather*} \frac {45}{4} \, x^{2} - 18 \, x \log \relax (2) - 9 \, x \log \left (-x^{2} e^{4} + x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*log(-x^2*exp(4)+x^2)-18*log(2)+45/2*x-18,x, algorithm="maxima")

[Out]

45/4*x^2 - 18*x*log(2) - 9*x*log(-x^2*e^4 + x^2)

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mupad [B]  time = 8.01, size = 23, normalized size = 0.82 \begin {gather*} -\frac {9\,x\,\left (4\,\ln \left (x^2-x^2\,{\mathrm {e}}^4\right )-5\,x+\ln \left (256\right )\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((45*x)/2 - 9*log(x^2 - x^2*exp(4)) - 18*log(2) - 18,x)

[Out]

-(9*x*(4*log(x^2 - x^2*exp(4)) - 5*x + log(256)))/4

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sympy [A]  time = 0.10, size = 27, normalized size = 0.96 \begin {gather*} \frac {45 x^{2}}{4} - 9 x \log {\left (- x^{2} e^{4} + x^{2} \right )} - 18 x \log {\relax (2 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*ln(-x**2*exp(4)+x**2)-18*ln(2)+45/2*x-18,x)

[Out]

45*x**2/4 - 9*x*log(-x**2*exp(4) + x**2) - 18*x*log(2)

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