3.58.18 \(\int \frac {e (3-46 x)-92 x \log (\frac {1}{46} (-3+46 x))}{e (-3 x+46 x^2)} \, dx\)

Optimal. Leaf size=21 \[ 3-\frac {-2+\log ^2\left (-\frac {3}{46}+x\right )}{e}-\log (x) \]

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Rubi [A]  time = 0.18, antiderivative size = 18, normalized size of antiderivative = 0.86, 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 = {12, 1593, 6742, 2390, 2301} \begin {gather*} -\frac {\log ^2\left (x-\frac {3}{46}\right )}{e}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E*(3 - 46*x) - 92*x*Log[(-3 + 46*x)/46])/(E*(-3*x + 46*x^2)),x]

[Out]

-(Log[-3/46 + x]^2/E) - 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 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e (3-46 x)-92 x \log \left (\frac {1}{46} (-3+46 x)\right )}{-3 x+46 x^2} \, dx}{e}\\ &=\frac {\int \frac {e (3-46 x)-92 x \log \left (\frac {1}{46} (-3+46 x)\right )}{x (-3+46 x)} \, dx}{e}\\ &=\frac {\int \left (-\frac {e}{x}-\frac {92 \log \left (-\frac {3}{46}+x\right )}{-3+46 x}\right ) \, dx}{e}\\ &=-\log (x)-\frac {92 \int \frac {\log \left (-\frac {3}{46}+x\right )}{-3+46 x} \, dx}{e}\\ &=-\log (x)-\frac {92 \operatorname {Subst}\left (\int \frac {\log (x)}{46 x} \, dx,x,-\frac {3}{46}+x\right )}{e}\\ &=-\log (x)-\frac {2 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-\frac {3}{46}+x\right )}{e}\\ &=-\frac {\log ^2\left (-\frac {3}{46}+x\right )}{e}-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 20, normalized size = 0.95 \begin {gather*} \frac {-\log ^2\left (-\frac {3}{46}+x\right )-e \log (x)}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E*(3 - 46*x) - 92*x*Log[(-3 + 46*x)/46])/(E*(-3*x + 46*x^2)),x]

[Out]

(-Log[-3/46 + x]^2 - E*Log[x])/E

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fricas [A]  time = 0.74, size = 16, normalized size = 0.76 \begin {gather*} -{\left (\log \left (x - \frac {3}{46}\right )^{2} + e \log \relax (x)\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-92*x*log(x-3/46)+(-46*x+3)*exp(1))/(46*x^2-3*x)/exp(1),x, algorithm="fricas")

[Out]

-(log(x - 3/46)^2 + e*log(x))*e^(-1)

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giac [A]  time = 0.14, size = 16, normalized size = 0.76 \begin {gather*} -{\left (\log \left (x - \frac {3}{46}\right )^{2} + e \log \relax (x)\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-92*x*log(x-3/46)+(-46*x+3)*exp(1))/(46*x^2-3*x)/exp(1),x, algorithm="giac")

[Out]

-(log(x - 3/46)^2 + e*log(x))*e^(-1)

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maple [A]  time = 0.26, size = 16, normalized size = 0.76

method	result	size
risch	\(-{\mathrm e}^{-1} \ln \left (x -\frac {3}{46}\right )^{2}-\ln \relax (x )\)	\(16\)
norman	\(-{\mathrm e}^{-1} \ln \left (x -\frac {3}{46}\right )^{2}-\ln \relax (x )\)	\(18\)
default	\({\mathrm e}^{-1} \left (-\ln \left (x -\frac {3}{46}\right )^{2}-{\mathrm e} \ln \left (46 x \right )\right )\)	\(23\)
derivativedivides	\(2116 \,{\mathrm e}^{-1} \left (-\frac {\ln \left (x -\frac {3}{46}\right )^{2}}{2116}-\frac {{\mathrm e} \ln \left (46 x \right )}{2116}\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-92*x*ln(x-3/46)+(-46*x+3)*exp(1))/(46*x^2-3*x)/exp(1),x,method=_RETURNVERBOSE)

[Out]

-exp(-1)*ln(x-3/46)^2-ln(x)

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maxima [B]  time = 0.45, size = 51, normalized size = 2.43 \begin {gather*} {\left ({\left (\log \left (46 \, x - 3\right ) - \log \relax (x)\right )} e + 2 \, {\left (\log \left (23\right ) + \log \relax (2)\right )} \log \left (46 \, x - 3\right ) - e \log \left (46 \, x - 3\right ) - \log \left (46 \, x - 3\right )^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-92*x*log(x-3/46)+(-46*x+3)*exp(1))/(46*x^2-3*x)/exp(1),x, algorithm="maxima")

[Out]

((log(46*x - 3) - log(x))*e + 2*(log(23) + log(2))*log(46*x - 3) - e*log(46*x - 3) - log(46*x - 3)^2)*e^(-1)

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mupad [B]  time = 0.14, size = 15, normalized size = 0.71 \begin {gather*} -{\mathrm {e}}^{-1}\,{\ln \left (x-\frac {3}{46}\right )}^2-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-1)*(92*x*log(x - 3/46) + exp(1)*(46*x - 3)))/(3*x - 46*x^2),x)

[Out]

- log(x) - log(x - 3/46)^2*exp(-1)

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sympy [A]  time = 0.13, size = 15, normalized size = 0.71 \begin {gather*} - \log {\relax (x )} - \frac {\log {\left (x - \frac {3}{46} \right )}^{2}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-92*x*ln(x-3/46)+(-46*x+3)*exp(1))/(46*x**2-3*x)/exp(1),x)

[Out]

-log(x) - exp(-1)*log(x - 3/46)**2

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