\(\int \frac {45+(72 x^2+16 x^3) \log ^2(2)+(15 x^4+4 x^5) \log ^4(2)+\log (6)}{225 x^2+240 x^4 \log ^2(2)+94 x^6 \log ^4(2)+16 x^8 \log ^6(2)+x^{10} \log ^8(2)+(-30 x-16 x^3 \log ^2(2)-2 x^5 \log ^4(2)) \log (6)+\log ^2(6)} \, dx\) [5638]

Optimal result

Integrand size = 109, antiderivative size = 25 \[ \int \frac {45+\left (72 x^2+16 x^3\right ) \log ^2(2)+\left (15 x^4+4 x^5\right ) \log ^4(2)+\log (6)}{225 x^2+240 x^4 \log ^2(2)+94 x^6 \log ^4(2)+16 x^8 \log ^6(2)+x^{10} \log ^8(2)+\left (-30 x-16 x^3 \log ^2(2)-2 x^5 \log ^4(2)\right ) \log (6)+\log ^2(6)} \, dx=\frac {3+x}{x-x \left (4+x^2 \log ^2(2)\right )^2+\log (6)} \]

[Out]

(3+x)/(x-(x^2*ln(2)^2+4)^2*x+ln(6))

Rubi [F]

\[ \int \frac {45+\left (72 x^2+16 x^3\right ) \log ^2(2)+\left (15 x^4+4 x^5\right ) \log ^4(2)+\log (6)}{225 x^2+240 x^4 \log ^2(2)+94 x^6 \log ^4(2)+16 x^8 \log ^6(2)+x^{10} \log ^8(2)+\left (-30 x-16 x^3 \log ^2(2)-2 x^5 \log ^4(2)\right ) \log (6)+\log ^2(6)} \, dx=\int \frac {45+\left (72 x^2+16 x^3\right ) \log ^2(2)+\left (15 x^4+4 x^5\right ) \log ^4(2)+\log (6)}{225 x^2+240 x^4 \log ^2(2)+94 x^6 \log ^4(2)+16 x^8 \log ^6(2)+x^{10} \log ^8(2)+\left (-30 x-16 x^3 \log ^2(2)-2 x^5 \log ^4(2)\right ) \log (6)+\log ^2(6)} \, dx \]

[In]

Int[(45 + (72*x^2 + 16*x^3)*Log[2]^2 + (15*x^4 + 4*x^5)*Log[2]^4 + Log[6])/(225*x^2 + 240*x^4*Log[2]^2 + 94*x^
6*Log[2]^4 + 16*x^8*Log[2]^6 + x^10*Log[2]^8 + (-30*x - 16*x^3*Log[2]^2 - 2*x^5*Log[2]^4)*Log[6] + Log[6]^2),x
]

[Out]

-3/(15*x + 8*x^3*Log[2]^2 + x^5*Log[2]^4 - Log[6]) + 5*Log[6]*Defer[Int][(15*x + 8*x^3*Log[2]^2 + x^5*Log[2]^4
- Log[6])^(-2), x] - 60*Defer[Int][x/(15*x + 8*x^3*Log[2]^2 + x^5*Log[2]^4 - Log[6])^2, x] - 16*Log[2]^2*Defe
r[Int][x^3/(15*x + 8*x^3*Log[2]^2 + x^5*Log[2]^4 - Log[6])^2, x] + 4*Defer[Int][(15*x + 8*x^3*Log[2]^2 + x^5*L
og[2]^4 - Log[6])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4}{15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)}+\frac {-60 x+72 x^2 \log ^2(2)-16 x^3 \log ^2(2)+15 x^4 \log ^4(2)+5 (9+\log (6))}{\left (15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)\right )^2}\right ) \, dx \\ & = 4 \int \frac {1}{15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)} \, dx+\int \frac {-60 x+72 x^2 \log ^2(2)-16 x^3 \log ^2(2)+15 x^4 \log ^4(2)+5 (9+\log (6))}{\left (15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)\right )^2} \, dx \\ & = -\frac {3}{15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)}+4 \int \frac {1}{15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)} \, dx+\frac {\int \frac {-300 x \log ^4(2)-80 x^3 \log ^6(2)+25 \log ^4(2) \log (6)}{\left (15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)\right )^2} \, dx}{5 \log ^4(2)} \\ & = -\frac {3}{15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)}+4 \int \frac {1}{15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)} \, dx+\frac {\int \left (-\frac {300 x \log ^4(2)}{\left (15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)\right )^2}-\frac {80 x^3 \log ^6(2)}{\left (15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)\right )^2}+\frac {25 \log ^4(2) \log (6)}{\left (15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)\right )^2}\right ) \, dx}{5 \log ^4(2)} \\ & = -\frac {3}{15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)}+4 \int \frac {1}{15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)} \, dx-60 \int \frac {x}{\left (15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)\right )^2} \, dx-\left (16 \log ^2(2)\right ) \int \frac {x^3}{\left (15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)\right )^2} \, dx+(5 \log (6)) \int \frac {1}{\left (15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {45+\left (72 x^2+16 x^3\right ) \log ^2(2)+\left (15 x^4+4 x^5\right ) \log ^4(2)+\log (6)}{225 x^2+240 x^4 \log ^2(2)+94 x^6 \log ^4(2)+16 x^8 \log ^6(2)+x^{10} \log ^8(2)+\left (-30 x-16 x^3 \log ^2(2)-2 x^5 \log ^4(2)\right ) \log (6)+\log ^2(6)} \, dx=\frac {-3-x}{15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)} \]

[In]

Integrate[(45 + (72*x^2 + 16*x^3)*Log[2]^2 + (15*x^4 + 4*x^5)*Log[2]^4 + Log[6])/(225*x^2 + 240*x^4*Log[2]^2 +
94*x^6*Log[2]^4 + 16*x^8*Log[2]^6 + x^10*Log[2]^8 + (-30*x - 16*x^3*Log[2]^2 - 2*x^5*Log[2]^4)*Log[6] + Log[6
]^2),x]

[Out]

(-3 - x)/(15*x + 8*x^3*Log[2]^2 + x^5*Log[2]^4 - Log[6])

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32

method	result	size
gosper	\(-\frac {3+x}{x^{5} \ln \left (2\right )^{4}+8 x^{3} \ln \left (2\right )^{2}-\ln \left (6\right )+15 x}\)	\(33\)
default	\(\frac {-3-x}{x^{5} \ln \left (2\right )^{4}+8 x^{3} \ln \left (2\right )^{2}-\ln \left (6\right )+15 x}\)	\(34\)
risch	\(\frac {-3-x}{x^{5} \ln \left (2\right )^{4}+8 x^{3} \ln \left (2\right )^{2}-\ln \left (2\right )-\ln \left (3\right )+15 x}\)	\(38\)
parallelrisch	\(\frac {-45+x^{5} \ln \left (2\right )^{4}+8 x^{3} \ln \left (2\right )^{2}-\ln \left (6\right )}{15 x^{5} \ln \left (2\right )^{4}+120 x^{3} \ln \left (2\right )^{2}-15 \ln \left (6\right )+225 x}\)	\(53\)
norman	\(\frac {-\frac {\left (\ln \left (6\right )+45\right ) x}{\ln \left (6\right )}-\frac {24 \ln \left (2\right )^{2} x^{3}}{\ln \left (6\right )}-\frac {3 \ln \left (2\right )^{4} x^{5}}{\ln \left (6\right )}}{x^{5} \ln \left (2\right )^{4}+8 x^{3} \ln \left (2\right )^{2}-\ln \left (6\right )+15 x}\)	\(67\)

[In]

int((ln(6)+(4*x^5+15*x^4)*ln(2)^4+(16*x^3+72*x^2)*ln(2)^2+45)/(ln(6)^2+(-2*x^5*ln(2)^4-16*x^3*ln(2)^2-30*x)*ln
(6)+x^10*ln(2)^8+16*x^8*ln(2)^6+94*x^6*ln(2)^4+240*x^4*ln(2)^2+225*x^2),x,method=_RETURNVERBOSE)

[Out]

-(3+x)/(x^5*ln(2)^4+8*x^3*ln(2)^2-ln(6)+15*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {45+\left (72 x^2+16 x^3\right ) \log ^2(2)+\left (15 x^4+4 x^5\right ) \log ^4(2)+\log (6)}{225 x^2+240 x^4 \log ^2(2)+94 x^6 \log ^4(2)+16 x^8 \log ^6(2)+x^{10} \log ^8(2)+\left (-30 x-16 x^3 \log ^2(2)-2 x^5 \log ^4(2)\right ) \log (6)+\log ^2(6)} \, dx=-\frac {x + 3}{x^{5} \log \left (2\right )^{4} + 8 \, x^{3} \log \left (2\right )^{2} + 15 \, x - \log \left (6\right )} \]

[In]

integrate((log(6)+(4*x^5+15*x^4)*log(2)^4+(16*x^3+72*x^2)*log(2)^2+45)/(log(6)^2+(-2*x^5*log(2)^4-16*x^3*log(2
)^2-30*x)*log(6)+x^10*log(2)^8+16*x^8*log(2)^6+94*x^6*log(2)^4+240*x^4*log(2)^2+225*x^2),x, algorithm="fricas"
)

[Out]

-(x + 3)/(x^5*log(2)^4 + 8*x^3*log(2)^2 + 15*x - log(6))

Sympy [A] (verification not implemented)

Time = 8.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {45+\left (72 x^2+16 x^3\right ) \log ^2(2)+\left (15 x^4+4 x^5\right ) \log ^4(2)+\log (6)}{225 x^2+240 x^4 \log ^2(2)+94 x^6 \log ^4(2)+16 x^8 \log ^6(2)+x^{10} \log ^8(2)+\left (-30 x-16 x^3 \log ^2(2)-2 x^5 \log ^4(2)\right ) \log (6)+\log ^2(6)} \, dx=\frac {- x - 3}{x^{5} \log {\left (2 \right )}^{4} + 8 x^{3} \log {\left (2 \right )}^{2} + 15 x - \log {\left (6 \right )}} \]

[In]

integrate((ln(6)+(4*x**5+15*x**4)*ln(2)**4+(16*x**3+72*x**2)*ln(2)**2+45)/(ln(6)**2+(-2*x**5*ln(2)**4-16*x**3*
ln(2)**2-30*x)*ln(6)+x**10*ln(2)**8+16*x**8*ln(2)**6+94*x**6*ln(2)**4+240*x**4*ln(2)**2+225*x**2),x)

[Out]

(-x - 3)/(x**5*log(2)**4 + 8*x**3*log(2)**2 + 15*x - log(6))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {45+\left (72 x^2+16 x^3\right ) \log ^2(2)+\left (15 x^4+4 x^5\right ) \log ^4(2)+\log (6)}{225 x^2+240 x^4 \log ^2(2)+94 x^6 \log ^4(2)+16 x^8 \log ^6(2)+x^{10} \log ^8(2)+\left (-30 x-16 x^3 \log ^2(2)-2 x^5 \log ^4(2)\right ) \log (6)+\log ^2(6)} \, dx=-\frac {x + 3}{x^{5} \log \left (2\right )^{4} + 8 \, x^{3} \log \left (2\right )^{2} + 15 \, x - \log \left (6\right )} \]

[In]

integrate((log(6)+(4*x^5+15*x^4)*log(2)^4+(16*x^3+72*x^2)*log(2)^2+45)/(log(6)^2+(-2*x^5*log(2)^4-16*x^3*log(2
)^2-30*x)*log(6)+x^10*log(2)^8+16*x^8*log(2)^6+94*x^6*log(2)^4+240*x^4*log(2)^2+225*x^2),x, algorithm="maxima"
)

[Out]

-(x + 3)/(x^5*log(2)^4 + 8*x^3*log(2)^2 + 15*x - log(6))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {45+\left (72 x^2+16 x^3\right ) \log ^2(2)+\left (15 x^4+4 x^5\right ) \log ^4(2)+\log (6)}{225 x^2+240 x^4 \log ^2(2)+94 x^6 \log ^4(2)+16 x^8 \log ^6(2)+x^{10} \log ^8(2)+\left (-30 x-16 x^3 \log ^2(2)-2 x^5 \log ^4(2)\right ) \log (6)+\log ^2(6)} \, dx=-\frac {x + 3}{x^{5} \log \left (2\right )^{4} + 8 \, x^{3} \log \left (2\right )^{2} + 15 \, x - \log \left (6\right )} \]

[In]

integrate((log(6)+(4*x^5+15*x^4)*log(2)^4+(16*x^3+72*x^2)*log(2)^2+45)/(log(6)^2+(-2*x^5*log(2)^4-16*x^3*log(2
)^2-30*x)*log(6)+x^10*log(2)^8+16*x^8*log(2)^6+94*x^6*log(2)^4+240*x^4*log(2)^2+225*x^2),x, algorithm="giac")

[Out]

-(x + 3)/(x^5*log(2)^4 + 8*x^3*log(2)^2 + 15*x - log(6))

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {45+\left (72 x^2+16 x^3\right ) \log ^2(2)+\left (15 x^4+4 x^5\right ) \log ^4(2)+\log (6)}{225 x^2+240 x^4 \log ^2(2)+94 x^6 \log ^4(2)+16 x^8 \log ^6(2)+x^{10} \log ^8(2)+\left (-30 x-16 x^3 \log ^2(2)-2 x^5 \log ^4(2)\right ) \log (6)+\log ^2(6)} \, dx=-\frac {x+3}{{\ln \left (2\right )}^4\,x^5+8\,{\ln \left (2\right )}^2\,x^3+15\,x-\ln \left (6\right )} \]

[In]

int((log(6) + log(2)^4*(15*x^4 + 4*x^5) + log(2)^2*(72*x^2 + 16*x^3) + 45)/(240*x^4*log(2)^2 + 94*x^6*log(2)^4
+ 16*x^8*log(2)^6 + x^10*log(2)^8 - log(6)*(30*x + 16*x^3*log(2)^2 + 2*x^5*log(2)^4) + log(6)^2 + 225*x^2),x)

[Out]

-(x + 3)/(15*x - log(6) + 8*x^3*log(2)^2 + x^5*log(2)^4)