3.16.58 \(\int \frac {9-x+3 x^3-132 x^4+15 x^5+(-1+15 x^4) \log (2 x)+(-108 x^3+12 x^4+12 x^3 \log (2 x)) \log (-9+x+\log (2 x))}{-9+x+\log (2 x)} \, dx\)

Optimal. Leaf size=19 \[ -x+3 x^4 (x+\log (-9+x+\log (2 x))) \]

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Rubi [F]  time = 0.74, 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 {9-x+3 x^3-132 x^4+15 x^5+\left (-1+15 x^4\right ) \log (2 x)+\left (-108 x^3+12 x^4+12 x^3 \log (2 x)\right ) \log (-9+x+\log (2 x))}{-9+x+\log (2 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 - x + 3*x^3 - 132*x^4 + 15*x^5 + (-1 + 15*x^4)*Log[2*x] + (-108*x^3 + 12*x^4 + 12*x^3*Log[2*x])*Log[-9
+ x + Log[2*x]])/(-9 + x + Log[2*x]),x]

[Out]

-x + 3*x^5 + 3*Defer[Int][x^3/(-9 + x + Log[2*x]), x] + 3*Defer[Int][x^4/(-9 + x + Log[2*x]), x] + 12*Defer[In
t][x^3*Log[-9 + x + Log[2*x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {9}{-9+x+\log (2 x)}-\frac {x}{-9+x+\log (2 x)}+\frac {3 x^3}{-9+x+\log (2 x)}-\frac {132 x^4}{-9+x+\log (2 x)}+\frac {15 x^5}{-9+x+\log (2 x)}+\frac {\left (-1+15 x^4\right ) \log (2 x)}{-9+x+\log (2 x)}+12 x^3 \log (-9+x+\log (2 x))\right ) \, dx\\ &=3 \int \frac {x^3}{-9+x+\log (2 x)} \, dx+9 \int \frac {1}{-9+x+\log (2 x)} \, dx+12 \int x^3 \log (-9+x+\log (2 x)) \, dx+15 \int \frac {x^5}{-9+x+\log (2 x)} \, dx-132 \int \frac {x^4}{-9+x+\log (2 x)} \, dx-\int \frac {x}{-9+x+\log (2 x)} \, dx+\int \frac {\left (-1+15 x^4\right ) \log (2 x)}{-9+x+\log (2 x)} \, dx\\ &=3 \int \frac {x^3}{-9+x+\log (2 x)} \, dx+9 \int \frac {1}{-9+x+\log (2 x)} \, dx+12 \int x^3 \log (-9+x+\log (2 x)) \, dx+15 \int \frac {x^5}{-9+x+\log (2 x)} \, dx-132 \int \frac {x^4}{-9+x+\log (2 x)} \, dx-\int \frac {x}{-9+x+\log (2 x)} \, dx+\int \left (-1+15 x^4+\frac {-9+x+135 x^4-15 x^5}{-9+x+\log (2 x)}\right ) \, dx\\ &=-x+3 x^5+3 \int \frac {x^3}{-9+x+\log (2 x)} \, dx+9 \int \frac {1}{-9+x+\log (2 x)} \, dx+12 \int x^3 \log (-9+x+\log (2 x)) \, dx+15 \int \frac {x^5}{-9+x+\log (2 x)} \, dx-132 \int \frac {x^4}{-9+x+\log (2 x)} \, dx-\int \frac {x}{-9+x+\log (2 x)} \, dx+\int \frac {-9+x+135 x^4-15 x^5}{-9+x+\log (2 x)} \, dx\\ &=-x+3 x^5+3 \int \frac {x^3}{-9+x+\log (2 x)} \, dx+9 \int \frac {1}{-9+x+\log (2 x)} \, dx+12 \int x^3 \log (-9+x+\log (2 x)) \, dx+15 \int \frac {x^5}{-9+x+\log (2 x)} \, dx-132 \int \frac {x^4}{-9+x+\log (2 x)} \, dx-\int \frac {x}{-9+x+\log (2 x)} \, dx+\int \left (-\frac {9}{-9+x+\log (2 x)}+\frac {x}{-9+x+\log (2 x)}+\frac {135 x^4}{-9+x+\log (2 x)}-\frac {15 x^5}{-9+x+\log (2 x)}\right ) \, dx\\ &=-x+3 x^5+3 \int \frac {x^3}{-9+x+\log (2 x)} \, dx+12 \int x^3 \log (-9+x+\log (2 x)) \, dx-132 \int \frac {x^4}{-9+x+\log (2 x)} \, dx+135 \int \frac {x^4}{-9+x+\log (2 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 22, normalized size = 1.16 \begin {gather*} -x+3 x^5+3 x^4 \log (-9+x+\log (2 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 - x + 3*x^3 - 132*x^4 + 15*x^5 + (-1 + 15*x^4)*Log[2*x] + (-108*x^3 + 12*x^4 + 12*x^3*Log[2*x])*L
og[-9 + x + Log[2*x]])/(-9 + x + Log[2*x]),x]

[Out]

-x + 3*x^5 + 3*x^4*Log[-9 + x + Log[2*x]]

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fricas [A]  time = 0.68, size = 22, normalized size = 1.16 \begin {gather*} 3 \, x^{5} + 3 \, x^{4} \log \left (x + \log \left (2 \, x\right ) - 9\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^3*log(2*x)+12*x^4-108*x^3)*log(log(2*x)+x-9)+(15*x^4-1)*log(2*x)+15*x^5-132*x^4+3*x^3-x+9)/(l
og(2*x)+x-9),x, algorithm="fricas")

[Out]

3*x^5 + 3*x^4*log(x + log(2*x) - 9) - x

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giac [A]  time = 0.17, size = 22, normalized size = 1.16 \begin {gather*} 3 \, x^{5} + 3 \, x^{4} \log \left (x + \log \left (2 \, x\right ) - 9\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^3*log(2*x)+12*x^4-108*x^3)*log(log(2*x)+x-9)+(15*x^4-1)*log(2*x)+15*x^5-132*x^4+3*x^3-x+9)/(l
og(2*x)+x-9),x, algorithm="giac")

[Out]

3*x^5 + 3*x^4*log(x + log(2*x) - 9) - x

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maple [A]  time = 0.07, size = 23, normalized size = 1.21

method	result	size
risch	\(3 x^{5}+3 x^{4} \ln \left (\ln \left (2 x \right )+x -9\right )-x\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^3*ln(2*x)+12*x^4-108*x^3)*ln(ln(2*x)+x-9)+(15*x^4-1)*ln(2*x)+15*x^5-132*x^4+3*x^3-x+9)/(ln(2*x)+x-9
),x,method=_RETURNVERBOSE)

[Out]

3*x^5+3*x^4*ln(ln(2*x)+x-9)-x

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maxima [A]  time = 0.51, size = 22, normalized size = 1.16 \begin {gather*} 3 \, x^{5} + 3 \, x^{4} \log \left (x + \log \relax (2) + \log \relax (x) - 9\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^3*log(2*x)+12*x^4-108*x^3)*log(log(2*x)+x-9)+(15*x^4-1)*log(2*x)+15*x^5-132*x^4+3*x^3-x+9)/(l
og(2*x)+x-9),x, algorithm="maxima")

[Out]

3*x^5 + 3*x^4*log(x + log(2) + log(x) - 9) - x

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mupad [B]  time = 1.09, size = 22, normalized size = 1.16 \begin {gather*} 3\,x^4\,\ln \left (x+\ln \left (2\,x\right )-9\right )-x+3\,x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + log(2*x) - 9)*(12*x^3*log(2*x) - 108*x^3 + 12*x^4) - x + log(2*x)*(15*x^4 - 1) + 3*x^3 - 132*x^4
+ 15*x^5 + 9)/(x + log(2*x) - 9),x)

[Out]

3*x^4*log(x + log(2*x) - 9) - x + 3*x^5

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sympy [A]  time = 0.39, size = 20, normalized size = 1.05 \begin {gather*} 3 x^{5} + 3 x^{4} \log {\left (x + \log {\left (2 x \right )} - 9 \right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**3*ln(2*x)+12*x**4-108*x**3)*ln(ln(2*x)+x-9)+(15*x**4-1)*ln(2*x)+15*x**5-132*x**4+3*x**3-x+9)
/(ln(2*x)+x-9),x)

[Out]

3*x**5 + 3*x**4*log(x + log(2*x) - 9) - x

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