3.17.5 \(\int \frac {-10 x-x^5+(15 x+2 x^4) \log (x^2)+(-10-x^3) \log ^2(x^2)}{2 x^5-4 x^4 \log (x^2)+2 x^3 \log ^2(x^2)} \, dx\)

Optimal. Leaf size=30 \[ \frac {1}{2} \left (-x+\frac {5 \left (x-\frac {x^2}{x-\log \left (x^2\right )}\right )}{x^3}\right ) \]

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Rubi [F]  time = 0.53, 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 {-10 x-x^5+\left (15 x+2 x^4\right ) \log \left (x^2\right )+\left (-10-x^3\right ) \log ^2\left (x^2\right )}{2 x^5-4 x^4 \log \left (x^2\right )+2 x^3 \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-10*x - x^5 + (15*x + 2*x^4)*Log[x^2] + (-10 - x^3)*Log[x^2]^2)/(2*x^5 - 4*x^4*Log[x^2] + 2*x^3*Log[x^2]^
2),x]

[Out]

5/(2*x^2) - x/2 - 5*Defer[Int][1/(x^2*(x - Log[x^2])^2), x] + (5*Defer[Int][1/(x*(x - Log[x^2])^2), x])/2 + (5
*Defer[Int][1/(x^2*(x - Log[x^2])), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-10 x-x^5+\left (15 x+2 x^4\right ) \log \left (x^2\right )+\left (-10-x^3\right ) \log ^2\left (x^2\right )}{2 x^3 \left (x-\log \left (x^2\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {-10 x-x^5+\left (15 x+2 x^4\right ) \log \left (x^2\right )+\left (-10-x^3\right ) \log ^2\left (x^2\right )}{x^3 \left (x-\log \left (x^2\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {-10-x^3}{x^3}+\frac {5 (-2+x)}{x^2 \left (x-\log \left (x^2\right )\right )^2}+\frac {5}{x^2 \left (x-\log \left (x^2\right )\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-10-x^3}{x^3} \, dx+\frac {5}{2} \int \frac {-2+x}{x^2 \left (x-\log \left (x^2\right )\right )^2} \, dx+\frac {5}{2} \int \frac {1}{x^2 \left (x-\log \left (x^2\right )\right )} \, dx\\ &=\frac {1}{2} \int \left (-1-\frac {10}{x^3}\right ) \, dx+\frac {5}{2} \int \left (-\frac {2}{x^2 \left (x-\log \left (x^2\right )\right )^2}+\frac {1}{x \left (x-\log \left (x^2\right )\right )^2}\right ) \, dx+\frac {5}{2} \int \frac {1}{x^2 \left (x-\log \left (x^2\right )\right )} \, dx\\ &=\frac {5}{2 x^2}-\frac {x}{2}+\frac {5}{2} \int \frac {1}{x \left (x-\log \left (x^2\right )\right )^2} \, dx+\frac {5}{2} \int \frac {1}{x^2 \left (x-\log \left (x^2\right )\right )} \, dx-5 \int \frac {1}{x^2 \left (x-\log \left (x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 28, normalized size = 0.93 \begin {gather*} \frac {1}{2} \left (\frac {5}{x^2}-x+\frac {5}{x \left (-x+\log \left (x^2\right )\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10*x - x^5 + (15*x + 2*x^4)*Log[x^2] + (-10 - x^3)*Log[x^2]^2)/(2*x^5 - 4*x^4*Log[x^2] + 2*x^3*Log
[x^2]^2),x]

[Out]

(5/x^2 - x + 5/(x*(-x + Log[x^2])))/2

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fricas [A]  time = 0.55, size = 32, normalized size = 1.07 \begin {gather*} -\frac {x^{4} - {\left (x^{3} - 5\right )} \log \left (x^{2}\right )}{2 \, {\left (x^{3} - x^{2} \log \left (x^{2}\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-10)*log(x^2)^2+(2*x^4+15*x)*log(x^2)-x^5-10*x)/(2*x^3*log(x^2)^2-4*x^4*log(x^2)+2*x^5),x, alg
orithm="fricas")

[Out]

-1/2*(x^4 - (x^3 - 5)*log(x^2))/(x^3 - x^2*log(x^2))

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giac [A]  time = 0.28, size = 24, normalized size = 0.80 \begin {gather*} -\frac {1}{2} \, x - \frac {5}{2 \, {\left (x^{2} - x \log \left (x^{2}\right )\right )}} + \frac {5}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-10)*log(x^2)^2+(2*x^4+15*x)*log(x^2)-x^5-10*x)/(2*x^3*log(x^2)^2-4*x^4*log(x^2)+2*x^5),x, alg
orithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 27, normalized size = 0.90

method	result	size
risch	\(-\frac {x^{3}-5}{2 x^{2}}-\frac {5}{2 x \left (-\ln \left (x^{2}\right )+x \right )}\)	\(27\)
norman	\(\frac {\frac {x^{3} \ln \left (x^{2}\right )}{2}-\frac {x^{4}}{2}-\frac {5 \ln \left (x^{2}\right )}{2}}{x^{2} \left (-\ln \left (x^{2}\right )+x \right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3-10)*ln(x^2)^2+(2*x^4+15*x)*ln(x^2)-x^5-10*x)/(2*x^3*ln(x^2)^2-4*x^4*ln(x^2)+2*x^5),x,method=_RETURN
VERBOSE)

[Out]

-1/2*(x^3-5)/x^2-5/2/x/(-ln(x^2)+x)

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maxima [A]  time = 0.38, size = 28, normalized size = 0.93 \begin {gather*} -\frac {x^{4} - 2 \, {\left (x^{3} - 5\right )} \log \relax (x)}{2 \, {\left (x^{3} - 2 \, x^{2} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-10)*log(x^2)^2+(2*x^4+15*x)*log(x^2)-x^5-10*x)/(2*x^3*log(x^2)^2-4*x^4*log(x^2)+2*x^5),x, alg
orithm="maxima")

[Out]

-1/2*(x^4 - 2*(x^3 - 5)*log(x))/(x^3 - 2*x^2*log(x))

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mupad [B]  time = 1.15, size = 23, normalized size = 0.77 \begin {gather*} -\frac {x}{2}-\frac {5\,\ln \left (x^2\right )}{2\,x^2\,\left (x-\ln \left (x^2\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*x - log(x^2)*(15*x + 2*x^4) + log(x^2)^2*(x^3 + 10) + x^5)/(2*x^5 - 4*x^4*log(x^2) + 2*x^3*log(x^2)^2
),x)

[Out]

- x/2 - (5*log(x^2))/(2*x^2*(x - log(x^2)))

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sympy [A]  time = 0.12, size = 24, normalized size = 0.80 \begin {gather*} - \frac {x}{2} + \frac {5}{- 2 x^{2} + 2 x \log {\left (x^{2} \right )}} + \frac {5}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3-10)*ln(x**2)**2+(2*x**4+15*x)*ln(x**2)-x**5-10*x)/(2*x**3*ln(x**2)**2-4*x**4*ln(x**2)+2*x**5
),x)

[Out]

-x/2 + 5/(-2*x**2 + 2*x*log(x**2)) + 5/(2*x**2)

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