3.60.64 \(\int \frac {e (-30 x^4-5 x^5)+(x^3+30 x^5+5 x^6) \log ^2(x)+(e (60 x^2+10 x^3)+(-x-60 x^3-10 x^4) \log ^2(x)) \log (2 x)+(e (-30-5 x)+(30 x+5 x^2) \log ^2(x)) \log ^2(2 x)+(6+x-12 x^2-2 x^3) \log ^2(x) \log (6+x)}{(30 x^5+5 x^6) \log ^2(x)+(-60 x^3-10 x^4) \log ^2(x) \log (2 x)+(30 x+5 x^2) \log ^2(x) \log ^2(2 x)} \, dx\)

Optimal. Leaf size=28 \[ x+\frac {e}{\log (x)}+\frac {\log (6+x)}{5 \left (x^2-\log (2 x)\right )} \]

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Rubi [F]  time = 4.74, 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 {e \left (-30 x^4-5 x^5\right )+\left (x^3+30 x^5+5 x^6\right ) \log ^2(x)+\left (e \left (60 x^2+10 x^3\right )+\left (-x-60 x^3-10 x^4\right ) \log ^2(x)\right ) \log (2 x)+\left (e (-30-5 x)+\left (30 x+5 x^2\right ) \log ^2(x)\right ) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log ^2(x) \log (6+x)}{\left (30 x^5+5 x^6\right ) \log ^2(x)+\left (-60 x^3-10 x^4\right ) \log ^2(x) \log (2 x)+\left (30 x+5 x^2\right ) \log ^2(x) \log ^2(2 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E*(-30*x^4 - 5*x^5) + (x^3 + 30*x^5 + 5*x^6)*Log[x]^2 + (E*(60*x^2 + 10*x^3) + (-x - 60*x^3 - 10*x^4)*Log
[x]^2)*Log[2*x] + (E*(-30 - 5*x) + (30*x + 5*x^2)*Log[x]^2)*Log[2*x]^2 + (6 + x - 12*x^2 - 2*x^3)*Log[x]^2*Log
[6 + x])/((30*x^5 + 5*x^6)*Log[x]^2 + (-60*x^3 - 10*x^4)*Log[x]^2*Log[2*x] + (30*x + 5*x^2)*Log[x]^2*Log[2*x]^
2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\frac {5 e}{\log ^2(x)}+\frac {x^3+30 x^5+5 x^6-\left (x+60 x^3+10 x^4\right ) \log (2 x)+5 x (6+x) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log (6+x)}{(6+x) \left (x^2-\log (2 x)\right )^2}}{5 x} \, dx\\ &=\frac {1}{5} \int \frac {-\frac {5 e}{\log ^2(x)}+\frac {x^3+30 x^5+5 x^6-\left (x+60 x^3+10 x^4\right ) \log (2 x)+5 x (6+x) \log ^2(2 x)+\left (6+x-12 x^2-2 x^3\right ) \log (6+x)}{(6+x) \left (x^2-\log (2 x)\right )^2}}{x} \, dx\\ &=\frac {1}{5} \int \left (-\frac {5 e}{x \log ^2(x)}+\frac {x^2}{(6+x) \left (x^2-\log (2 x)\right )^2}+\frac {30 x^4}{(6+x) \left (x^2-\log (2 x)\right )^2}+\frac {5 x^5}{(6+x) \left (x^2-\log (2 x)\right )^2}-\frac {\left (1+60 x^2+10 x^3\right ) \log (2 x)}{(6+x) \left (x^2-\log (2 x)\right )^2}+\frac {5 \log ^2(2 x)}{\left (x^2-\log (2 x)\right )^2}-\frac {\left (-1+2 x^2\right ) \log (6+x)}{x \left (x^2-\log (2 x)\right )^2}\right ) \, dx\\ &=\frac {1}{5} \int \frac {x^2}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx-\frac {1}{5} \int \frac {\left (1+60 x^2+10 x^3\right ) \log (2 x)}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx-\frac {1}{5} \int \frac {\left (-1+2 x^2\right ) \log (6+x)}{x \left (x^2-\log (2 x)\right )^2} \, dx+6 \int \frac {x^4}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx-e \int \frac {1}{x \log ^2(x)} \, dx+\int \frac {x^5}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx+\int \frac {\log ^2(2 x)}{\left (x^2-\log (2 x)\right )^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {6}{\left (x^2-\log (2 x)\right )^2}+\frac {x}{\left (x^2-\log (2 x)\right )^2}+\frac {36}{(6+x) \left (x^2-\log (2 x)\right )^2}\right ) \, dx-\frac {1}{5} \int \left (\frac {x^2 \left (1+60 x^2+10 x^3\right )}{(6+x) \left (x^2-\log (2 x)\right )^2}+\frac {-1-60 x^2-10 x^3}{(6+x) \left (x^2-\log (2 x)\right )}\right ) \, dx-\frac {1}{5} \int \left (-\frac {\log (6+x)}{x \left (x^2-\log (2 x)\right )^2}+\frac {2 x \log (6+x)}{\left (x^2-\log (2 x)\right )^2}\right ) \, dx+6 \int \left (-\frac {216}{\left (x^2-\log (2 x)\right )^2}+\frac {36 x}{\left (x^2-\log (2 x)\right )^2}-\frac {6 x^2}{\left (x^2-\log (2 x)\right )^2}+\frac {x^3}{\left (x^2-\log (2 x)\right )^2}+\frac {1296}{(6+x) \left (x^2-\log (2 x)\right )^2}\right ) \, dx-e \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+\int \left (\frac {1296}{\left (x^2-\log (2 x)\right )^2}-\frac {216 x}{\left (x^2-\log (2 x)\right )^2}+\frac {36 x^2}{\left (x^2-\log (2 x)\right )^2}-\frac {6 x^3}{\left (x^2-\log (2 x)\right )^2}+\frac {x^4}{\left (x^2-\log (2 x)\right )^2}-\frac {7776}{(6+x) \left (x^2-\log (2 x)\right )^2}\right ) \, dx+\int \left (1+\frac {x^4}{\left (x^2-\log (2 x)\right )^2}-\frac {2 x^2}{x^2-\log (2 x)}\right ) \, dx\\ &=x+\frac {e}{\log (x)}+\frac {1}{5} \int \frac {x}{\left (x^2-\log (2 x)\right )^2} \, dx-\frac {1}{5} \int \frac {x^2 \left (1+60 x^2+10 x^3\right )}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx-\frac {1}{5} \int \frac {-1-60 x^2-10 x^3}{(6+x) \left (x^2-\log (2 x)\right )} \, dx+\frac {1}{5} \int \frac {\log (6+x)}{x \left (x^2-\log (2 x)\right )^2} \, dx-\frac {2}{5} \int \frac {x \log (6+x)}{\left (x^2-\log (2 x)\right )^2} \, dx-\frac {6}{5} \int \frac {1}{\left (x^2-\log (2 x)\right )^2} \, dx-2 \int \frac {x^2}{x^2-\log (2 x)} \, dx+\frac {36}{5} \int \frac {1}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx+2 \int \frac {x^4}{\left (x^2-\log (2 x)\right )^2} \, dx\\ &=x+\frac {e}{\log (x)}-\frac {1}{5} \int \left (-\frac {6}{\left (x^2-\log (2 x)\right )^2}+\frac {x}{\left (x^2-\log (2 x)\right )^2}+\frac {10 x^4}{\left (x^2-\log (2 x)\right )^2}+\frac {36}{(6+x) \left (x^2-\log (2 x)\right )^2}\right ) \, dx-\frac {1}{5} \int \left (-\frac {10 x^2}{x^2-\log (2 x)}-\frac {1}{(6+x) \left (x^2-\log (2 x)\right )}\right ) \, dx+\frac {1}{5} \int \frac {x}{\left (x^2-\log (2 x)\right )^2} \, dx+\frac {1}{5} \int \frac {\log (6+x)}{x \left (x^2-\log (2 x)\right )^2} \, dx-\frac {2}{5} \int \frac {x \log (6+x)}{\left (x^2-\log (2 x)\right )^2} \, dx-\frac {6}{5} \int \frac {1}{\left (x^2-\log (2 x)\right )^2} \, dx-2 \int \frac {x^2}{x^2-\log (2 x)} \, dx+\frac {36}{5} \int \frac {1}{(6+x) \left (x^2-\log (2 x)\right )^2} \, dx+2 \int \frac {x^4}{\left (x^2-\log (2 x)\right )^2} \, dx\\ &=x+\frac {e}{\log (x)}+\frac {1}{5} \int \frac {1}{(6+x) \left (x^2-\log (2 x)\right )} \, dx+\frac {1}{5} \int \frac {\log (6+x)}{x \left (x^2-\log (2 x)\right )^2} \, dx-\frac {2}{5} \int \frac {x \log (6+x)}{\left (x^2-\log (2 x)\right )^2} \, dx-2 \int \frac {x^4}{\left (x^2-\log (2 x)\right )^2} \, dx+2 \int \frac {x^4}{\left (x^2-\log (2 x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 33, normalized size = 1.18 \begin {gather*} \frac {1}{5} \left (5 x+\frac {5 e}{\log (x)}-\frac {\log (6+x)}{-x^2+\log (2 x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E*(-30*x^4 - 5*x^5) + (x^3 + 30*x^5 + 5*x^6)*Log[x]^2 + (E*(60*x^2 + 10*x^3) + (-x - 60*x^3 - 10*x^
4)*Log[x]^2)*Log[2*x] + (E*(-30 - 5*x) + (30*x + 5*x^2)*Log[x]^2)*Log[2*x]^2 + (6 + x - 12*x^2 - 2*x^3)*Log[x]
^2*Log[6 + x])/((30*x^5 + 5*x^6)*Log[x]^2 + (-60*x^3 - 10*x^4)*Log[x]^2*Log[2*x] + (30*x + 5*x^2)*Log[x]^2*Log
[2*x]^2),x]

[Out]

(5*x + (5*E)/Log[x] - Log[6 + x]/(-x^2 + Log[2*x]))/5

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fricas [B]  time = 0.68, size = 67, normalized size = 2.39 \begin {gather*} \frac {5 \, x^{2} e - 5 \, x \log \relax (x)^{2} - 5 \, e \log \relax (2) + 5 \, {\left (x^{3} - x \log \relax (2) - e\right )} \log \relax (x) + \log \left (x + 6\right ) \log \relax (x)}{5 \, {\left ({\left (x^{2} - \log \relax (2)\right )} \log \relax (x) - \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^2+30*x)*log(x)^2+(-5*x-30)*exp(1))*log(2*x)^2+((-10*x^4-60*x^3-x)*log(x)^2+(10*x^3+60*x^2)*ex
p(1))*log(2*x)+(-2*x^3-12*x^2+x+6)*log(x)^2*log(x+6)+(5*x^6+30*x^5+x^3)*log(x)^2+(-5*x^5-30*x^4)*exp(1))/((5*x
^2+30*x)*log(x)^2*log(2*x)^2+(-10*x^4-60*x^3)*log(x)^2*log(2*x)+(5*x^6+30*x^5)*log(x)^2),x, algorithm="fricas"
)

[Out]

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

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giac [B]  time = 0.25, size = 71, normalized size = 2.54 \begin {gather*} \frac {5 \, x^{3} \log \relax (x) + 5 \, x^{2} e - 5 \, x \log \relax (2) \log \relax (x) - 5 \, x \log \relax (x)^{2} - 5 \, e \log \relax (2) - 5 \, e \log \relax (x) + \log \left (x + 6\right ) \log \relax (x)}{5 \, {\left (x^{2} \log \relax (x) - \log \relax (2) \log \relax (x) - \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^2+30*x)*log(x)^2+(-5*x-30)*exp(1))*log(2*x)^2+((-10*x^4-60*x^3-x)*log(x)^2+(10*x^3+60*x^2)*ex
p(1))*log(2*x)+(-2*x^3-12*x^2+x+6)*log(x)^2*log(x+6)+(5*x^6+30*x^5+x^3)*log(x)^2+(-5*x^5-30*x^4)*exp(1))/((5*x
^2+30*x)*log(x)^2*log(2*x)^2+(-10*x^4-60*x^3)*log(x)^2*log(2*x)+(5*x^6+30*x^5)*log(x)^2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.37, size = 40, normalized size = 1.43

method	result	size
risch	\(-\frac {2 i \ln \left (x +6\right )}{5 \left (-2 i x^{2}+2 i \ln \relax (2)+2 i \ln \relax (x )\right )}+\frac {x \ln \relax (x )+{\mathrm e}}{\ln \relax (x )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((5*x^2+30*x)*ln(x)^2+(-5*x-30)*exp(1))*ln(2*x)^2+((-10*x^4-60*x^3-x)*ln(x)^2+(10*x^3+60*x^2)*exp(1))*ln(
2*x)+(-2*x^3-12*x^2+x+6)*ln(x)^2*ln(x+6)+(5*x^6+30*x^5+x^3)*ln(x)^2+(-5*x^5-30*x^4)*exp(1))/((5*x^2+30*x)*ln(x
)^2*ln(2*x)^2+(-10*x^4-60*x^3)*ln(x)^2*ln(2*x)+(5*x^6+30*x^5)*ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

-2/5*I/(-2*I*x^2+2*I*ln(2)+2*I*ln(x))*ln(x+6)+(x*ln(x)+exp(1))/ln(x)

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maxima [B]  time = 0.48, size = 67, normalized size = 2.39 \begin {gather*} \frac {5 \, x^{2} e - 5 \, x \log \relax (x)^{2} - 5 \, e \log \relax (2) + 5 \, {\left (x^{3} - x \log \relax (2) - e\right )} \log \relax (x) + \log \left (x + 6\right ) \log \relax (x)}{5 \, {\left ({\left (x^{2} - \log \relax (2)\right )} \log \relax (x) - \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^2+30*x)*log(x)^2+(-5*x-30)*exp(1))*log(2*x)^2+((-10*x^4-60*x^3-x)*log(x)^2+(10*x^3+60*x^2)*ex
p(1))*log(2*x)+(-2*x^3-12*x^2+x+6)*log(x)^2*log(x+6)+(5*x^6+30*x^5+x^3)*log(x)^2+(-5*x^5-30*x^4)*exp(1))/((5*x
^2+30*x)*log(x)^2*log(2*x)^2+(-10*x^4-60*x^3)*log(x)^2*log(2*x)+(5*x^6+30*x^5)*log(x)^2),x, algorithm="maxima"
)

[Out]

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

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mupad [B]  time = 4.74, size = 29, normalized size = 1.04 \begin {gather*} x+\frac {\mathrm {e}}{\ln \relax (x)}-\frac {\ln \left (x+6\right )}{5\,\left (\ln \left (2\,x\right )-x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(x^3 + 30*x^5 + 5*x^6) + log(2*x)^2*(log(x)^2*(30*x + 5*x^2) - exp(1)*(5*x + 30)) + log(2*x)*(ex
p(1)*(60*x^2 + 10*x^3) - log(x)^2*(x + 60*x^3 + 10*x^4)) - exp(1)*(30*x^4 + 5*x^5) + log(x + 6)*log(x)^2*(x -
12*x^2 - 2*x^3 + 6))/(log(x)^2*(30*x^5 + 5*x^6) + log(2*x)^2*log(x)^2*(30*x + 5*x^2) - log(2*x)*log(x)^2*(60*x
^3 + 10*x^4)),x)

[Out]

x + exp(1)/log(x) - log(x + 6)/(5*(log(2*x) - x^2))

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sympy [A]  time = 0.65, size = 27, normalized size = 0.96 \begin {gather*} x + \frac {e}{\log {\relax (x )}} + \frac {\log {\left (x + 6 \right )}}{5 x^{2} - 5 \log {\relax (x )} - 5 \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x**2+30*x)*ln(x)**2+(-5*x-30)*exp(1))*ln(2*x)**2+((-10*x**4-60*x**3-x)*ln(x)**2+(10*x**3+60*x**
2)*exp(1))*ln(2*x)+(-2*x**3-12*x**2+x+6)*ln(x)**2*ln(x+6)+(5*x**6+30*x**5+x**3)*ln(x)**2+(-5*x**5-30*x**4)*exp
(1))/((5*x**2+30*x)*ln(x)**2*ln(2*x)**2+(-10*x**4-60*x**3)*ln(x)**2*ln(2*x)+(5*x**6+30*x**5)*ln(x)**2),x)

[Out]

x + E/log(x) + log(x + 6)/(5*x**2 - 5*log(x) - 5*log(2))

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