3.37.88 \(\int \frac {(16-x)^{\frac {1}{2-4 x-8 x^2+2 \log (5)}} (1-2 x-4 x^2+\log (5)+(-32-126 x+8 x^2) \log (16-x))}{-32+130 x+120 x^2-520 x^3-480 x^4+32 x^5+(-64+132 x+248 x^2-16 x^3) \log (5)+(-32+2 x) \log ^2(5)} \, dx\)

Optimal. Leaf size=24 \[ (16-x)^{\frac {1}{2 \left (1-2 x-4 x^2+\log (5)\right )}} \]

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Rubi [F]  time = 8.25, 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 {(16-x)^{\frac {1}{2-4 x-8 x^2+2 \log (5)}} \left (1-2 x-4 x^2+\log (5)+\left (-32-126 x+8 x^2\right ) \log (16-x)\right )}{-32+130 x+120 x^2-520 x^3-480 x^4+32 x^5+\left (-64+132 x+248 x^2-16 x^3\right ) \log (5)+(-32+2 x) \log ^2(5)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((16 - x)^(2 - 4*x - 8*x^2 + 2*Log[5])^(-1)*(1 - 2*x - 4*x^2 + Log[5] + (-32 - 126*x + 8*x^2)*Log[16 - x])
)/(-32 + 130*x + 120*x^2 - 520*x^3 - 480*x^4 + 32*x^5 + (-64 + 132*x + 248*x^2 - 16*x^3)*Log[5] + (-32 + 2*x)*
Log[5]^2),x]

[Out]

((16 - x)^(2 - 4*x - 8*x^2 + Log[25])^(-1)*Hypergeometric2F1[1, (2 - 4*x - 8*x^2 + Log[25])^(-1), 1 + (2 - 4*x
- 8*x^2 + Log[25])^(-1), (4*(16 - x))/(65 - Sqrt[5 + Log[625]])]*(2 - 4*x - 8*x^2 + Log[25]))/(5 + Log[625] -
65*Sqrt[5 + Log[625]]) + ((16 - x)^(2 - 4*x - 8*x^2 + Log[25])^(-1)*Hypergeometric2F1[1, (2 - 4*x - 8*x^2 + L
og[25])^(-1), 1 + (2 - 4*x - 8*x^2 + Log[25])^(-1), (4*(16 - x))/(65 + Sqrt[5 + Log[625]])]*(2 - 4*x - 8*x^2 +
Log[25]))/(5 + Log[625] + 65*Sqrt[5 + Log[625]]) + Defer[Int][((16 - x)^(2 - 4*x - 8*x^2 + Log[25])^(-1)*Log[
16 - x])/(-1 + 2*x + 4*x^2 - Log[5])^2, x] + 4*Defer[Int][((16 - x)^(2 - 4*x - 8*x^2 + Log[25])^(-1)*x*Log[16
- x])/(-1 + 2*x + 4*x^2 - Log[5])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} \left (-1+2 x+4 x^2-\log (5)-2 \left (-16-63 x+4 x^2\right ) \log (16-x)\right )}{2 \left (1-2 x-4 x^2+\log (5)\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} \left (-1+2 x+4 x^2-\log (5)-2 \left (-16-63 x+4 x^2\right ) \log (16-x)\right )}{\left (1-2 x-4 x^2+\log (5)\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {2 (16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} x}{\left (-1+2 x+4 x^2-\log (5)\right )^2}+\frac {4 (16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} x^2}{\left (-1+2 x+4 x^2-\log (5)\right )^2}-\frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} (1+\log (5))}{\left (-1+2 x+4 x^2-\log (5)\right )^2}+\frac {2 (16-x)^{\frac {1}{2-4 x-8 x^2+\log (25)}} (1+4 x) \log (16-x)}{\left (-1+2 x+4 x^2-\log (5)\right )^2}\right ) \, dx\\ &=2 \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} x^2}{\left (-1+2 x+4 x^2-\log (5)\right )^2} \, dx+\frac {1}{2} (-1-\log (5)) \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{\left (-1+2 x+4 x^2-\log (5)\right )^2} \, dx+\int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} x}{\left (-1+2 x+4 x^2-\log (5)\right )^2} \, dx+\int \frac {(16-x)^{\frac {1}{2-4 x-8 x^2+\log (25)}} (1+4 x) \log (16-x)}{\left (-1+2 x+4 x^2-\log (5)\right )^2} \, dx\\ &=2 \int \left (\frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} (1-2 x+\log (5))}{4 \left (-1+2 x+4 x^2-\log (5)\right )^2}+\frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{-4+8 x+16 x^2-\log (625)}\right ) \, dx+\frac {1}{2} (-1-\log (5)) \int \left (\frac {16 (16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{(5+\log (625)) \left (-2-8 x+2 \sqrt {5+\log (625)}\right )^2}+\frac {8 (16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{(5+\log (625))^{3/2} \left (-2-8 x+2 \sqrt {5+\log (625)}\right )}+\frac {16 (16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{(5+\log (625)) \left (2+8 x+2 \sqrt {5+\log (625)}\right )^2}+\frac {8 (16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{(5+\log (625))^{3/2} \left (2+8 x+2 \sqrt {5+\log (625)}\right )}\right ) \, dx+\int \left (\frac {2 (16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} \left (-2+2 \sqrt {5+\log (625)}\right )}{(5+\log (625)) \left (-2-8 x+2 \sqrt {5+\log (625)}\right )^2}-\frac {2 (16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{(5+\log (625))^{3/2} \left (-2-8 x+2 \sqrt {5+\log (625)}\right )}+\frac {2 (16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} \left (-2-2 \sqrt {5+\log (625)}\right )}{(5+\log (625)) \left (2+8 x+2 \sqrt {5+\log (625)}\right )^2}-\frac {2 (16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{(5+\log (625))^{3/2} \left (2+8 x+2 \sqrt {5+\log (625)}\right )}\right ) \, dx+\int \left (\frac {(16-x)^{\frac {1}{2-4 x-8 x^2+\log (25)}} \log (16-x)}{\left (-1+2 x+4 x^2-\log (5)\right )^2}+\frac {4 (16-x)^{\frac {1}{2-4 x-8 x^2+\log (25)}} x \log (16-x)}{\left (-1+2 x+4 x^2-\log (5)\right )^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}} (1-2 x+\log (5))}{\left (-1+2 x+4 x^2-\log (5)\right )^2} \, dx+2 \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{-4+8 x+16 x^2-\log (625)} \, dx+4 \int \frac {(16-x)^{\frac {1}{2-4 x-8 x^2+\log (25)}} x \log (16-x)}{\left (-1+2 x+4 x^2-\log (5)\right )^2} \, dx-\frac {2 \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{-2-8 x+2 \sqrt {5+\log (625)}} \, dx}{(5+\log (625))^{3/2}}-\frac {2 \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{2+8 x+2 \sqrt {5+\log (625)}} \, dx}{(5+\log (625))^{3/2}}-\frac {(4 (1+\log (5))) \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{-2-8 x+2 \sqrt {5+\log (625)}} \, dx}{(5+\log (625))^{3/2}}-\frac {(4 (1+\log (5))) \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{2+8 x+2 \sqrt {5+\log (625)}} \, dx}{(5+\log (625))^{3/2}}-\frac {(8 (1+\log (5))) \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{\left (-2-8 x+2 \sqrt {5+\log (625)}\right )^2} \, dx}{5+\log (625)}-\frac {(8 (1+\log (5))) \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{\left (2+8 x+2 \sqrt {5+\log (625)}\right )^2} \, dx}{5+\log (625)}-\frac {\left (4 \left (1-\sqrt {5+\log (625)}\right )\right ) \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{\left (-2-8 x+2 \sqrt {5+\log (625)}\right )^2} \, dx}{5+\log (625)}-\frac {\left (4 \left (1+\sqrt {5+\log (625)}\right )\right ) \int \frac {(16-x)^{-1+\frac {1}{2-4 x-8 x^2+\log (25)}}}{\left (2+8 x+2 \sqrt {5+\log (625)}\right )^2} \, dx}{5+\log (625)}+\int \frac {(16-x)^{\frac {1}{2-4 x-8 x^2+\log (25)}} \log (16-x)}{\left (-1+2 x+4 x^2-\log (5)\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.65, size = 52, normalized size = 2.17 \begin {gather*} \frac {(16-x)^{\frac {1}{2-4 x-8 x^2+\log (25)}} \left (2-4 x-8 x^2+\log (25)\right )^2}{4 \left (1-2 x-4 x^2+\log (5)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((16 - x)^(2 - 4*x - 8*x^2 + 2*Log[5])^(-1)*(1 - 2*x - 4*x^2 + Log[5] + (-32 - 126*x + 8*x^2)*Log[16
- x]))/(-32 + 130*x + 120*x^2 - 520*x^3 - 480*x^4 + 32*x^5 + (-64 + 132*x + 248*x^2 - 16*x^3)*Log[5] + (-32 +
2*x)*Log[5]^2),x]

[Out]

((16 - x)^(2 - 4*x - 8*x^2 + Log[25])^(-1)*(2 - 4*x - 8*x^2 + Log[25])^2)/(4*(1 - 2*x - 4*x^2 + Log[5])^2)

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fricas [A]  time = 0.69, size = 26, normalized size = 1.08 \begin {gather*} \frac {1}{{\left (-x + 16\right )}^{\frac {1}{2 \, {\left (4 \, x^{2} + 2 \, x - \log \relax (5) - 1\right )}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2-126*x-32)*log(16-x)+log(5)-4*x^2-2*x+1)*exp(log(16-x)/(2*log(5)-8*x^2-4*x+2))/((2*x-32)*log(
5)^2+(-16*x^3+248*x^2+132*x-64)*log(5)+32*x^5-480*x^4-520*x^3+120*x^2+130*x-32),x, algorithm="fricas")

[Out]

1/((-x + 16)^(1/2/(4*x^2 + 2*x - log(5) - 1)))

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giac [A]  time = 0.43, size = 28, normalized size = 1.17 \begin {gather*} \frac {1}{{\left (-x + 16\right )}^{\frac {1}{2 \, {\left (4 \, {\left (x - 16\right )}^{2} + 130 \, x - \log \relax (5) - 1025\right )}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2-126*x-32)*log(16-x)+log(5)-4*x^2-2*x+1)*exp(log(16-x)/(2*log(5)-8*x^2-4*x+2))/((2*x-32)*log(
5)^2+(-16*x^3+248*x^2+132*x-64)*log(5)+32*x^5-480*x^4-520*x^3+120*x^2+130*x-32),x, algorithm="giac")

[Out]

1/((-x + 16)^(1/2/(4*(x - 16)^2 + 130*x - log(5) - 1025)))

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maple [A]  time = 0.49, size = 23, normalized size = 0.96

method	result	size
risch	\(\left (16-x \right )^{\frac {1}{2 \ln \relax (5)-8 x^{2}-4 x +2}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^2-126*x-32)*ln(16-x)+ln(5)-4*x^2-2*x+1)*exp(ln(16-x)/(2*ln(5)-8*x^2-4*x+2))/((2*x-32)*ln(5)^2+(-16*x
^3+248*x^2+132*x-64)*ln(5)+32*x^5-480*x^4-520*x^3+120*x^2+130*x-32),x,method=_RETURNVERBOSE)

[Out]

(16-x)^(1/2/(-4*x^2+ln(5)-2*x+1))

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maxima [A]  time = 0.56, size = 26, normalized size = 1.08 \begin {gather*} \frac {1}{{\left (-x + 16\right )}^{\frac {1}{2 \, {\left (4 \, x^{2} + 2 \, x - \log \relax (5) - 1\right )}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2-126*x-32)*log(16-x)+log(5)-4*x^2-2*x+1)*exp(log(16-x)/(2*log(5)-8*x^2-4*x+2))/((2*x-32)*log(
5)^2+(-16*x^3+248*x^2+132*x-64)*log(5)+32*x^5-480*x^4-520*x^3+120*x^2+130*x-32),x, algorithm="maxima")

[Out]

1/((-x + 16)^(1/2/(4*x^2 + 2*x - log(5) - 1)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {{\mathrm {e}}^{-\frac {\ln \left (16-x\right )}{8\,x^2+4\,x-2\,\ln \relax (5)-2}}\,\left (2\,x-\ln \relax (5)+\ln \left (16-x\right )\,\left (-8\,x^2+126\,x+32\right )+4\,x^2-1\right )}{130\,x+{\ln \relax (5)}^2\,\left (2\,x-32\right )+\ln \relax (5)\,\left (-16\,x^3+248\,x^2+132\,x-64\right )+120\,x^2-520\,x^3-480\,x^4+32\,x^5-32} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-log(16 - x)/(4*x - 2*log(5) + 8*x^2 - 2))*(2*x - log(5) + log(16 - x)*(126*x - 8*x^2 + 32) + 4*x^2
- 1))/(130*x + log(5)^2*(2*x - 32) + log(5)*(132*x + 248*x^2 - 16*x^3 - 64) + 120*x^2 - 520*x^3 - 480*x^4 + 32
*x^5 - 32),x)

[Out]

-int((exp(-log(16 - x)/(4*x - 2*log(5) + 8*x^2 - 2))*(2*x - log(5) + log(16 - x)*(126*x - 8*x^2 + 32) + 4*x^2
- 1))/(130*x + log(5)^2*(2*x - 32) + log(5)*(132*x + 248*x^2 - 16*x^3 - 64) + 120*x^2 - 520*x^3 - 480*x^4 + 32
*x^5 - 32), x)

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sympy [A]  time = 0.77, size = 20, normalized size = 0.83 \begin {gather*} e^{\frac {\log {\left (16 - x \right )}}{- 8 x^{2} - 4 x + 2 + 2 \log {\relax (5 )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**2-126*x-32)*ln(16-x)+ln(5)-4*x**2-2*x+1)*exp(ln(16-x)/(2*ln(5)-8*x**2-4*x+2))/((2*x-32)*ln(5)
**2+(-16*x**3+248*x**2+132*x-64)*ln(5)+32*x**5-480*x**4-520*x**3+120*x**2+130*x-32),x)

[Out]

exp(log(16 - x)/(-8*x**2 - 4*x + 2 + 2*log(5)))

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