3.35.75 \(\int \frac {4 e-6 x+(-3 e x+3 x^2) \log ^3(e x^2-x^3)}{(e x-x^2) \log ^3(e x^2-x^3)} \, dx\)

Optimal. Leaf size=19 \[ 2-3 x-\frac {1}{\log ^2\left ((e-x) x^2\right )} \]

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Rubi [F]  time = 0.63, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {4 e-6 x+\left (-3 e x+3 x^2\right ) \log ^3\left (e x^2-x^3\right )}{\left (e x-x^2\right ) \log ^3\left (e x^2-x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*E - 6*x + (-3*E*x + 3*x^2)*Log[E*x^2 - x^3]^3)/((E*x - x^2)*Log[E*x^2 - x^3]^3),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e-6 x+\left (-3 e x+3 x^2\right ) \log ^3\left (e x^2-x^3\right )}{(e-x) x \log ^3\left (e x^2-x^3\right )} \, dx\\ &=\int \frac {4 e-6 x+\left (-3 e x+3 x^2\right ) \log ^3\left (e x^2-x^3\right )}{(e-x) x \log ^3\left ((e-x) x^2\right )} \, dx\\ &=\int \left (-3+\frac {2 (2 e-3 x)}{(e-x) x \log ^3\left ((e-x) x^2\right )}\right ) \, dx\\ &=-3 x+2 \int \frac {2 e-3 x}{(e-x) x \log ^3\left ((e-x) x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 18, normalized size = 0.95 \begin {gather*} -3 x-\frac {1}{\log ^2\left ((e-x) x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E - 6*x + (-3*E*x + 3*x^2)*Log[E*x^2 - x^3]^3)/((E*x - x^2)*Log[E*x^2 - x^3]^3),x]

[Out]

-3*x - Log[(E - x)*x^2]^(-2)

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fricas [A]  time = 1.07, size = 37, normalized size = 1.95 \begin {gather*} -\frac {3 \, x \log \left (-x^{3} + x^{2} e\right )^{2} + 1}{\log \left (-x^{3} + x^{2} e\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x*exp(1)+3*x^2)*log(x^2*exp(1)-x^3)^3+4*exp(1)-6*x)/(x*exp(1)-x^2)/log(x^2*exp(1)-x^3)^3,x, alg
orithm="fricas")

[Out]

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

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giac [A]  time = 0.55, size = 37, normalized size = 1.95 \begin {gather*} -\frac {3 \, x \log \left (-x^{3} + x^{2} e\right )^{2} + 1}{\log \left (-x^{3} + x^{2} e\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x*exp(1)+3*x^2)*log(x^2*exp(1)-x^3)^3+4*exp(1)-6*x)/(x*exp(1)-x^2)/log(x^2*exp(1)-x^3)^3,x, alg
orithm="giac")

[Out]

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

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maple [A]  time = 0.13, size = 22, normalized size = 1.16

method	result	size
default	\(-3 x -\frac {1}{\ln \left (x^{2} {\mathrm e}-x^{3}\right )^{2}}\)	\(22\)
risch	\(-3 x -\frac {1}{\ln \left (x^{2} {\mathrm e}-x^{3}\right )^{2}}\)	\(22\)
norman	\(\frac {-1-3 x \ln \left (x^{2} {\mathrm e}-x^{3}\right )^{2}}{\ln \left (x^{2} {\mathrm e}-x^{3}\right )^{2}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x*exp(1)+3*x^2)*ln(x^2*exp(1)-x^3)^3+4*exp(1)-6*x)/(x*exp(1)-x^2)/ln(x^2*exp(1)-x^3)^3,x,method=_RETU
RNVERBOSE)

[Out]

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

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maxima [B]  time = 0.69, size = 64, normalized size = 3.37 \begin {gather*} -\frac {12 \, x \log \relax (x)^{2} + 12 \, x \log \relax (x) \log \left (-x + e\right ) + 3 \, x \log \left (-x + e\right )^{2} + 1}{4 \, \log \relax (x)^{2} + 4 \, \log \relax (x) \log \left (-x + e\right ) + \log \left (-x + e\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x*exp(1)+3*x^2)*log(x^2*exp(1)-x^3)^3+4*exp(1)-6*x)/(x*exp(1)-x^2)/log(x^2*exp(1)-x^3)^3,x, alg
orithm="maxima")

[Out]

-(12*x*log(x)^2 + 12*x*log(x)*log(-x + e) + 3*x*log(-x + e)^2 + 1)/(4*log(x)^2 + 4*log(x)*log(-x + e) + log(-x
+ e)^2)

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mupad [B]  time = 2.24, size = 21, normalized size = 1.11 \begin {gather*} -3\,x-\frac {1}{{\ln \left (x^2\,\mathrm {e}-x^3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x - 4*exp(1) + log(x^2*exp(1) - x^3)^3*(3*x*exp(1) - 3*x^2))/(log(x^2*exp(1) - x^3)^3*(x*exp(1) - x^2)
),x)

[Out]

- 3*x - 1/log(x^2*exp(1) - x^3)^2

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sympy [A]  time = 0.15, size = 19, normalized size = 1.00 \begin {gather*} - 3 x - \frac {1}{\log {\left (- x^{3} + e x^{2} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x*exp(1)+3*x**2)*ln(x**2*exp(1)-x**3)**3+4*exp(1)-6*x)/(x*exp(1)-x**2)/ln(x**2*exp(1)-x**3)**3,
x)

[Out]

-3*x - 1/log(-x**3 + E*x**2)**2

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