3.47.40 \(\int \frac {6 x^2+x^{\frac {1}{x}} (3+x+4 x^2+x^3+(-3-x) \log (x))}{6 x^2+x^{\frac {1}{x}} (3 x^2+x^3)} \, dx\) [4640]

Optimal. Leaf size=16 \[ -1+x+\log \left (-6-x^{\frac {1}{x}} (3+x)\right ) \]

[Out]

ln(-6-(3+x)*exp(ln(x)/x))+x-1

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Rubi [F]
time = 3.42, 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 {6 x^2+x^{\frac {1}{x}} \left (3+x+4 x^2+x^3+(-3-x) \log (x)\right )}{6 x^2+x^{\frac {1}{x}} \left (3 x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6*x^2 + x^x^(-1)*(3 + x + 4*x^2 + x^3 + (-3 - x)*Log[x]))/(6*x^2 + x^x^(-1)*(3*x^2 + x^3)),x]

[Out]

x + Log[x]/x + Log[3 + x] - 6*Defer[Int][1/(x^2*(6 + x^(1 + x^(-1)) + 3*x^x^(-1))), x] + 6*Log[x]*Defer[Int][1
/(x^2*(6 + x^(1 + x^(-1)) + 3*x^x^(-1))), x] - 6*Defer[Int][1/((3 + x)*(6 + x^(1 + x^(-1)) + 3*x^x^(-1))), x]
- 6*Defer[Int][Defer[Int][1/(x^2*(6 + x^(1 + x^(-1)) + 3*x^x^(-1))), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {6 \left (3+x+x^2-3 \log (x)-x \log (x)\right )}{x^2 (3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}+\frac {3+x+4 x^2+x^3-3 \log (x)-x \log (x)}{x^2 (3+x)}\right ) \, dx\\ &=-\left (6 \int \frac {3+x+x^2-3 \log (x)-x \log (x)}{x^2 (3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx\right )+\int \frac {3+x+4 x^2+x^3-3 \log (x)-x \log (x)}{x^2 (3+x)} \, dx\\ &=-\left (6 \int \left (\frac {3+x+x^2-3 \log (x)-x \log (x)}{3 x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}-\frac {3+x+x^2-3 \log (x)-x \log (x)}{9 x \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}+\frac {3+x+x^2-3 \log (x)-x \log (x)}{9 (3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}\right ) \, dx\right )+\int \left (\frac {3+x+4 x^2+x^3}{x^2 (3+x)}-\frac {\log (x)}{x^2}\right ) \, dx\\ &=\frac {2}{3} \int \frac {3+x+x^2-3 \log (x)-x \log (x)}{x \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-\frac {2}{3} \int \frac {3+x+x^2-3 \log (x)-x \log (x)}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-2 \int \frac {3+x+x^2-3 \log (x)-x \log (x)}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx+\int \frac {3+x+4 x^2+x^3}{x^2 (3+x)} \, dx-\int \frac {\log (x)}{x^2} \, dx\\ &=\frac {1}{x}+\frac {\log (x)}{x}+\frac {2}{3} \int \left (\frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}}+\frac {3}{x \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}+\frac {x}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}}-\frac {\log (x)}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}}-\frac {3 \log (x)}{x \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}\right ) \, dx-\frac {2}{3} \int \left (\frac {3}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}+\frac {x}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}+\frac {x^2}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}-\frac {3 \log (x)}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}-\frac {x \log (x)}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}\right ) \, dx-2 \int \left (\frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}}+\frac {3}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}+\frac {1}{x \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}-\frac {3 \log (x)}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}-\frac {\log (x)}{x \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}\right ) \, dx+\int \left (1+\frac {1}{x^2}+\frac {1}{3+x}\right ) \, dx\\ &=x+\frac {\log (x)}{x}+\log (3+x)+\frac {2}{3} \int \frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx+\frac {2}{3} \int \frac {x}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx-\frac {2}{3} \int \frac {x}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-\frac {2}{3} \int \frac {x^2}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-\frac {2}{3} \int \frac {\log (x)}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx+\frac {2}{3} \int \frac {x \log (x)}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-2 \int \frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx-2 \int \frac {1}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx+2 \int \frac {\log (x)}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-6 \int \frac {1}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx+6 \int \frac {\log (x)}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx\\ &=x+\frac {\log (x)}{x}+\log (3+x)+\frac {2}{3} \int \frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx+\frac {2}{3} \int \frac {x}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx-\frac {2}{3} \int \left (\frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}}-\frac {3}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}\right ) \, dx-\frac {2}{3} \int \left (-\frac {3}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}}+\frac {x}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}}+\frac {9}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )}\right ) \, dx+\frac {2}{3} \int \frac {\int \frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx}{x} \, dx-\frac {2}{3} \int \frac {\int \frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx-3 \int \frac {1}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx}{x} \, dx-2 \int \frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx-2 \int \frac {1}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-2 \int \frac {\int \frac {1}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx}{x} \, dx-6 \int \frac {1}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-6 \int \frac {\int \frac {1}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx}{x} \, dx+(6 \log (x)) \int \frac {1}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx\\ &=x+\frac {\log (x)}{x}+\log (3+x)+\frac {2}{3} \int \frac {\int \frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx}{x} \, dx-\frac {2}{3} \int \left (\frac {\int \frac {1}{6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}} \, dx}{x}-\frac {3 \int \frac {1}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx}{x}\right ) \, dx-2 \int \frac {\int \frac {1}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx}{x} \, dx-6 \int \frac {1}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-6 \int \frac {1}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-6 \int \frac {\int \frac {1}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx}{x} \, dx+(6 \log (x)) \int \frac {1}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx\\ &=x+\frac {\log (x)}{x}+\log (3+x)-6 \int \frac {1}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-6 \int \frac {1}{(3+x) \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx-6 \int \frac {\int \frac {1}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx}{x} \, dx+(6 \log (x)) \int \frac {1}{x^2 \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.11, size = 19, normalized size = 1.19 \begin {gather*} x+\log \left (6+x^{1+\frac {1}{x}}+3 x^{\frac {1}{x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*x^2 + x^x^(-1)*(3 + x + 4*x^2 + x^3 + (-3 - x)*Log[x]))/(6*x^2 + x^x^(-1)*(3*x^2 + x^3)),x]

[Out]

x + Log[6 + x^(1 + x^(-1)) + 3*x^x^(-1)]

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Maple [A]
time = 0.11, size = 21, normalized size = 1.31

method	result	size
risch	\(x +\ln \left (3+x \right )+\ln \left (x^{\frac {1}{x}}+\frac {6}{3+x}\right )\)	\(21\)
norman	\(x +\ln \left ({\mathrm e}^{\frac {\ln \left (x \right )}{x}} x +3 \,{\mathrm e}^{\frac {\ln \left (x \right )}{x}}+6\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+ln(3+x)+ln(x^(1/x)+6/(3+x))

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Maxima [A]
time = 0.31, size = 24, normalized size = 1.50 \begin {gather*} x + \log \left (x + 3\right ) + \log \left (\frac {{\left (x + 3\right )} x^{\left (\frac {1}{x}\right )} + 6}{x + 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x + 3) + log(((x + 3)*x^(1/x) + 6)/(x + 3))

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Fricas [A]
time = 0.37, size = 24, normalized size = 1.50 \begin {gather*} x + \log \left (x + 3\right ) + \log \left (\frac {{\left (x + 3\right )} x^{\left (\frac {1}{x}\right )} + 6}{x + 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x + 3) + log(((x + 3)*x^(1/x) + 6)/(x + 3))

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Sympy [A]
time = 0.24, size = 19, normalized size = 1.19 \begin {gather*} x + \log {\left (x + 3 \right )} + \log {\left (e^{\frac {\log {\left (x \right )}}{x}} + \frac {6}{x + 3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x + 3) + log(exp(log(x)/x) + 6/(x + 3))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (16) = 32\).
time = 0.50, size = 47, normalized size = 2.94 \begin {gather*} x + \frac {x \log \left (x x^{\left (\frac {1}{x}\right )} + 3 \, x^{\left (\frac {1}{x}\right )} + 6\right ) - x \log \left (x + 3\right ) - \log \left (x\right )}{x} + \frac {\log \left (x\right )}{x} + \log \left (x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + (x*log(x*x^(1/x) + 3*x^(1/x) + 6) - x*log(x + 3) - log(x))/x + log(x)/x + log(x + 3)

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Mupad [B]
time = 3.63, size = 29, normalized size = 1.81 \begin {gather*} x+\ln \left (x+3\right )+\ln \left (\frac {x\,x^{1/x}+3\,x^{1/x}+6}{x+3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x + 3) + log((x*x^(1/x) + 3*x^(1/x) + 6)/(x + 3))

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