3.45.11 \(\int \frac {15+27 x+12 x^2-3 x^3-3 x^4+(27 x+24 x^2-9 x^3-12 x^4) \log (x)+(25 x+40 x^2+16 x^3-10 x^4-8 x^5+x^7) \log ^2(x)}{(-15 x-27 x^2-12 x^3+3 x^4+3 x^5) \log (x)+(475 x+1735 x^2+2364 x^3+1234 x^4-222 x^5-512 x^6-141 x^7+39 x^8+20 x^9) \log ^2(x)} \, dx\)

Optimal. Leaf size=34 \[ \log \left (-5+\frac {\frac {3}{5+x \left (4-x^2\right )}+\log (x)}{4 (1+x) \log (x)}\right ) \]

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Rubi [F]  time = 4.99, 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 {15+27 x+12 x^2-3 x^3-3 x^4+\left (27 x+24 x^2-9 x^3-12 x^4\right ) \log (x)+\left (25 x+40 x^2+16 x^3-10 x^4-8 x^5+x^7\right ) \log ^2(x)}{\left (-15 x-27 x^2-12 x^3+3 x^4+3 x^5\right ) \log (x)+\left (475 x+1735 x^2+2364 x^3+1234 x^4-222 x^5-512 x^6-141 x^7+39 x^8+20 x^9\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(15 + 27*x + 12*x^2 - 3*x^3 - 3*x^4 + (27*x + 24*x^2 - 9*x^3 - 12*x^4)*Log[x] + (25*x + 40*x^2 + 16*x^3 -
10*x^4 - 8*x^5 + x^7)*Log[x]^2)/((-15*x - 27*x^2 - 12*x^3 + 3*x^4 + 3*x^5)*Log[x] + (475*x + 1735*x^2 + 2364*x
^3 + 1234*x^4 - 222*x^5 - 512*x^6 - 141*x^7 + 39*x^8 + 20*x^9)*Log[x]^2),x]

[Out]

-Log[1 + x] + Log[19 + 20*x] - Log[Log[x]] - 176*Defer[Int][(3 - 95*Log[x] - 176*x*Log[x] - 80*x^2*Log[x] + 19
*x^3*Log[x] + 20*x^4*Log[x])^(-1), x] - 95*Defer[Int][1/(x*(3 - 95*Log[x] - 176*x*Log[x] - 80*x^2*Log[x] + 19*
x^3*Log[x] + 20*x^4*Log[x])), x] - 80*Defer[Int][x/(3 - 95*Log[x] - 176*x*Log[x] - 80*x^2*Log[x] + 19*x^3*Log[
x] + 20*x^4*Log[x]), x] + 19*Defer[Int][x^2/(3 - 95*Log[x] - 176*x*Log[x] - 80*x^2*Log[x] + 19*x^3*Log[x] + 20
*x^4*Log[x]), x] + 20*Defer[Int][x^3/(3 - 95*Log[x] - 176*x*Log[x] - 80*x^2*Log[x] + 19*x^3*Log[x] + 20*x^4*Lo
g[x]), x] - 60*Defer[Int][1/((19 + 20*x)*(3 - 95*Log[x] - 176*x*Log[x] - 80*x^2*Log[x] + 19*x^3*Log[x] + 20*x^
4*Log[x])), x] + 12*Defer[Int][1/((-5 - 4*x + x^3)*(3 - 95*Log[x] - 176*x*Log[x] - 80*x^2*Log[x] + 19*x^3*Log[
x] + 20*x^4*Log[x])), x] - 9*Defer[Int][x^2/((-5 - 4*x + x^3)*(3 - 95*Log[x] - 176*x*Log[x] - 80*x^2*Log[x] +
19*x^3*Log[x] + 20*x^4*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-5-9 x-4 x^2+x^3+x^4\right )+3 x \left (-9-8 x+3 x^2+4 x^3\right ) \log (x)-x \left (-5-4 x+x^3\right )^2 \log ^2(x)}{x \left (5+9 x+4 x^2-x^3-x^4\right ) \log (x) \left (3+\left (-95-176 x-80 x^2+19 x^3+20 x^4\right ) \log (x)\right )} \, dx\\ &=\int \left (\frac {1}{19+39 x+20 x^2}-\frac {1}{x \log (x)}-\frac {3 \left (-176-160 x+57 x^2+80 x^3\right )}{(19+20 x) \left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}+\frac {-95-176 x-80 x^2+19 x^3+20 x^4}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}\right ) \, dx\\ &=-\left (3 \int \frac {-176-160 x+57 x^2+80 x^3}{(19+20 x) \left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx\right )+\int \frac {1}{19+39 x+20 x^2} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {-95-176 x-80 x^2+19 x^3+20 x^4}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx\\ &=-\left (3 \int \left (\frac {20}{(19+20 x) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}+\frac {-4+3 x^2}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}\right ) \, dx\right )+20 \int \frac {1}{19+20 x} \, dx-20 \int \frac {1}{20+20 x} \, dx+\int \left (-\frac {176}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)}-\frac {95}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}-\frac {80 x}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)}+\frac {19 x^2}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)}+\frac {20 x^3}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)}\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\log (1+x)+\log (19+20 x)-\log (\log (x))-3 \int \frac {-4+3 x^2}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx+19 \int \frac {x^2}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx+20 \int \frac {x^3}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-60 \int \frac {1}{(19+20 x) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-80 \int \frac {x}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-95 \int \frac {1}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-176 \int \frac {1}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx\\ &=-\log (1+x)+\log (19+20 x)-\log (\log (x))-3 \int \left (-\frac {4}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}+\frac {3 x^2}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}\right ) \, dx+19 \int \frac {x^2}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx+20 \int \frac {x^3}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-60 \int \frac {1}{(19+20 x) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-80 \int \frac {x}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-95 \int \frac {1}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-176 \int \frac {1}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx\\ &=-\log (1+x)+\log (19+20 x)-\log (\log (x))-9 \int \frac {x^2}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx+12 \int \frac {1}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx+19 \int \frac {x^2}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx+20 \int \frac {x^3}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-60 \int \frac {1}{(19+20 x) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-80 \int \frac {x}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-95 \int \frac {1}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-176 \int \frac {1}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 11.71, size = 70, normalized size = 2.06 \begin {gather*} -\log (1+x)+\log (19+20 x)-\log \left ((19+20 x) \left (5+4 x-x^3\right )\right )-\log (\log (x))+\log \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15 + 27*x + 12*x^2 - 3*x^3 - 3*x^4 + (27*x + 24*x^2 - 9*x^3 - 12*x^4)*Log[x] + (25*x + 40*x^2 + 16*
x^3 - 10*x^4 - 8*x^5 + x^7)*Log[x]^2)/((-15*x - 27*x^2 - 12*x^3 + 3*x^4 + 3*x^5)*Log[x] + (475*x + 1735*x^2 +
2364*x^3 + 1234*x^4 - 222*x^5 - 512*x^6 - 141*x^7 + 39*x^8 + 20*x^9)*Log[x]^2),x]

[Out]

-Log[1 + x] + Log[19 + 20*x] - Log[(19 + 20*x)*(5 + 4*x - x^3)] - Log[Log[x]] + Log[3 - 95*Log[x] - 176*x*Log[
x] - 80*x^2*Log[x] + 19*x^3*Log[x] + 20*x^4*Log[x]]

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fricas [B]  time = 0.52, size = 67, normalized size = 1.97 \begin {gather*} \log \left (20 \, x + 19\right ) - \log \left (x + 1\right ) + \log \left (\frac {{\left (20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95\right )} \log \relax (x) + 3}{20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95}\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^7-8*x^5-10*x^4+16*x^3+40*x^2+25*x)*log(x)^2+(-12*x^4-9*x^3+24*x^2+27*x)*log(x)-3*x^4-3*x^3+12*x^
2+27*x+15)/((20*x^9+39*x^8-141*x^7-512*x^6-222*x^5+1234*x^4+2364*x^3+1735*x^2+475*x)*log(x)^2+(3*x^5+3*x^4-12*
x^3-27*x^2-15*x)*log(x)),x, algorithm="fricas")

[Out]

log(20*x + 19) - log(x + 1) + log(((20*x^4 + 19*x^3 - 80*x^2 - 176*x - 95)*log(x) + 3)/(20*x^4 + 19*x^3 - 80*x
^2 - 176*x - 95)) - log(log(x))

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giac [A]  time = 0.16, size = 58, normalized size = 1.71 \begin {gather*} \log \left (20 \, x^{4} \log \relax (x) + 19 \, x^{3} \log \relax (x) - 80 \, x^{2} \log \relax (x) - 176 \, x \log \relax (x) - 95 \, \log \relax (x) + 3\right ) - \log \left (x^{4} + x^{3} - 4 \, x^{2} - 9 \, x - 5\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^7-8*x^5-10*x^4+16*x^3+40*x^2+25*x)*log(x)^2+(-12*x^4-9*x^3+24*x^2+27*x)*log(x)-3*x^4-3*x^3+12*x^
2+27*x+15)/((20*x^9+39*x^8-141*x^7-512*x^6-222*x^5+1234*x^4+2364*x^3+1735*x^2+475*x)*log(x)^2+(3*x^5+3*x^4-12*
x^3-27*x^2-15*x)*log(x)),x, algorithm="giac")

[Out]

log(20*x^4*log(x) + 19*x^3*log(x) - 80*x^2*log(x) - 176*x*log(x) - 95*log(x) + 3) - log(x^4 + x^3 - 4*x^2 - 9*
x - 5) - log(log(x))

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maple [A]  time = 0.04, size = 47, normalized size = 1.38

method	result	size
risch	\(-\ln \left (x +1\right )+\ln \left (20 x +19\right )-\ln \left (\ln \relax (x )\right )+\ln \left (\ln \relax (x )+\frac {3}{20 x^{4}+19 x^{3}-80 x^{2}-176 x -95}\right )\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^7-8*x^5-10*x^4+16*x^3+40*x^2+25*x)*ln(x)^2+(-12*x^4-9*x^3+24*x^2+27*x)*ln(x)-3*x^4-3*x^3+12*x^2+27*x+1
5)/((20*x^9+39*x^8-141*x^7-512*x^6-222*x^5+1234*x^4+2364*x^3+1735*x^2+475*x)*ln(x)^2+(3*x^5+3*x^4-12*x^3-27*x^
2-15*x)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-ln(x+1)+ln(20*x+19)-ln(ln(x))+ln(ln(x)+3/(20*x^4+19*x^3-80*x^2-176*x-95))

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maxima [B]  time = 0.40, size = 67, normalized size = 1.97 \begin {gather*} \log \left (20 \, x + 19\right ) - \log \left (x + 1\right ) + \log \left (\frac {{\left (20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95\right )} \log \relax (x) + 3}{20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95}\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^7-8*x^5-10*x^4+16*x^3+40*x^2+25*x)*log(x)^2+(-12*x^4-9*x^3+24*x^2+27*x)*log(x)-3*x^4-3*x^3+12*x^
2+27*x+15)/((20*x^9+39*x^8-141*x^7-512*x^6-222*x^5+1234*x^4+2364*x^3+1735*x^2+475*x)*log(x)^2+(3*x^5+3*x^4-12*
x^3-27*x^2-15*x)*log(x)),x, algorithm="maxima")

[Out]

log(20*x + 19) - log(x + 1) + log(((20*x^4 + 19*x^3 - 80*x^2 - 176*x - 95)*log(x) + 3)/(20*x^4 + 19*x^3 - 80*x
^2 - 176*x - 95)) - log(log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {27\,x+\ln \relax (x)\,\left (-12\,x^4-9\,x^3+24\,x^2+27\,x\right )+{\ln \relax (x)}^2\,\left (x^7-8\,x^5-10\,x^4+16\,x^3+40\,x^2+25\,x\right )+12\,x^2-3\,x^3-3\,x^4+15}{\ln \relax (x)\,\left (-3\,x^5-3\,x^4+12\,x^3+27\,x^2+15\,x\right )-{\ln \relax (x)}^2\,\left (20\,x^9+39\,x^8-141\,x^7-512\,x^6-222\,x^5+1234\,x^4+2364\,x^3+1735\,x^2+475\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(27*x + log(x)*(27*x + 24*x^2 - 9*x^3 - 12*x^4) + log(x)^2*(25*x + 40*x^2 + 16*x^3 - 10*x^4 - 8*x^5 + x^7
) + 12*x^2 - 3*x^3 - 3*x^4 + 15)/(log(x)*(15*x + 27*x^2 + 12*x^3 - 3*x^4 - 3*x^5) - log(x)^2*(475*x + 1735*x^2
+ 2364*x^3 + 1234*x^4 - 222*x^5 - 512*x^6 - 141*x^7 + 39*x^8 + 20*x^9)),x)

[Out]

int(-(27*x + log(x)*(27*x + 24*x^2 - 9*x^3 - 12*x^4) + log(x)^2*(25*x + 40*x^2 + 16*x^3 - 10*x^4 - 8*x^5 + x^7
) + 12*x^2 - 3*x^3 - 3*x^4 + 15)/(log(x)*(15*x + 27*x^2 + 12*x^3 - 3*x^4 - 3*x^5) - log(x)^2*(475*x + 1735*x^2
+ 2364*x^3 + 1234*x^4 - 222*x^5 - 512*x^6 - 141*x^7 + 39*x^8 + 20*x^9)), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**7-8*x**5-10*x**4+16*x**3+40*x**2+25*x)*ln(x)**2+(-12*x**4-9*x**3+24*x**2+27*x)*ln(x)-3*x**4-3*x
**3+12*x**2+27*x+15)/((20*x**9+39*x**8-141*x**7-512*x**6-222*x**5+1234*x**4+2364*x**3+1735*x**2+475*x)*ln(x)**
2+(3*x**5+3*x**4-12*x**3-27*x**2-15*x)*ln(x)),x)

[Out]

Exception raised: PolynomialError

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