3.21.14 \(\int \frac {256+16 e-64 x+24 x^2-2 x^3}{(-4096+2944 x-704 x^2+52 x^3+x^4+e (-16+4 x)) \log ^2(\frac {1024+4 e-480 x+56 x^2+x^3}{64-32 x+4 x^2})} \, dx\)

Optimal. Leaf size=25 \[ \frac {2}{\log \left (16+\frac {x}{4}+\frac {e+4 x}{(4-x)^2}\right )} \]

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Rubi [F]  time = 1.23, 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 {256+16 e-64 x+24 x^2-2 x^3}{\left (-4096+2944 x-704 x^2+52 x^3+x^4+e (-16+4 x)\right ) \log ^2\left (\frac {1024+4 e-480 x+56 x^2+x^3}{64-32 x+4 x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(256 + 16*E - 64*x + 24*x^2 - 2*x^3)/((-4096 + 2944*x - 704*x^2 + 52*x^3 + x^4 + E*(-16 + 4*x))*Log[(1024
+ 4*E - 480*x + 56*x^2 + x^3)/(64 - 32*x + 4*x^2)]^2),x]

[Out]

4*Defer[Int][1/((-4 + x)*Log[(4*(256 + E) - 480*x + 56*x^2 + x^3)/(4*(-4 + x)^2)]^2), x] + 224*Defer[Int][x/((
-4*(256 + E) + 480*x - 56*x^2 - x^3)*Log[(4*(256 + E) - 480*x + 56*x^2 + x^3)/(4*(-4 + x)^2)]^2), x] + 6*Defer
[Int][x^2/((-4*(256 + E) + 480*x - 56*x^2 - x^3)*Log[(4*(256 + E) - 480*x + 56*x^2 + x^3)/(4*(-4 + x)^2)]^2),
x] + 960*Defer[Int][1/((4*(256 + E) - 480*x + 56*x^2 + x^3)*Log[(4*(256 + E) - 480*x + 56*x^2 + x^3)/(4*(-4 +
x)^2)]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-16 (16+e)+64 x-24 x^2+2 x^3}{\left (16 (256+e)-4 (736+e) x+704 x^2-52 x^3-x^4\right ) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{64-32 x+4 x^2}\right )} \, dx\\ &=\int \left (\frac {4}{(-4+x) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )}+\frac {2 \left (480-112 x-3 x^2\right )}{\left (4 (256+e)-480 x+56 x^2+x^3\right ) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )}\right ) \, dx\\ &=2 \int \frac {480-112 x-3 x^2}{\left (4 (256+e)-480 x+56 x^2+x^3\right ) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )} \, dx+4 \int \frac {1}{(-4+x) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )} \, dx\\ &=2 \int \left (\frac {112 x}{\left (-4 (256+e)+480 x-56 x^2-x^3\right ) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )}+\frac {3 x^2}{\left (-4 (256+e)+480 x-56 x^2-x^3\right ) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )}+\frac {480}{\left (4 (256+e)-480 x+56 x^2+x^3\right ) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )}\right ) \, dx+4 \int \frac {1}{(-4+x) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )} \, dx\\ &=4 \int \frac {1}{(-4+x) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )} \, dx+6 \int \frac {x^2}{\left (-4 (256+e)+480 x-56 x^2-x^3\right ) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )} \, dx+224 \int \frac {x}{\left (-4 (256+e)+480 x-56 x^2-x^3\right ) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )} \, dx+960 \int \frac {1}{\left (4 (256+e)-480 x+56 x^2+x^3\right ) \log ^2\left (\frac {4 (256+e)-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 30, normalized size = 1.20 \begin {gather*} \frac {2}{\log \left (\frac {1024+4 e-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(256 + 16*E - 64*x + 24*x^2 - 2*x^3)/((-4096 + 2944*x - 704*x^2 + 52*x^3 + x^4 + E*(-16 + 4*x))*Log[
(1024 + 4*E - 480*x + 56*x^2 + x^3)/(64 - 32*x + 4*x^2)]^2),x]

[Out]

2/Log[(1024 + 4*E - 480*x + 56*x^2 + x^3)/(4*(-4 + x)^2)]

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fricas [A]  time = 0.54, size = 34, normalized size = 1.36 \begin {gather*} \frac {2}{\log \left (\frac {x^{3} + 56 \, x^{2} - 480 \, x + 4 \, e + 1024}{4 \, {\left (x^{2} - 8 \, x + 16\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp(1)-2*x^3+24*x^2-64*x+256)/((4*x-16)*exp(1)+x^4+52*x^3-704*x^2+2944*x-4096)/log((4*exp(1)+x^3
+56*x^2-480*x+1024)/(4*x^2-32*x+64))^2,x, algorithm="fricas")

[Out]

2/log(1/4*(x^3 + 56*x^2 - 480*x + 4*e + 1024)/(x^2 - 8*x + 16))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp(1)-2*x^3+24*x^2-64*x+256)/((4*x-16)*exp(1)+x^4+52*x^3-704*x^2+2944*x-4096)/log((4*exp(1)+x^3
+56*x^2-480*x+1024)/(4*x^2-32*x+64))^2,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.17, size = 36, normalized size = 1.44

method	result	size
norman	\(\frac {2}{\ln \left (\frac {4 \,{\mathrm e}+x^{3}+56 x^{2}-480 x +1024}{4 x^{2}-32 x +64}\right )}\)	\(36\)
risch	\(\frac {2}{\ln \left (\frac {4 \,{\mathrm e}+x^{3}+56 x^{2}-480 x +1024}{4 x^{2}-32 x +64}\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*exp(1)-2*x^3+24*x^2-64*x+256)/((4*x-16)*exp(1)+x^4+52*x^3-704*x^2+2944*x-4096)/ln((4*exp(1)+x^3+56*x^2
-480*x+1024)/(4*x^2-32*x+64))^2,x,method=_RETURNVERBOSE)

[Out]

2/ln((4*exp(1)+x^3+56*x^2-480*x+1024)/(4*x^2-32*x+64))

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maxima [A]  time = 0.66, size = 35, normalized size = 1.40 \begin {gather*} -\frac {2}{2 \, \log \relax (2) - \log \left (x^{3} + 56 \, x^{2} - 480 \, x + 4 \, e + 1024\right ) + 2 \, \log \left (x - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp(1)-2*x^3+24*x^2-64*x+256)/((4*x-16)*exp(1)+x^4+52*x^3-704*x^2+2944*x-4096)/log((4*exp(1)+x^3
+56*x^2-480*x+1024)/(4*x^2-32*x+64))^2,x, algorithm="maxima")

[Out]

-2/(2*log(2) - log(x^3 + 56*x^2 - 480*x + 4*e + 1024) + 2*log(x - 4))

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mupad [B]  time = 5.76, size = 35, normalized size = 1.40 \begin {gather*} \frac {2}{\ln \left (\frac {x^3+56\,x^2-480\,x+4\,\mathrm {e}+1024}{4\,x^2-32\,x+64}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*exp(1) - 64*x + 24*x^2 - 2*x^3 + 256)/(log((4*exp(1) - 480*x + 56*x^2 + x^3 + 1024)/(4*x^2 - 32*x + 64
))^2*(2944*x - 704*x^2 + 52*x^3 + x^4 + exp(1)*(4*x - 16) - 4096)),x)

[Out]

2/log((4*exp(1) - 480*x + 56*x^2 + x^3 + 1024)/(4*x^2 - 32*x + 64))

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sympy [A]  time = 0.29, size = 31, normalized size = 1.24 \begin {gather*} \frac {2}{\log {\left (\frac {x^{3} + 56 x^{2} - 480 x + 4 e + 1024}{4 x^{2} - 32 x + 64} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp(1)-2*x**3+24*x**2-64*x+256)/((4*x-16)*exp(1)+x**4+52*x**3-704*x**2+2944*x-4096)/ln((4*exp(1)
+x**3+56*x**2-480*x+1024)/(4*x**2-32*x+64))**2,x)

[Out]

2/log((x**3 + 56*x**2 - 480*x + 4*E + 1024)/(4*x**2 - 32*x + 64))

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