3.8.95 \(\int \frac {416+68 x-84 x^2+(-128-16 x+16 x^2) \log (8+x-x^2)}{-1352+247 x+189 x^2-56 x^3+4 x^4+(832-24 x-120 x^2+16 x^3) \log (8+x-x^2)+(-128-16 x+16 x^2) \log ^2(8+x-x^2)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x}{-\frac {19}{4}+\frac {3+x}{2}+\log \left (8+x-x^2\right )} \]

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Rubi [F]  time = 1.34, 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 {416+68 x-84 x^2+\left (-128-16 x+16 x^2\right ) \log \left (8+x-x^2\right )}{-1352+247 x+189 x^2-56 x^3+4 x^4+\left (832-24 x-120 x^2+16 x^3\right ) \log \left (8+x-x^2\right )+\left (-128-16 x+16 x^2\right ) \log ^2\left (8+x-x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(416 + 68*x - 84*x^2 + (-128 - 16*x + 16*x^2)*Log[8 + x - x^2])/(-1352 + 247*x + 189*x^2 - 56*x^3 + 4*x^4
+ (832 - 24*x - 120*x^2 + 16*x^3)*Log[8 + x - x^2] + (-128 - 16*x + 16*x^2)*Log[8 + x - x^2]^2),x]

[Out]

-32*Defer[Int][(-13 + 2*x + 4*Log[8 + x - x^2])^(-2), x] + (512*Defer[Int][1/((1 + Sqrt[33] - 2*x)*(-13 + 2*x
+ 4*Log[8 + x - x^2])^2), x])/Sqrt[33] - 8*Defer[Int][x/(-13 + 2*x + 4*Log[8 + x - x^2])^2, x] - (16*(33 + Sqr
t[33])*Defer[Int][1/((-1 - Sqrt[33] + 2*x)*(-13 + 2*x + 4*Log[8 + x - x^2])^2), x])/33 + (512*Defer[Int][1/((-
1 + Sqrt[33] + 2*x)*(-13 + 2*x + 4*Log[8 + x - x^2])^2), x])/Sqrt[33] - (16*(33 - Sqrt[33])*Defer[Int][1/((-1
+ Sqrt[33] + 2*x)*(-13 + 2*x + 4*Log[8 + x - x^2])^2), x])/33 + 4*Defer[Int][(-13 + 2*x + 4*Log[8 + x - x^2])^
(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-416-68 x+84 x^2-16 \left (-8-x+x^2\right ) \log \left (8+x-x^2\right )}{\left (8+x-x^2\right ) \left (13-2 x-4 \log \left (8+x-x^2\right )\right )^2} \, dx\\ &=\int \left (-\frac {8 x \left (-10+3 x+x^2\right )}{\left (-8-x+x^2\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2}+\frac {4}{-13+2 x+4 \log \left (8+x-x^2\right )}\right ) \, dx\\ &=4 \int \frac {1}{-13+2 x+4 \log \left (8+x-x^2\right )} \, dx-8 \int \frac {x \left (-10+3 x+x^2\right )}{\left (-8-x+x^2\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx\\ &=4 \int \frac {1}{-13+2 x+4 \log \left (8+x-x^2\right )} \, dx-8 \int \left (\frac {4}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2}+\frac {x}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2}+\frac {2 (16+x)}{\left (-8-x+x^2\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2}\right ) \, dx\\ &=4 \int \frac {1}{-13+2 x+4 \log \left (8+x-x^2\right )} \, dx-8 \int \frac {x}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx-16 \int \frac {16+x}{\left (-8-x+x^2\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx-32 \int \frac {1}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx\\ &=4 \int \frac {1}{-13+2 x+4 \log \left (8+x-x^2\right )} \, dx-8 \int \frac {x}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx-16 \int \left (\frac {16}{\left (-8-x+x^2\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2}+\frac {x}{\left (-8-x+x^2\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2}\right ) \, dx-32 \int \frac {1}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx\\ &=4 \int \frac {1}{-13+2 x+4 \log \left (8+x-x^2\right )} \, dx-8 \int \frac {x}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx-16 \int \frac {x}{\left (-8-x+x^2\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx-32 \int \frac {1}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx-256 \int \frac {1}{\left (-8-x+x^2\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx\\ &=4 \int \frac {1}{-13+2 x+4 \log \left (8+x-x^2\right )} \, dx-8 \int \frac {x}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx-16 \int \left (\frac {1+\frac {1}{\sqrt {33}}}{\left (-1-\sqrt {33}+2 x\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2}+\frac {1-\frac {1}{\sqrt {33}}}{\left (-1+\sqrt {33}+2 x\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2}\right ) \, dx-32 \int \frac {1}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx-256 \int \left (-\frac {2}{\sqrt {33} \left (1+\sqrt {33}-2 x\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2}-\frac {2}{\sqrt {33} \left (-1+\sqrt {33}+2 x\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2}\right ) \, dx\\ &=4 \int \frac {1}{-13+2 x+4 \log \left (8+x-x^2\right )} \, dx-8 \int \frac {x}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx-32 \int \frac {1}{\left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx+\frac {512 \int \frac {1}{\left (1+\sqrt {33}-2 x\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx}{\sqrt {33}}+\frac {512 \int \frac {1}{\left (-1+\sqrt {33}+2 x\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx}{\sqrt {33}}-\frac {1}{33} \left (16 \left (33-\sqrt {33}\right )\right ) \int \frac {1}{\left (-1+\sqrt {33}+2 x\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx-\frac {1}{33} \left (16 \left (33+\sqrt {33}\right )\right ) \int \frac {1}{\left (-1-\sqrt {33}+2 x\right ) \left (-13+2 x+4 \log \left (8+x-x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.20, size = 21, normalized size = 0.88 \begin {gather*} \frac {4 x}{-13+2 x+4 \log \left (8+x-x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(416 + 68*x - 84*x^2 + (-128 - 16*x + 16*x^2)*Log[8 + x - x^2])/(-1352 + 247*x + 189*x^2 - 56*x^3 +
4*x^4 + (832 - 24*x - 120*x^2 + 16*x^3)*Log[8 + x - x^2] + (-128 - 16*x + 16*x^2)*Log[8 + x - x^2]^2),x]

[Out]

(4*x)/(-13 + 2*x + 4*Log[8 + x - x^2])

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fricas [A]  time = 1.09, size = 21, normalized size = 0.88 \begin {gather*} \frac {4 \, x}{2 \, x + 4 \, \log \left (-x^{2} + x + 8\right ) - 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2-16*x-128)*log(-x^2+x+8)-84*x^2+68*x+416)/((16*x^2-16*x-128)*log(-x^2+x+8)^2+(16*x^3-120*x^2
-24*x+832)*log(-x^2+x+8)+4*x^4-56*x^3+189*x^2+247*x-1352),x, algorithm="fricas")

[Out]

4*x/(2*x + 4*log(-x^2 + x + 8) - 13)

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giac [A]  time = 0.57, size = 21, normalized size = 0.88 \begin {gather*} \frac {4 \, x}{2 \, x + 4 \, \log \left (-x^{2} + x + 8\right ) - 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2-16*x-128)*log(-x^2+x+8)-84*x^2+68*x+416)/((16*x^2-16*x-128)*log(-x^2+x+8)^2+(16*x^3-120*x^2
-24*x+832)*log(-x^2+x+8)+4*x^4-56*x^3+189*x^2+247*x-1352),x, algorithm="giac")

[Out]

4*x/(2*x + 4*log(-x^2 + x + 8) - 13)

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maple [A]  time = 0.06, size = 22, normalized size = 0.92

method	result	size
risch	\(\frac {4 x}{4 \ln \left (-x^{2}+x +8\right )+2 x -13}\)	\(22\)
norman	\(\frac {-8 \ln \left (-x^{2}+x +8\right )+26}{4 \ln \left (-x^{2}+x +8\right )+2 x -13}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^2-16*x-128)*ln(-x^2+x+8)-84*x^2+68*x+416)/((16*x^2-16*x-128)*ln(-x^2+x+8)^2+(16*x^3-120*x^2-24*x+83
2)*ln(-x^2+x+8)+4*x^4-56*x^3+189*x^2+247*x-1352),x,method=_RETURNVERBOSE)

[Out]

4*x/(4*ln(-x^2+x+8)+2*x-13)

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maxima [A]  time = 0.64, size = 21, normalized size = 0.88 \begin {gather*} \frac {4 \, x}{2 \, x + 4 \, \log \left (-x^{2} + x + 8\right ) - 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2-16*x-128)*log(-x^2+x+8)-84*x^2+68*x+416)/((16*x^2-16*x-128)*log(-x^2+x+8)^2+(16*x^3-120*x^2
-24*x+832)*log(-x^2+x+8)+4*x^4-56*x^3+189*x^2+247*x-1352),x, algorithm="maxima")

[Out]

4*x/(2*x + 4*log(-x^2 + x + 8) - 13)

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mupad [B]  time = 0.94, size = 21, normalized size = 0.88 \begin {gather*} \frac {4\,x}{2\,x+4\,\ln \left (-x^2+x+8\right )-13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(68*x - log(x - x^2 + 8)*(16*x - 16*x^2 + 128) - 84*x^2 + 416)/(log(x - x^2 + 8)*(24*x + 120*x^2 - 16*x^3
- 832) - 247*x + log(x - x^2 + 8)^2*(16*x - 16*x^2 + 128) - 189*x^2 + 56*x^3 - 4*x^4 + 1352),x)

[Out]

(4*x)/(2*x + 4*log(x - x^2 + 8) - 13)

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sympy [A]  time = 0.18, size = 15, normalized size = 0.62 \begin {gather*} \frac {x}{\frac {x}{2} + \log {\left (- x^{2} + x + 8 \right )} - \frac {13}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**2-16*x-128)*ln(-x**2+x+8)-84*x**2+68*x+416)/((16*x**2-16*x-128)*ln(-x**2+x+8)**2+(16*x**3-12
0*x**2-24*x+832)*ln(-x**2+x+8)+4*x**4-56*x**3+189*x**2+247*x-1352),x)

[Out]

x/(x/2 + log(-x**2 + x + 8) - 13/4)

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