3.38.97 \(\int \frac {-28+13 x^2+8 x^3-2 x^4+(28+16 x-15 x^2-4 x^3+2 x^4) \log (\frac {7+4 x-2 x^2}{-4 x+x^3}) \log (\log (\frac {7+4 x-2 x^2}{-4 x+x^3}))}{(28+16 x-15 x^2-4 x^3+2 x^4) \log (\frac {7+4 x-2 x^2}{-4 x+x^3})} \, dx\) [3797]

Optimal. Leaf size=25 \[ x \log \left (\log \left (-\frac {2}{x}+\frac {4-\frac {1}{x}}{-4+x^2}\right )\right ) \]

[Out]

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

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Rubi [F]
time = 4.91, 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 {-28+13 x^2+8 x^3-2 x^4+\left (28+16 x-15 x^2-4 x^3+2 x^4\right ) \log \left (\frac {7+4 x-2 x^2}{-4 x+x^3}\right ) \log \left (\log \left (\frac {7+4 x-2 x^2}{-4 x+x^3}\right )\right )}{\left (28+16 x-15 x^2-4 x^3+2 x^4\right ) \log \left (\frac {7+4 x-2 x^2}{-4 x+x^3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-28 + 13*x^2 + 8*x^3 - 2*x^4 + (28 + 16*x - 15*x^2 - 4*x^3 + 2*x^4)*Log[(7 + 4*x - 2*x^2)/(-4*x + x^3)]*L
og[Log[(7 + 4*x - 2*x^2)/(-4*x + x^3)]])/((28 + 16*x - 15*x^2 - 4*x^3 + 2*x^4)*Log[(7 + 4*x - 2*x^2)/(-4*x + x
^3)]),x]

[Out]

-Defer[Int][Log[(7 + 4*x - 2*x^2)/(x*(-4 + x^2))]^(-1), x] - (14*Sqrt[2]*Defer[Int][1/((4 + 6*Sqrt[2] - 4*x)*L
og[(7 + 4*x - 2*x^2)/(x*(-4 + x^2))]), x])/3 - 2*Defer[Int][1/((-2 + x)*Log[(7 + 4*x - 2*x^2)/(x*(-4 + x^2))])
, x] + 2*Defer[Int][1/((2 + x)*Log[(7 + 4*x - 2*x^2)/(x*(-4 + x^2))]), x] + (4*(3 + Sqrt[2])*Defer[Int][1/((-4
- 6*Sqrt[2] + 4*x)*Log[(7 + 4*x - 2*x^2)/(x*(-4 + x^2))]), x])/3 - (14*Sqrt[2]*Defer[Int][1/((-4 + 6*Sqrt[2]
+ 4*x)*Log[(7 + 4*x - 2*x^2)/(x*(-4 + x^2))]), x])/3 + (4*(3 - Sqrt[2])*Defer[Int][1/((-4 + 6*Sqrt[2] + 4*x)*L
og[(7 + 4*x - 2*x^2)/(x*(-4 + x^2))]), x])/3 + Defer[Int][Log[Log[(7 + 4*x - 2*x^2)/(x*(-4 + x^2))]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-28+13 x^2+8 x^3-2 x^4+\left (28+16 x-15 x^2-4 x^3+2 x^4\right ) \log \left (\frac {7+4 x-2 x^2}{-4 x+x^3}\right ) \log \left (\log \left (\frac {7+4 x-2 x^2}{-4 x+x^3}\right )\right )}{\left (28+16 x-15 x^2-4 x^3+2 x^4\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx\\ &=\int \left (-\frac {28}{(-2+x) (2+x) \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {13 x^2}{(-2+x) (2+x) \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {8 x^3}{(-2+x) (2+x) \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}-\frac {2 x^4}{(-2+x) (2+x) \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\log \left (\log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )\right )\right ) \, dx\\ &=-\left (2 \int \frac {x^4}{(-2+x) (2+x) \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx\right )+8 \int \frac {x^3}{(-2+x) (2+x) \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+13 \int \frac {x^2}{(-2+x) (2+x) \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-28 \int \frac {1}{(-2+x) (2+x) \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\int \log \left (\log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )\right ) \, dx\\ &=-\left (2 \int \left (\frac {1}{2 \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}-\frac {4}{7 (-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}-\frac {4}{9 (2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {497+508 x}{126 \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}\right ) \, dx\right )+8 \int \left (-\frac {2}{7 (-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {2}{9 (2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {112+71 x}{63 \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}\right ) \, dx+13 \int \left (-\frac {1}{7 (-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}-\frac {1}{9 (2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {7+32 x}{63 \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}\right ) \, dx-28 \int \left (-\frac {1}{28 (-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}-\frac {1}{36 (2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {2 (-7+4 x)}{63 \left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}\right ) \, dx+\int \log \left (\log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )\right ) \, dx\\ &=-\left (\frac {1}{63} \int \frac {497+508 x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx\right )+\frac {8}{63} \int \frac {112+71 x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {13}{63} \int \frac {7+32 x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {7}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {8}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {8}{9} \int \frac {-7+4 x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {8}{7} \int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {13}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {16}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {13}{7} \int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {16}{7} \int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\int \frac {1}{\log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\int \log \left (\log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )\right ) \, dx\\ &=-\left (\frac {1}{63} \int \left (\frac {497}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {508 x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}\right ) \, dx\right )+\frac {8}{63} \int \left (\frac {112}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {71 x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}\right ) \, dx+\frac {13}{63} \int \left (\frac {7}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {32 x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}\right ) \, dx+\frac {7}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {8}{9} \int \left (-\frac {7}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}+\frac {4 x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )}\right ) \, dx+\frac {8}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {8}{7} \int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {13}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {16}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {13}{7} \int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {16}{7} \int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\int \frac {1}{\log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\int \log \left (\log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )\right ) \, dx\\ &=\frac {7}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {8}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {8}{7} \int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {13}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {13}{9} \int \frac {1}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {16}{9} \int \frac {1}{(2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {13}{7} \int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {16}{7} \int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {32}{9} \int \frac {x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {56}{9} \int \frac {1}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {416}{63} \int \frac {x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {71}{9} \int \frac {1}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\frac {508}{63} \int \frac {x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {568}{63} \int \frac {x}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\frac {128}{9} \int \frac {1}{\left (-7-4 x+2 x^2\right ) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx-\int \frac {1}{\log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\int \frac {1}{(-2+x) \log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )} \, dx+\int \log \left (\log \left (\frac {7+4 x-2 x^2}{x \left (-4+x^2\right )}\right )\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.04, size = 24, normalized size = 0.96 \begin {gather*} x \log \left (\log \left (\frac {7+4 x-2 x^2}{-4 x+x^3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-28 + 13*x^2 + 8*x^3 - 2*x^4 + (28 + 16*x - 15*x^2 - 4*x^3 + 2*x^4)*Log[(7 + 4*x - 2*x^2)/(-4*x + x
^3)]*Log[Log[(7 + 4*x - 2*x^2)/(-4*x + x^3)]])/((28 + 16*x - 15*x^2 - 4*x^3 + 2*x^4)*Log[(7 + 4*x - 2*x^2)/(-4
*x + x^3)]),x]

[Out]

x*Log[Log[(7 + 4*x - 2*x^2)/(-4*x + x^3)]]

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (2 x^{4}-4 x^{3}-15 x^{2}+16 x +28\right ) \ln \left (\frac {-2 x^{2}+4 x +7}{x^{3}-4 x}\right ) \ln \left (\ln \left (\frac {-2 x^{2}+4 x +7}{x^{3}-4 x}\right )\right )-2 x^{4}+8 x^{3}+13 x^{2}-28}{\left (2 x^{4}-4 x^{3}-15 x^{2}+16 x +28\right ) \ln \left (\frac {-2 x^{2}+4 x +7}{x^{3}-4 x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4-4*x^3-15*x^2+16*x+28)*ln((-2*x^2+4*x+7)/(x^3-4*x))*ln(ln((-2*x^2+4*x+7)/(x^3-4*x)))-2*x^4+8*x^3+13
*x^2-28)/(2*x^4-4*x^3-15*x^2+16*x+28)/ln((-2*x^2+4*x+7)/(x^3-4*x)),x)

[Out]

int(((2*x^4-4*x^3-15*x^2+16*x+28)*ln((-2*x^2+4*x+7)/(x^3-4*x))*ln(ln((-2*x^2+4*x+7)/(x^3-4*x)))-2*x^4+8*x^3+13
*x^2-28)/(2*x^4-4*x^3-15*x^2+16*x+28)/ln((-2*x^2+4*x+7)/(x^3-4*x)),x)

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Maxima [A]
time = 0.30, size = 31, normalized size = 1.24 \begin {gather*} x \log \left (\log \left (-2 \, x^{2} + 4 \, x + 7\right ) - \log \left (x + 2\right ) - \log \left (x - 2\right ) - \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-4*x^3-15*x^2+16*x+28)*log((-2*x^2+4*x+7)/(x^3-4*x))*log(log((-2*x^2+4*x+7)/(x^3-4*x)))-2*x^4
+8*x^3+13*x^2-28)/(2*x^4-4*x^3-15*x^2+16*x+28)/log((-2*x^2+4*x+7)/(x^3-4*x)),x, algorithm="maxima")

[Out]

x*log(log(-2*x^2 + 4*x + 7) - log(x + 2) - log(x - 2) - log(x))

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Fricas [A]
time = 0.37, size = 25, normalized size = 1.00 \begin {gather*} x \log \left (\log \left (-\frac {2 \, x^{2} - 4 \, x - 7}{x^{3} - 4 \, x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-4*x^3-15*x^2+16*x+28)*log((-2*x^2+4*x+7)/(x^3-4*x))*log(log((-2*x^2+4*x+7)/(x^3-4*x)))-2*x^4
+8*x^3+13*x^2-28)/(2*x^4-4*x^3-15*x^2+16*x+28)/log((-2*x^2+4*x+7)/(x^3-4*x)),x, algorithm="fricas")

[Out]

x*log(log(-(2*x^2 - 4*x - 7)/(x^3 - 4*x)))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).
time = 0.74, size = 46, normalized size = 1.84 \begin {gather*} \left (x - \frac {1}{15}\right ) \log {\left (\log {\left (\frac {- 2 x^{2} + 4 x + 7}{x^{3} - 4 x} \right )} \right )} + \frac {\log {\left (\log {\left (\frac {- 2 x^{2} + 4 x + 7}{x^{3} - 4 x} \right )} \right )}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4-4*x**3-15*x**2+16*x+28)*ln((-2*x**2+4*x+7)/(x**3-4*x))*ln(ln((-2*x**2+4*x+7)/(x**3-4*x)))-2
*x**4+8*x**3+13*x**2-28)/(2*x**4-4*x**3-15*x**2+16*x+28)/ln((-2*x**2+4*x+7)/(x**3-4*x)),x)

[Out]

(x - 1/15)*log(log((-2*x**2 + 4*x + 7)/(x**3 - 4*x))) + log(log((-2*x**2 + 4*x + 7)/(x**3 - 4*x)))/15

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Giac [A]
time = 0.77, size = 25, normalized size = 1.00 \begin {gather*} x \log \left (\log \left (-\frac {2 \, x^{2} - 4 \, x - 7}{x^{3} - 4 \, x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-4*x^3-15*x^2+16*x+28)*log((-2*x^2+4*x+7)/(x^3-4*x))*log(log((-2*x^2+4*x+7)/(x^3-4*x)))-2*x^4
+8*x^3+13*x^2-28)/(2*x^4-4*x^3-15*x^2+16*x+28)/log((-2*x^2+4*x+7)/(x^3-4*x)),x, algorithm="giac")

[Out]

x*log(log(-(2*x^2 - 4*x - 7)/(x^3 - 4*x)))

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Mupad [B]
time = 2.92, size = 27, normalized size = 1.08 \begin {gather*} x\,\ln \left (\ln \left (-\frac {-2\,x^2+4\,x+7}{4\,x-x^3}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((13*x^2 + 8*x^3 - 2*x^4 + log(-(4*x - 2*x^2 + 7)/(4*x - x^3))*log(log(-(4*x - 2*x^2 + 7)/(4*x - x^3)))*(16
*x - 15*x^2 - 4*x^3 + 2*x^4 + 28) - 28)/(log(-(4*x - 2*x^2 + 7)/(4*x - x^3))*(16*x - 15*x^2 - 4*x^3 + 2*x^4 +
28)),x)

[Out]

x*log(log(-(4*x - 2*x^2 + 7)/(4*x - x^3)))

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