3.79.30 \(\int \frac {-90 x-3 x^2+3 x^3+(-44 x-33 x^2) \log (4+3 x)+(-120-94 x+x^2+3 x^3) \log (4+3 x) \log (\frac {6-x}{(10+2 x) \log (4+3 x)})}{(-120-94 x+x^2+3 x^3) \log (4+3 x) \log ^2(\frac {6-x}{(10+2 x) \log (4+3 x)})} \, dx\)

Optimal. Leaf size=30 \[ \frac {x}{\log \left (\frac {\frac {3 (2-x)}{2}+x}{(5+x) \log (4+3 x)}\right )} \]

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Rubi [F]  time = 6.35, 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 {-90 x-3 x^2+3 x^3+\left (-44 x-33 x^2\right ) \log (4+3 x)+\left (-120-94 x+x^2+3 x^3\right ) \log (4+3 x) \log \left (\frac {6-x}{(10+2 x) \log (4+3 x)}\right )}{\left (-120-94 x+x^2+3 x^3\right ) \log (4+3 x) \log ^2\left (\frac {6-x}{(10+2 x) \log (4+3 x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-90*x - 3*x^2 + 3*x^3 + (-44*x - 33*x^2)*Log[4 + 3*x] + (-120 - 94*x + x^2 + 3*x^3)*Log[4 + 3*x]*Log[(6 -
x)/((10 + 2*x)*Log[4 + 3*x])])/((-120 - 94*x + x^2 + 3*x^3)*Log[4 + 3*x]*Log[(6 - x)/((10 + 2*x)*Log[4 + 3*x]
)]^2),x]

[Out]

-6*Defer[Int][1/((-6 + x)*Log[-1/2*(-6 + x)/((5 + x)*Log[4 + 3*x])]^2), x] - 5*Defer[Int][1/((5 + x)*Log[-1/2*
(-6 + x)/((5 + x)*Log[4 + 3*x])]^2), x] + Defer[Int][1/(Log[4 + 3*x]*Log[-1/2*(-6 + x)/((5 + x)*Log[4 + 3*x])]
^2), x] - 4*Defer[Int][1/((4 + 3*x)*Log[4 + 3*x]*Log[-1/2*(-6 + x)/((5 + x)*Log[4 + 3*x])]^2), x] + Defer[Int]
[Log[-1/2*(-6 + x)/((5 + x)*Log[4 + 3*x])]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {x \left (-90-3 x+3 x^2-44 \log (4+3 x)-33 x \log (4+3 x)\right )}{(-6+x) (5+x) (4+3 x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}+\frac {1}{\log \left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}\right ) \, dx\\ &=\int \frac {x \left (-90-3 x+3 x^2-44 \log (4+3 x)-33 x \log (4+3 x)\right )}{(-6+x) (5+x) (4+3 x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx+\int \frac {1}{\log \left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx\\ &=\int \left (\frac {3 \left (-90-3 x+3 x^2-44 \log (4+3 x)-33 x \log (4+3 x)\right )}{121 (-6+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}-\frac {5 \left (-90-3 x+3 x^2-44 \log (4+3 x)-33 x \log (4+3 x)\right )}{121 (5+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}+\frac {6 \left (-90-3 x+3 x^2-44 \log (4+3 x)-33 x \log (4+3 x)\right )}{121 (4+3 x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}\right ) \, dx+\int \frac {1}{\log \left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx\\ &=\frac {3}{121} \int \frac {-90-3 x+3 x^2-44 \log (4+3 x)-33 x \log (4+3 x)}{(-6+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx-\frac {5}{121} \int \frac {-90-3 x+3 x^2-44 \log (4+3 x)-33 x \log (4+3 x)}{(5+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx+\frac {6}{121} \int \frac {-90-3 x+3 x^2-44 \log (4+3 x)-33 x \log (4+3 x)}{(4+3 x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx+\int \frac {1}{\log \left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx\\ &=\frac {3}{121} \int \left (-\frac {44}{(-6+x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}-\frac {33 x}{(-6+x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}-\frac {90}{(-6+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}-\frac {3 x}{(-6+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}+\frac {3 x^2}{(-6+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}\right ) \, dx-\frac {5}{121} \int \left (-\frac {44}{(5+x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}-\frac {33 x}{(5+x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}-\frac {90}{(5+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}-\frac {3 x}{(5+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}+\frac {3 x^2}{(5+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}\right ) \, dx+\frac {6}{121} \int \left (-\frac {44}{(4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}-\frac {33 x}{(4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}-\frac {90}{(4+3 x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}-\frac {3 x}{(4+3 x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}+\frac {3 x^2}{(4+3 x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )}\right ) \, dx+\int \frac {1}{\log \left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx\\ &=-\left (\frac {9}{121} \int \frac {x}{(-6+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx\right )+\frac {9}{121} \int \frac {x^2}{(-6+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx+\frac {15}{121} \int \frac {x}{(5+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx-\frac {15}{121} \int \frac {x^2}{(5+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx-\frac {18}{121} \int \frac {x}{(4+3 x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx+\frac {18}{121} \int \frac {x^2}{(4+3 x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx-\frac {9}{11} \int \frac {x}{(-6+x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx-\frac {12}{11} \int \frac {1}{(-6+x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx+\frac {15}{11} \int \frac {x}{(5+x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx-\frac {18}{11} \int \frac {x}{(4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx+\frac {20}{11} \int \frac {1}{(5+x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx-\frac {24}{11} \int \frac {1}{(4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx-\frac {270}{121} \int \frac {1}{(-6+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx+\frac {450}{121} \int \frac {1}{(5+x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx-\frac {540}{121} \int \frac {1}{(4+3 x) \log (4+3 x) \log ^2\left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx+\int \frac {1}{\log \left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 25, normalized size = 0.83 \begin {gather*} \frac {x}{\log \left (-\frac {-6+x}{2 (5+x) \log (4+3 x)}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-90*x - 3*x^2 + 3*x^3 + (-44*x - 33*x^2)*Log[4 + 3*x] + (-120 - 94*x + x^2 + 3*x^3)*Log[4 + 3*x]*Lo
g[(6 - x)/((10 + 2*x)*Log[4 + 3*x])])/((-120 - 94*x + x^2 + 3*x^3)*Log[4 + 3*x]*Log[(6 - x)/((10 + 2*x)*Log[4
+ 3*x])]^2),x]

[Out]

x/Log[-1/2*(-6 + x)/((5 + x)*Log[4 + 3*x])]

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fricas [A]  time = 0.71, size = 23, normalized size = 0.77 \begin {gather*} \frac {x}{\log \left (-\frac {x - 6}{2 \, {\left (x + 5\right )} \log \left (3 \, x + 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+x^2-94*x-120)*log(4+3*x)*log((-x+6)/(2*x+10)/log(4+3*x))+(-33*x^2-44*x)*log(4+3*x)+3*x^3-3*x
^2-90*x)/(3*x^3+x^2-94*x-120)/log(4+3*x)/log((-x+6)/(2*x+10)/log(4+3*x))^2,x, algorithm="fricas")

[Out]

x/log(-1/2*(x - 6)/((x + 5)*log(3*x + 4)))

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giac [A]  time = 0.44, size = 33, normalized size = 1.10 \begin {gather*} -\frac {x}{\log \left (2 \, x \log \left (3 \, x + 4\right ) + 10 \, \log \left (3 \, x + 4\right )\right ) - \log \left (-x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+x^2-94*x-120)*log(4+3*x)*log((-x+6)/(2*x+10)/log(4+3*x))+(-33*x^2-44*x)*log(4+3*x)+3*x^3-3*x
^2-90*x)/(3*x^3+x^2-94*x-120)/log(4+3*x)/log((-x+6)/(2*x+10)/log(4+3*x))^2,x, algorithm="giac")

[Out]

-x/(log(2*x*log(3*x + 4) + 10*log(3*x + 4)) - log(-x + 6))

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maple [C]  time = 0.38, size = 304, normalized size = 10.13

method	result	size
risch	\(-\frac {2 i x}{\pi \,\mathrm {csgn}\left (i \left (x -6\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -6\right )}{5+x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (x -6\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -6\right )}{5+x}\right ) \mathrm {csgn}\left (\frac {i}{5+x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (4+3 x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (x -6\right )}{5+x}\right ) \mathrm {csgn}\left (\frac {i \left (x -6\right )}{\ln \left (4+3 x \right ) \left (5+x \right )}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (4+3 x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (x -6\right )}{\ln \left (4+3 x \right ) \left (5+x \right )}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left (x -6\right )}{5+x}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left (x -6\right )}{5+x}\right )^{2} \mathrm {csgn}\left (\frac {i}{5+x}\right )+\pi \,\mathrm {csgn}\left (\frac {i \left (x -6\right )}{5+x}\right ) \mathrm {csgn}\left (\frac {i \left (x -6\right )}{\ln \left (4+3 x \right ) \left (5+x \right )}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (x -6\right )}{\ln \left (4+3 x \right ) \left (5+x \right )}\right )^{3}-2 \pi \mathrm {csgn}\left (\frac {i \left (x -6\right )}{\ln \left (4+3 x \right ) \left (5+x \right )}\right )^{2}+2 \pi +2 i \ln \relax (2)-2 i \ln \left (x -6\right )+2 i \ln \left (\ln \left (4+3 x \right )\right )+2 i \ln \left (5+x \right )}\)	\(304\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^3+x^2-94*x-120)*ln(4+3*x)*ln((-x+6)/(2*x+10)/ln(4+3*x))+(-33*x^2-44*x)*ln(4+3*x)+3*x^3-3*x^2-90*x)/(
3*x^3+x^2-94*x-120)/ln(4+3*x)/ln((-x+6)/(2*x+10)/ln(4+3*x))^2,x,method=_RETURNVERBOSE)

[Out]

-2*I*x/(Pi*csgn(I*(x-6))*csgn(I/(5+x)*(x-6))^2-Pi*csgn(I*(x-6))*csgn(I/(5+x)*(x-6))*csgn(I/(5+x))-Pi*csgn(I/ln
(4+3*x))*csgn(I/(5+x)*(x-6))*csgn(I/ln(4+3*x)*(x-6)/(5+x))+Pi*csgn(I/ln(4+3*x))*csgn(I/ln(4+3*x)*(x-6)/(5+x))^
2-Pi*csgn(I/(5+x)*(x-6))^3+Pi*csgn(I/(5+x)*(x-6))^2*csgn(I/(5+x))+Pi*csgn(I/(5+x)*(x-6))*csgn(I/ln(4+3*x)*(x-6
)/(5+x))^2+Pi*csgn(I/ln(4+3*x)*(x-6)/(5+x))^3-2*Pi*csgn(I/ln(4+3*x)*(x-6)/(5+x))^2+2*Pi+2*I*ln(2)-2*I*ln(x-6)+
2*I*ln(ln(4+3*x))+2*I*ln(5+x))

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maxima [A]  time = 0.52, size = 27, normalized size = 0.90 \begin {gather*} -\frac {x}{\log \relax (2) + \log \left (x + 5\right ) - \log \left (-x + 6\right ) + \log \left (\log \left (3 \, x + 4\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+x^2-94*x-120)*log(4+3*x)*log((-x+6)/(2*x+10)/log(4+3*x))+(-33*x^2-44*x)*log(4+3*x)+3*x^3-3*x
^2-90*x)/(3*x^3+x^2-94*x-120)/log(4+3*x)/log((-x+6)/(2*x+10)/log(4+3*x))^2,x, algorithm="maxima")

[Out]

-x/(log(2) + log(x + 5) - log(-x + 6) + log(log(3*x + 4)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {90\,x+\ln \left (3\,x+4\right )\,\left (33\,x^2+44\,x\right )+3\,x^2-3\,x^3+\ln \left (-\frac {x-6}{\ln \left (3\,x+4\right )\,\left (2\,x+10\right )}\right )\,\ln \left (3\,x+4\right )\,\left (-3\,x^3-x^2+94\,x+120\right )}{{\ln \left (-\frac {x-6}{\ln \left (3\,x+4\right )\,\left (2\,x+10\right )}\right )}^2\,\ln \left (3\,x+4\right )\,\left (-3\,x^3-x^2+94\,x+120\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((90*x + log(3*x + 4)*(44*x + 33*x^2) + 3*x^2 - 3*x^3 + log(-(x - 6)/(log(3*x + 4)*(2*x + 10)))*log(3*x + 4
)*(94*x - x^2 - 3*x^3 + 120))/(log(-(x - 6)/(log(3*x + 4)*(2*x + 10)))^2*log(3*x + 4)*(94*x - x^2 - 3*x^3 + 12
0)),x)

[Out]

int((90*x + log(3*x + 4)*(44*x + 33*x^2) + 3*x^2 - 3*x^3 + log(-(x - 6)/(log(3*x + 4)*(2*x + 10)))*log(3*x + 4
)*(94*x - x^2 - 3*x^3 + 120))/(log(-(x - 6)/(log(3*x + 4)*(2*x + 10)))^2*log(3*x + 4)*(94*x - x^2 - 3*x^3 + 12
0)), x)

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sympy [A]  time = 0.49, size = 17, normalized size = 0.57 \begin {gather*} \frac {x}{\log {\left (\frac {6 - x}{\left (2 x + 10\right ) \log {\left (3 x + 4 \right )}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**3+x**2-94*x-120)*ln(4+3*x)*ln((-x+6)/(2*x+10)/ln(4+3*x))+(-33*x**2-44*x)*ln(4+3*x)+3*x**3-3*x
**2-90*x)/(3*x**3+x**2-94*x-120)/ln(4+3*x)/ln((-x+6)/(2*x+10)/ln(4+3*x))**2,x)

[Out]

x/log((6 - x)/((2*x + 10)*log(3*x + 4)))

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