3.39.67 \(\int \frac {7 x+e^{\log ^2(x^2)} (3 x+(4-12 x) \log (x^2))}{12 x+3 e^{2 \log ^2(x^2)} x+12 x^2+3 x^3+e^{\log ^2(x^2)} (12 x+6 x^2)} \, dx\)

Optimal. Leaf size=24 \[ \frac {-1+3 x}{3 \left (2+e^{\log ^2\left (x^2\right )}\right )+3 x} \]

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Rubi [F]  time = 2.61, 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 {7 x+e^{\log ^2\left (x^2\right )} \left (3 x+(4-12 x) \log \left (x^2\right )\right )}{12 x+3 e^{2 \log ^2\left (x^2\right )} x+12 x^2+3 x^3+e^{\log ^2\left (x^2\right )} \left (12 x+6 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(7*x + E^Log[x^2]^2*(3*x + (4 - 12*x)*Log[x^2]))/(12*x + 3*E^(2*Log[x^2]^2)*x + 12*x^2 + 3*x^3 + E^Log[x^2
]^2*(12*x + 6*x^2)),x]

[Out]

Defer[Int][(2 + E^Log[x^2]^2 + x)^(-2), x]/3 - Defer[Int][x/(2 + E^Log[x^2]^2 + x)^2, x] + Defer[Int][(2 + E^L
og[x^2]^2 + x)^(-1), x] + (20*Defer[Int][Log[x^2]/(2 + E^Log[x^2]^2 + x)^2, x])/3 - (8*Defer[Int][Log[x^2]/(x*
(2 + E^Log[x^2]^2 + x)^2), x])/3 + 4*Defer[Int][(x*Log[x^2])/(2 + E^Log[x^2]^2 + x)^2, x] - 4*Defer[Int][Log[x
^2]/(2 + E^Log[x^2]^2 + x), x] + (4*Defer[Int][Log[x^2]/(x*(2 + E^Log[x^2]^2 + x)), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (7+3 e^{\log ^2\left (x^2\right )}\right ) x-4 e^{\log ^2\left (x^2\right )} (-1+3 x) \log \left (x^2\right )}{3 x \left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {\left (7+3 e^{\log ^2\left (x^2\right )}\right ) x-4 e^{\log ^2\left (x^2\right )} (-1+3 x) \log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {(-1+3 x) \left (-x+8 \log \left (x^2\right )+4 x \log \left (x^2\right )\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )^2}-\frac {-3 x-4 \log \left (x^2\right )+12 x \log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {(-1+3 x) \left (-x+8 \log \left (x^2\right )+4 x \log \left (x^2\right )\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx-\frac {1}{3} \int \frac {-3 x-4 \log \left (x^2\right )+12 x \log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )} \, dx\\ &=\frac {1}{3} \int \frac {(1-3 x) \left (x-4 (2+x) \log \left (x^2\right )\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx-\frac {1}{3} \int \left (-\frac {3}{2+e^{\log ^2\left (x^2\right )}+x}+\frac {12 \log \left (x^2\right )}{2+e^{\log ^2\left (x^2\right )}+x}-\frac {4 \log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )}\right ) \, dx\\ &=\frac {1}{3} \int \left (\frac {3 \left (-x+8 \log \left (x^2\right )+4 x \log \left (x^2\right )\right )}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2}-\frac {-x+8 \log \left (x^2\right )+4 x \log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )^2}\right ) \, dx+\frac {4}{3} \int \frac {\log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )} \, dx-4 \int \frac {\log \left (x^2\right )}{2+e^{\log ^2\left (x^2\right )}+x} \, dx+\int \frac {1}{2+e^{\log ^2\left (x^2\right )}+x} \, dx\\ &=-\left (\frac {1}{3} \int \frac {-x+8 \log \left (x^2\right )+4 x \log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx\right )+\frac {4}{3} \int \frac {\log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )} \, dx-4 \int \frac {\log \left (x^2\right )}{2+e^{\log ^2\left (x^2\right )}+x} \, dx+\int \frac {1}{2+e^{\log ^2\left (x^2\right )}+x} \, dx+\int \frac {-x+8 \log \left (x^2\right )+4 x \log \left (x^2\right )}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx\\ &=-\left (\frac {1}{3} \int \left (-\frac {1}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2}+\frac {4 \log \left (x^2\right )}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2}+\frac {8 \log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )^2}\right ) \, dx\right )+\frac {4}{3} \int \frac {\log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )} \, dx-4 \int \frac {\log \left (x^2\right )}{2+e^{\log ^2\left (x^2\right )}+x} \, dx+\int \frac {1}{2+e^{\log ^2\left (x^2\right )}+x} \, dx+\int \left (-\frac {x}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2}+\frac {8 \log \left (x^2\right )}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2}+\frac {4 x \log \left (x^2\right )}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {1}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx-\frac {4}{3} \int \frac {\log \left (x^2\right )}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx+\frac {4}{3} \int \frac {\log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )} \, dx-\frac {8}{3} \int \frac {\log \left (x^2\right )}{x \left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx+4 \int \frac {x \log \left (x^2\right )}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx-4 \int \frac {\log \left (x^2\right )}{2+e^{\log ^2\left (x^2\right )}+x} \, dx+8 \int \frac {\log \left (x^2\right )}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx-\int \frac {x}{\left (2+e^{\log ^2\left (x^2\right )}+x\right )^2} \, dx+\int \frac {1}{2+e^{\log ^2\left (x^2\right )}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.50, size = 22, normalized size = 0.92 \begin {gather*} \frac {-1+3 x}{3 \left (2+e^{\log ^2\left (x^2\right )}+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7*x + E^Log[x^2]^2*(3*x + (4 - 12*x)*Log[x^2]))/(12*x + 3*E^(2*Log[x^2]^2)*x + 12*x^2 + 3*x^3 + E^L
og[x^2]^2*(12*x + 6*x^2)),x]

[Out]

(-1 + 3*x)/(3*(2 + E^Log[x^2]^2 + x))

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fricas [A]  time = 0.50, size = 19, normalized size = 0.79 \begin {gather*} \frac {3 \, x - 1}{3 \, {\left (x + e^{\left (\log \left (x^{2}\right )^{2}\right )} + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*x+4)*log(x^2)+3*x)*exp(log(x^2)^2)+7*x)/(3*x*exp(log(x^2)^2)^2+(6*x^2+12*x)*exp(log(x^2)^2)+3
*x^3+12*x^2+12*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.42, size = 19, normalized size = 0.79 \begin {gather*} \frac {3 \, x - 1}{3 \, {\left (x + e^{\left (\log \left (x^{2}\right )^{2}\right )} + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*x+4)*log(x^2)+3*x)*exp(log(x^2)^2)+7*x)/(3*x*exp(log(x^2)^2)^2+(6*x^2+12*x)*exp(log(x^2)^2)+3
*x^3+12*x^2+12*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.18, size = 20, normalized size = 0.83

method	result	size
risch	\(\frac {3 x -1}{3 \,{\mathrm e}^{\ln \left (x^{2}\right )^{2}}+6+3 x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-12*x+4)*ln(x^2)+3*x)*exp(ln(x^2)^2)+7*x)/(3*x*exp(ln(x^2)^2)^2+(6*x^2+12*x)*exp(ln(x^2)^2)+3*x^3+12*x^
2+12*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.47, size = 19, normalized size = 0.79 \begin {gather*} \frac {3 \, x - 1}{3 \, {\left (x + e^{\left (4 \, \log \relax (x)^{2}\right )} + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*x+4)*log(x^2)+3*x)*exp(log(x^2)^2)+7*x)/(3*x*exp(log(x^2)^2)^2+(6*x^2+12*x)*exp(log(x^2)^2)+3
*x^3+12*x^2+12*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.41, size = 72, normalized size = 3.00 \begin {gather*} \frac {20\,x^2\,\ln \left (x^2\right )-8\,x\,\ln \left (x^2\right )+12\,x^3\,\ln \left (x^2\right )+x^2-3\,x^3}{3\,\left (8\,x\,\ln \left (x^2\right )+4\,x^2\,\ln \left (x^2\right )-x^2\right )\,\left (x+{\mathrm {e}}^{{\ln \left (x^2\right )}^2}+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x + exp(log(x^2)^2)*(3*x - log(x^2)*(12*x - 4)))/(12*x + 3*x*exp(2*log(x^2)^2) + exp(log(x^2)^2)*(12*x
+ 6*x^2) + 12*x^2 + 3*x^3),x)

[Out]

(20*x^2*log(x^2) - 8*x*log(x^2) + 12*x^3*log(x^2) + x^2 - 3*x^3)/(3*(8*x*log(x^2) + 4*x^2*log(x^2) - x^2)*(x +
exp(log(x^2)^2) + 2))

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sympy [A]  time = 0.27, size = 19, normalized size = 0.79 \begin {gather*} \frac {3 x - 1}{3 x + 3 e^{\log {\left (x^{2} \right )}^{2}} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*x+4)*ln(x**2)+3*x)*exp(ln(x**2)**2)+7*x)/(3*x*exp(ln(x**2)**2)**2+(6*x**2+12*x)*exp(ln(x**2)*
*2)+3*x**3+12*x**2+12*x),x)

[Out]

(3*x - 1)/(3*x + 3*exp(log(x**2)**2) + 6)

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