3.62.100 \(\int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+(-18 x^2+72 x^3+e^4 (-18+72 x)) \log (-x+4 x^2)+(-x^2+4 x^3+e^4 (-1+4 x)) \log ^2(-x+4 x^2)}{-81 x^2+324 x^3+(-18 x^2+72 x^3) \log (-x+4 x^2)+(-x^2+4 x^3) \log ^2(-x+4 x^2)} \, dx\)

Optimal. Leaf size=27 \[ 5-\frac {e^4}{x}+x+\frac {5}{9+\log \left (-x+4 x^2\right )} \]

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Rubi [A]  time = 0.78, antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 4, integrand size = 147, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6688, 6742, 14, 6686} \begin {gather*} x-\frac {e^4}{x}+\frac {5}{\log (-((1-4 x) x))+9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5*x - 121*x^2 + 324*x^3 + E^4*(-81 + 324*x) + (-18*x^2 + 72*x^3 + E^4*(-18 + 72*x))*Log[-x + 4*x^2] + (-x
^2 + 4*x^3 + E^4*(-1 + 4*x))*Log[-x + 4*x^2]^2)/(-81*x^2 + 324*x^3 + (-18*x^2 + 72*x^3)*Log[-x + 4*x^2] + (-x^
2 + 4*x^3)*Log[-x + 4*x^2]^2),x]

[Out]

-(E^4/x) + x + 5/(9 + Log[-((1 - 4*x)*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 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-81 e^4 (-1+4 x)-x \left (5-121 x+324 x^2\right )-18 (-1+4 x) \left (e^4+x^2\right ) \log (x (-1+4 x))-(-1+4 x) \left (e^4+x^2\right ) \log ^2(x (-1+4 x))}{(1-4 x) x^2 (9+\log (x (-1+4 x)))^2} \, dx\\ &=\int \left (\frac {e^4+x^2}{x^2}-\frac {5 (-1+8 x)}{x (-1+4 x) (9+\log (x (-1+4 x)))^2}\right ) \, dx\\ &=-\left (5 \int \frac {-1+8 x}{x (-1+4 x) (9+\log (x (-1+4 x)))^2} \, dx\right )+\int \frac {e^4+x^2}{x^2} \, dx\\ &=\frac {5}{9+\log (-((1-4 x) x))}+\int \left (1+\frac {e^4}{x^2}\right ) \, dx\\ &=-\frac {e^4}{x}+x+\frac {5}{9+\log (-((1-4 x) x))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 24, normalized size = 0.89 \begin {gather*} -\frac {e^4}{x}+x+\frac {5}{9+\log (x (-1+4 x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5*x - 121*x^2 + 324*x^3 + E^4*(-81 + 324*x) + (-18*x^2 + 72*x^3 + E^4*(-18 + 72*x))*Log[-x + 4*x^2]
+ (-x^2 + 4*x^3 + E^4*(-1 + 4*x))*Log[-x + 4*x^2]^2)/(-81*x^2 + 324*x^3 + (-18*x^2 + 72*x^3)*Log[-x + 4*x^2]
+ (-x^2 + 4*x^3)*Log[-x + 4*x^2]^2),x]

[Out]

-(E^4/x) + x + 5/(9 + Log[x*(-1 + 4*x)])

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fricas [A]  time = 0.95, size = 51, normalized size = 1.89 \begin {gather*} \frac {9 \, x^{2} + {\left (x^{2} - e^{4}\right )} \log \left (4 \, x^{2} - x\right ) + 5 \, x - 9 \, e^{4}}{x \log \left (4 \, x^{2} - x\right ) + 9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x-1)*exp(4)+4*x^3-x^2)*log(4*x^2-x)^2+((72*x-18)*exp(4)+72*x^3-18*x^2)*log(4*x^2-x)+(324*x-81)*
exp(4)+324*x^3-121*x^2+5*x)/((4*x^3-x^2)*log(4*x^2-x)^2+(72*x^3-18*x^2)*log(4*x^2-x)+324*x^3-81*x^2),x, algori
thm="fricas")

[Out]

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

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giac [B]  time = 0.26, size = 60, normalized size = 2.22 \begin {gather*} \frac {x^{2} \log \left (4 \, x^{2} - x\right ) + 9 \, x^{2} - e^{4} \log \left (4 \, x^{2} - x\right ) + 5 \, x - 9 \, e^{4}}{x \log \left (4 \, x^{2} - x\right ) + 9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x-1)*exp(4)+4*x^3-x^2)*log(4*x^2-x)^2+((72*x-18)*exp(4)+72*x^3-18*x^2)*log(4*x^2-x)+(324*x-81)*
exp(4)+324*x^3-121*x^2+5*x)/((4*x^3-x^2)*log(4*x^2-x)^2+(72*x^3-18*x^2)*log(4*x^2-x)+324*x^3-81*x^2),x, algori
thm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 31, normalized size = 1.15

method	result	size
risch	\(-\frac {{\mathrm e}^{4}-x^{2}}{x}+\frac {5}{\ln \left (4 x^{2}-x \right )+9}\)	\(31\)
norman	\(\frac {5 x +\ln \left (4 x^{2}-x \right ) x^{2}+9 x^{2}-{\mathrm e}^{4} \ln \left (4 x^{2}-x \right )-9 \,{\mathrm e}^{4}}{x \left (\ln \left (4 x^{2}-x \right )+9\right )}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x-1)*exp(4)+4*x^3-x^2)*ln(4*x^2-x)^2+((72*x-18)*exp(4)+72*x^3-18*x^2)*ln(4*x^2-x)+(324*x-81)*exp(4)+3
24*x^3-121*x^2+5*x)/((4*x^3-x^2)*ln(4*x^2-x)^2+(72*x^3-18*x^2)*ln(4*x^2-x)+324*x^3-81*x^2),x,method=_RETURNVER
BOSE)

[Out]

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

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maxima [B]  time = 0.40, size = 58, normalized size = 2.15 \begin {gather*} \frac {9 \, x^{2} + {\left (x^{2} - e^{4}\right )} \log \left (4 \, x - 1\right ) + {\left (x^{2} - e^{4}\right )} \log \relax (x) + 5 \, x - 9 \, e^{4}}{x \log \left (4 \, x - 1\right ) + x \log \relax (x) + 9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x-1)*exp(4)+4*x^3-x^2)*log(4*x^2-x)^2+((72*x-18)*exp(4)+72*x^3-18*x^2)*log(4*x^2-x)+(324*x-81)*
exp(4)+324*x^3-121*x^2+5*x)/((4*x^3-x^2)*log(4*x^2-x)^2+(72*x^3-18*x^2)*log(4*x^2-x)+324*x^3-81*x^2),x, algori
thm="maxima")

[Out]

(9*x^2 + (x^2 - e^4)*log(4*x - 1) + (x^2 - e^4)*log(x) + 5*x - 9*e^4)/(x*log(4*x - 1) + x*log(x) + 9*x)

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mupad [B]  time = 4.38, size = 25, normalized size = 0.93 \begin {gather*} x+\frac {5}{\ln \left (4\,x^2-x\right )+9}-\frac {{\mathrm {e}}^4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + log(4*x^2 - x)*(72*x^3 - 18*x^2 + exp(4)*(72*x - 18)) + log(4*x^2 - x)^2*(4*x^3 - x^2 + exp(4)*(4*
x - 1)) - 121*x^2 + 324*x^3 + exp(4)*(324*x - 81))/(log(4*x^2 - x)*(18*x^2 - 72*x^3) + log(4*x^2 - x)^2*(x^2 -
4*x^3) + 81*x^2 - 324*x^3),x)

[Out]

x + 5/(log(4*x^2 - x) + 9) - exp(4)/x

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sympy [A]  time = 0.19, size = 17, normalized size = 0.63 \begin {gather*} x + \frac {5}{\log {\left (4 x^{2} - x \right )} + 9} - \frac {e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x-1)*exp(4)+4*x**3-x**2)*ln(4*x**2-x)**2+((72*x-18)*exp(4)+72*x**3-18*x**2)*ln(4*x**2-x)+(324*x
-81)*exp(4)+324*x**3-121*x**2+5*x)/((4*x**3-x**2)*ln(4*x**2-x)**2+(72*x**3-18*x**2)*ln(4*x**2-x)+324*x**3-81*x
**2),x)

[Out]

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

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