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

Optimal. Leaf size=25 \[ \frac {3+x}{x-x \left (4+x^2 \log ^2(2)\right )^2+\log (6)} \]

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Rubi [F]  time = 0.48, 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 {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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 33, normalized size = 1.32 \begin {gather*} \frac {-3-x}{15 x+8 x^3 \log ^2(2)+x^5 \log ^4(2)-\log (6)} \end {gather*}

Antiderivative was successfully verified.

[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])

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fricas [A]  time = 0.59, size = 32, normalized size = 1.28 \begin {gather*} -\frac {x + 3}{x^{5} \log \relax (2)^{4} + 8 \, x^{3} \log \relax (2)^{2} + 15 \, x - \log \relax (6)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.26, size = 32, normalized size = 1.28 \begin {gather*} -\frac {x + 3}{x^{5} \log \relax (2)^{4} + 8 \, x^{3} \log \relax (2)^{2} + 15 \, x - \log \relax (6)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.21, size = 31, normalized size = 1.24

method	result	size
gosper	\(\frac {3+x}{-x^{5} \ln \relax (2)^{4}-8 x^{3} \ln \relax (2)^{2}+\ln \relax (6)-15 x}\)	\(31\)
default	\(\frac {3+x}{-x^{5} \ln \relax (2)^{4}-8 x^{3} \ln \relax (2)^{2}+\ln \relax (6)-15 x}\)	\(31\)
risch	\(\frac {3+x}{-x^{5} \ln \relax (2)^{4}-8 x^{3} \ln \relax (2)^{2}+\ln \relax (3)+\ln \relax (2)-15 x}\)	\(33\)
norman	\(\frac {\frac {\left (\ln \relax (6)+45\right ) x}{\ln \relax (6)}+\frac {24 \ln \relax (2)^{2} x^{3}}{\ln \relax (6)}+\frac {3 \ln \relax (2)^{4} x^{5}}{\ln \relax (6)}}{-x^{5} \ln \relax (2)^{4}-8 x^{3} \ln \relax (2)^{2}+\ln \relax (6)-15 x}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.35, size = 32, normalized size = 1.28 \begin {gather*} -\frac {x + 3}{x^{5} \log \relax (2)^{4} + 8 \, x^{3} \log \relax (2)^{2} + 15 \, x - \log \relax (6)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 0.23, size = 32, normalized size = 1.28 \begin {gather*} -\frac {x+3}{{\ln \relax (2)}^4\,x^5+8\,{\ln \relax (2)}^2\,x^3+15\,x-\ln \relax (6)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 13.43, size = 29, normalized size = 1.16 \begin {gather*} \frac {- x - 3}{x^{5} \log {\relax (2 )}^{4} + 8 x^{3} \log {\relax (2 )}^{2} + 15 x - \log {\relax (6 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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