3.4.37 \(\int \frac {88 x-21 x^2-10 x^3+2 x^4+(-96+19 x+11 x^2-2 x^3) \log (3+x)+(-12 x-x^2+x^3+(12+x-x^2) \log (3+x)) \log (\frac {x-\log (3+x)}{(-4 x+x^2) \log (5)})}{60 x+5 x^2-5 x^3+(-60-5 x+5 x^2) \log (3+x)} \, dx\)

Optimal. Leaf size=35 \[ -5+x-\frac {1}{5} x \left (-2+x+\log \left (\frac {-x+\log (3+x)}{(4-x) x \log (5)}\right )\right ) \]

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Rubi [F]  time = 3.27, 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 {88 x-21 x^2-10 x^3+2 x^4+\left (-96+19 x+11 x^2-2 x^3\right ) \log (3+x)+\left (-12 x-x^2+x^3+\left (12+x-x^2\right ) \log (3+x)\right ) \log \left (\frac {x-\log (3+x)}{\left (-4 x+x^2\right ) \log (5)}\right )}{60 x+5 x^2-5 x^3+\left (-60-5 x+5 x^2\right ) \log (3+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(88*x - 21*x^2 - 10*x^3 + 2*x^4 + (-96 + 19*x + 11*x^2 - 2*x^3)*Log[3 + x] + (-12*x - x^2 + x^3 + (12 + x
- x^2)*Log[3 + x])*Log[(x - Log[3 + x])/((-4*x + x^2)*Log[5])])/(60*x + 5*x^2 - 5*x^3 + (-60 - 5*x + 5*x^2)*Lo
g[3 + x]),x]

[Out]

(7*x)/5 - x^2/5 - (x*Log[-((x - Log[3 + x])/((4 - x)*x*Log[5]))])/5 + Defer[Int][(x - Log[3 + x])^(-1), x]/5 +
Defer[Int][(-x + Log[3 + x])^(-1), x]/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {88 x-21 x^2-10 x^3+2 x^4+\left (-96+19 x+11 x^2-2 x^3\right ) \log (3+x)+\left (-12 x-x^2+x^3+\left (12+x-x^2\right ) \log (3+x)\right ) \log \left (\frac {x-\log (3+x)}{\left (-4 x+x^2\right ) \log (5)}\right )}{5 \left (12+x-x^2\right ) (x-\log (3+x))} \, dx\\ &=\frac {1}{5} \int \frac {88 x-21 x^2-10 x^3+2 x^4+\left (-96+19 x+11 x^2-2 x^3\right ) \log (3+x)+\left (-12 x-x^2+x^3+\left (12+x-x^2\right ) \log (3+x)\right ) \log \left (\frac {x-\log (3+x)}{\left (-4 x+x^2\right ) \log (5)}\right )}{\left (12+x-x^2\right ) (x-\log (3+x))} \, dx\\ &=\frac {1}{5} \int \left (-\frac {88 x}{(-4+x) (3+x) (x-\log (3+x))}+\frac {21 x^2}{(-4+x) (3+x) (x-\log (3+x))}+\frac {10 x^3}{(-4+x) (3+x) (x-\log (3+x))}-\frac {2 x^4}{(-4+x) (3+x) (x-\log (3+x))}+\frac {\left (32-17 x+2 x^2\right ) \log (3+x)}{(-4+x) (x-\log (3+x))}-\log \left (\frac {x-\log (3+x)}{(-4+x) x \log (5)}\right )\right ) \, dx\\ &=\frac {1}{5} \int \frac {\left (32-17 x+2 x^2\right ) \log (3+x)}{(-4+x) (x-\log (3+x))} \, dx-\frac {1}{5} \int \log \left (\frac {x-\log (3+x)}{(-4+x) x \log (5)}\right ) \, dx-\frac {2}{5} \int \frac {x^4}{(-4+x) (3+x) (x-\log (3+x))} \, dx+2 \int \frac {x^3}{(-4+x) (3+x) (x-\log (3+x))} \, dx+\frac {21}{5} \int \frac {x^2}{(-4+x) (3+x) (x-\log (3+x))} \, dx-\frac {88}{5} \int \frac {x}{(-4+x) (3+x) (x-\log (3+x))} \, dx\\ &=-\frac {1}{5} x \log \left (-\frac {x-\log (3+x)}{(4-x) x \log (5)}\right )+\frac {1}{5} \int \left (\frac {-32+17 x-2 x^2}{-4+x}+\frac {x \left (32-17 x+2 x^2\right )}{(-4+x) (x-\log (3+x))}\right ) \, dx+\frac {1}{5} \int \frac {-x \left (-4+4 x+x^2\right )+2 \left (-6+x+x^2\right ) \log (3+x)}{(-4+x) (3+x) (x-\log (3+x))} \, dx-\frac {2}{5} \int \left (\frac {13}{x-\log (3+x)}+\frac {256}{7 (-4+x) (x-\log (3+x))}+\frac {x}{x-\log (3+x)}+\frac {x^2}{x-\log (3+x)}-\frac {81}{7 (3+x) (x-\log (3+x))}\right ) \, dx+2 \int \left (\frac {1}{x-\log (3+x)}+\frac {64}{7 (-4+x) (x-\log (3+x))}+\frac {x}{x-\log (3+x)}+\frac {27}{7 (3+x) (x-\log (3+x))}\right ) \, dx+\frac {21}{5} \int \left (\frac {1}{x-\log (3+x)}+\frac {16}{7 (-4+x) (x-\log (3+x))}-\frac {9}{7 (3+x) (x-\log (3+x))}\right ) \, dx-\frac {88}{5} \int \left (\frac {4}{7 (-4+x) (x-\log (3+x))}+\frac {3}{7 (3+x) (x-\log (3+x))}\right ) \, dx\\ &=-\frac {1}{5} x \log \left (-\frac {x-\log (3+x)}{(4-x) x \log (5)}\right )+\frac {1}{5} \int \frac {-32+17 x-2 x^2}{-4+x} \, dx+\frac {1}{5} \int \left (-\frac {2 (-2+x)}{-4+x}+\frac {x (2+x)}{(3+x) (x-\log (3+x))}\right ) \, dx+\frac {1}{5} \int \frac {x \left (32-17 x+2 x^2\right )}{(-4+x) (x-\log (3+x))} \, dx-\frac {2}{5} \int \frac {x}{x-\log (3+x)} \, dx-\frac {2}{5} \int \frac {x^2}{x-\log (3+x)} \, dx+2 \int \frac {1}{x-\log (3+x)} \, dx+2 \int \frac {x}{x-\log (3+x)} \, dx+\frac {21}{5} \int \frac {1}{x-\log (3+x)} \, dx+\frac {162}{35} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx-\frac {26}{5} \int \frac {1}{x-\log (3+x)} \, dx-\frac {27}{5} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx-\frac {264}{35} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx+\frac {54}{7} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx+\frac {48}{5} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx-\frac {352}{35} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx-\frac {512}{35} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx+\frac {128}{7} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx\\ &=-\frac {1}{5} x \log \left (-\frac {x-\log (3+x)}{(4-x) x \log (5)}\right )+\frac {1}{5} \int \left (9+\frac {4}{-4+x}-2 x\right ) \, dx+\frac {1}{5} \int \left (-\frac {4}{x-\log (3+x)}-\frac {16}{(-4+x) (x-\log (3+x))}-\frac {9 x}{x-\log (3+x)}+\frac {2 x^2}{x-\log (3+x)}\right ) \, dx+\frac {1}{5} \int \frac {x (2+x)}{(3+x) (x-\log (3+x))} \, dx-\frac {2}{5} \int \frac {-2+x}{-4+x} \, dx-\frac {2}{5} \int \frac {x}{x-\log (3+x)} \, dx-\frac {2}{5} \int \frac {x^2}{x-\log (3+x)} \, dx+2 \int \frac {1}{x-\log (3+x)} \, dx+2 \int \frac {x}{x-\log (3+x)} \, dx+\frac {21}{5} \int \frac {1}{x-\log (3+x)} \, dx+\frac {162}{35} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx-\frac {26}{5} \int \frac {1}{x-\log (3+x)} \, dx-\frac {27}{5} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx-\frac {264}{35} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx+\frac {54}{7} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx+\frac {48}{5} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx-\frac {352}{35} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx-\frac {512}{35} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx+\frac {128}{7} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx\\ &=\frac {9 x}{5}-\frac {x^2}{5}+\frac {4}{5} \log (4-x)-\frac {1}{5} x \log \left (-\frac {x-\log (3+x)}{(4-x) x \log (5)}\right )+\frac {1}{5} \int \left (\frac {x}{x-\log (3+x)}+\frac {3}{(3+x) (x-\log (3+x))}+\frac {1}{-x+\log (3+x)}\right ) \, dx-\frac {2}{5} \int \left (1+\frac {2}{-4+x}\right ) \, dx-\frac {2}{5} \int \frac {x}{x-\log (3+x)} \, dx-\frac {4}{5} \int \frac {1}{x-\log (3+x)} \, dx-\frac {9}{5} \int \frac {x}{x-\log (3+x)} \, dx+2 \int \frac {1}{x-\log (3+x)} \, dx+2 \int \frac {x}{x-\log (3+x)} \, dx-\frac {16}{5} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx+\frac {21}{5} \int \frac {1}{x-\log (3+x)} \, dx+\frac {162}{35} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx-\frac {26}{5} \int \frac {1}{x-\log (3+x)} \, dx-\frac {27}{5} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx-\frac {264}{35} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx+\frac {54}{7} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx+\frac {48}{5} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx-\frac {352}{35} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx-\frac {512}{35} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx+\frac {128}{7} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx\\ &=\frac {7 x}{5}-\frac {x^2}{5}-\frac {1}{5} x \log \left (-\frac {x-\log (3+x)}{(4-x) x \log (5)}\right )+\frac {1}{5} \int \frac {x}{x-\log (3+x)} \, dx+\frac {1}{5} \int \frac {1}{-x+\log (3+x)} \, dx-\frac {2}{5} \int \frac {x}{x-\log (3+x)} \, dx+\frac {3}{5} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx-\frac {4}{5} \int \frac {1}{x-\log (3+x)} \, dx-\frac {9}{5} \int \frac {x}{x-\log (3+x)} \, dx+2 \int \frac {1}{x-\log (3+x)} \, dx+2 \int \frac {x}{x-\log (3+x)} \, dx-\frac {16}{5} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx+\frac {21}{5} \int \frac {1}{x-\log (3+x)} \, dx+\frac {162}{35} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx-\frac {26}{5} \int \frac {1}{x-\log (3+x)} \, dx-\frac {27}{5} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx-\frac {264}{35} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx+\frac {54}{7} \int \frac {1}{(3+x) (x-\log (3+x))} \, dx+\frac {48}{5} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx-\frac {352}{35} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx-\frac {512}{35} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx+\frac {128}{7} \int \frac {1}{(-4+x) (x-\log (3+x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 38, normalized size = 1.09 \begin {gather*} \frac {1}{5} \left (7 x-x^2-x \log \left (\frac {x-\log (3+x)}{(-4+x) x \log (5)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(88*x - 21*x^2 - 10*x^3 + 2*x^4 + (-96 + 19*x + 11*x^2 - 2*x^3)*Log[3 + x] + (-12*x - x^2 + x^3 + (1
2 + x - x^2)*Log[3 + x])*Log[(x - Log[3 + x])/((-4*x + x^2)*Log[5])])/(60*x + 5*x^2 - 5*x^3 + (-60 - 5*x + 5*x
^2)*Log[3 + x]),x]

[Out]

(7*x - x^2 - x*Log[(x - Log[3 + x])/((-4 + x)*x*Log[5])])/5

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fricas [A]  time = 0.76, size = 35, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, x^{2} - \frac {1}{5} \, x \log \left (\frac {x - \log \left (x + 3\right )}{{\left (x^{2} - 4 \, x\right )} \log \relax (5)}\right ) + \frac {7}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x+12)*log(3+x)+x^3-x^2-12*x)*log((-log(3+x)+x)/(x^2-4*x)/log(5))+(-2*x^3+11*x^2+19*x-96)*log
(3+x)+2*x^4-10*x^3-21*x^2+88*x)/((5*x^2-5*x-60)*log(3+x)-5*x^3+5*x^2+60*x),x, algorithm="fricas")

[Out]

-1/5*x^2 - 1/5*x*log((x - log(x + 3))/((x^2 - 4*x)*log(5))) + 7/5*x

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giac [A]  time = 0.50, size = 37, normalized size = 1.06 \begin {gather*} -\frac {1}{5} \, x^{2} + \frac {1}{5} \, x \log \left (x^{2} \log \relax (5) - 4 \, x \log \relax (5)\right ) - \frac {1}{5} \, x \log \left (x - \log \left (x + 3\right )\right ) + \frac {7}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x+12)*log(3+x)+x^3-x^2-12*x)*log((-log(3+x)+x)/(x^2-4*x)/log(5))+(-2*x^3+11*x^2+19*x-96)*log
(3+x)+2*x^4-10*x^3-21*x^2+88*x)/((5*x^2-5*x-60)*log(3+x)-5*x^3+5*x^2+60*x),x, algorithm="giac")

[Out]

-1/5*x^2 + 1/5*x*log(x^2*log(5) - 4*x*log(5)) - 1/5*x*log(x - log(x + 3)) + 7/5*x

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maple [C]  time = 0.26, size = 330, normalized size = 9.43

method	result	size
risch	\(-\frac {x \ln \left (-\ln \left (3+x \right )+x \right )}{5}+\frac {x \ln \left (x -4\right )}{5}+\frac {x \ln \relax (x )}{5}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{\left (x -4\right ) x}\right )}{10}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x -4}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right )^{2}}{10}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (\ln \left (3+x \right )-x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right )^{2}}{10}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{\left (x -4\right ) x}\right )^{3}}{10}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{\left (x -4\right ) x}\right )^{2}}{10}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right )^{3}}{10}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x -4}\right ) \mathrm {csgn}\left (i \left (\ln \left (3+x \right )-x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{x -4}\right )}{10}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (3+x \right )-x \right )}{\left (x -4\right ) x}\right )^{2}}{10}+\frac {x \ln \left (\ln \relax (5)\right )}{5}-\frac {x^{2}}{5}+\frac {7 x}{5}\)	\(330\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2+x+12)*ln(3+x)+x^3-x^2-12*x)*ln((-ln(3+x)+x)/(x^2-4*x)/ln(5))+(-2*x^3+11*x^2+19*x-96)*ln(3+x)+2*x^4
-10*x^3-21*x^2+88*x)/((5*x^2-5*x-60)*ln(3+x)-5*x^3+5*x^2+60*x),x,method=_RETURNVERBOSE)

[Out]

-1/5*x*ln(-ln(3+x)+x)+1/5*x*ln(x-4)+1/5*x*ln(x)+1/10*I*Pi*x*csgn(I/x)*csgn(I*(ln(3+x)-x)/(x-4))*csgn(I*(ln(3+x
)-x)/(x-4)/x)-1/10*I*Pi*x*csgn(I/(x-4))*csgn(I*(ln(3+x)-x)/(x-4))^2+1/10*I*Pi*x*csgn(I*(ln(3+x)-x))*csgn(I*(ln
(3+x)-x)/(x-4))^2-1/10*I*Pi*x*csgn(I*(ln(3+x)-x)/(x-4)/x)^3+1/10*I*Pi*x*csgn(I*(ln(3+x)-x)/(x-4))*csgn(I*(ln(3
+x)-x)/(x-4)/x)^2-1/10*I*Pi*x*csgn(I*(ln(3+x)-x)/(x-4))^3+1/10*I*Pi*x*csgn(I/(x-4))*csgn(I*(ln(3+x)-x))*csgn(I
*(ln(3+x)-x)/(x-4))-1/10*I*Pi*x*csgn(I/x)*csgn(I*(ln(3+x)-x)/(x-4)/x)^2+1/5*x*ln(ln(5))-1/5*x^2+7/5*x

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maxima [A]  time = 0.66, size = 38, normalized size = 1.09 \begin {gather*} -\frac {1}{5} \, x^{2} + \frac {1}{5} \, x {\left (\log \left (\log \relax (5)\right ) + 7\right )} - \frac {1}{5} \, x \log \left (x - \log \left (x + 3\right )\right ) + \frac {1}{5} \, x \log \left (x - 4\right ) + \frac {1}{5} \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x+12)*log(3+x)+x^3-x^2-12*x)*log((-log(3+x)+x)/(x^2-4*x)/log(5))+(-2*x^3+11*x^2+19*x-96)*log
(3+x)+2*x^4-10*x^3-21*x^2+88*x)/((5*x^2-5*x-60)*log(3+x)-5*x^3+5*x^2+60*x),x, algorithm="maxima")

[Out]

-1/5*x^2 + 1/5*x*(log(log(5)) + 7) - 1/5*x*log(x - log(x + 3)) + 1/5*x*log(x - 4) + 1/5*x*log(x)

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mupad [B]  time = 0.89, size = 32, normalized size = 0.91 \begin {gather*} -\frac {x\,\left (x+\ln \left (-\frac {x-\ln \left (x+3\right )}{\ln \relax (5)\,\left (4\,x-x^2\right )}\right )-7\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((88*x - log(-(x - log(x + 3))/(log(5)*(4*x - x^2)))*(12*x - log(x + 3)*(x - x^2 + 12) + x^2 - x^3) + log(x
+ 3)*(19*x + 11*x^2 - 2*x^3 - 96) - 21*x^2 - 10*x^3 + 2*x^4)/(60*x - log(x + 3)*(5*x - 5*x^2 + 60) + 5*x^2 -
5*x^3),x)

[Out]

-(x*(x + log(-(x - log(x + 3))/(log(5)*(4*x - x^2))) - 7))/5

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sympy [B]  time = 1.16, size = 54, normalized size = 1.54 \begin {gather*} - \frac {x^{2}}{5} + \frac {7 x}{5} + \left (\frac {1}{60} - \frac {x}{5}\right ) \log {\left (\frac {x - \log {\left (x + 3 \right )}}{\left (x^{2} - 4 x\right ) \log {\relax (5 )}} \right )} - \frac {\log {\left (- x + \log {\left (x + 3 \right )} \right )}}{60} + \frac {\log {\left (x^{2} - 4 x \right )}}{60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2+x+12)*ln(3+x)+x**3-x**2-12*x)*ln((-ln(3+x)+x)/(x**2-4*x)/ln(5))+(-2*x**3+11*x**2+19*x-96)*l
n(3+x)+2*x**4-10*x**3-21*x**2+88*x)/((5*x**2-5*x-60)*ln(3+x)-5*x**3+5*x**2+60*x),x)

[Out]

-x**2/5 + 7*x/5 + (1/60 - x/5)*log((x - log(x + 3))/((x**2 - 4*x)*log(5))) - log(-x + log(x + 3))/60 + log(x**
2 - 4*x)/60

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