3.31.64 \(\int \frac {e^{\frac {1}{\log ^2(4-40 x+36 x^2+320 x^3+256 x^4)}} (-20-44 x+64 x^2+(-1+5 x+8 x^2) \log ^3(4-40 x+36 x^2+320 x^3+256 x^4))}{(-1+7 x-3 x^2-11 x^3+8 x^4) \log ^3(4-40 x+36 x^2+320 x^3+256 x^4)} \, dx\)

Optimal. Leaf size=29 \[ \frac {e^{\frac {1}{\log ^2\left (\left (3-2 x-(1+4 x)^2\right )^2\right )}}}{1-x} \]

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Rubi [B]  time = 0.19, antiderivative size = 95, normalized size of antiderivative = 3.28, number of steps used = 1, number of rules used = 1, integrand size = 115, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {2288} \begin {gather*} \frac {\left (-16 x^2+11 x+5\right ) \left (64 x^4+80 x^3+9 x^2-10 x+1\right ) e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}}}{\left (-128 x^3-120 x^2-9 x+5\right ) \left (-8 x^4+11 x^3+3 x^2-7 x+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^Log[4 - 40*x + 36*x^2 + 320*x^3 + 256*x^4]^(-2)*(-20 - 44*x + 64*x^2 + (-1 + 5*x + 8*x^2)*Log[4 - 40*x
+ 36*x^2 + 320*x^3 + 256*x^4]^3))/((-1 + 7*x - 3*x^2 - 11*x^3 + 8*x^4)*Log[4 - 40*x + 36*x^2 + 320*x^3 + 256*x
^4]^3),x]

[Out]

(E^Log[4 - 40*x + 36*x^2 + 320*x^3 + 256*x^4]^(-2)*(5 + 11*x - 16*x^2)*(1 - 10*x + 9*x^2 + 80*x^3 + 64*x^4))/(
(5 - 9*x - 120*x^2 - 128*x^3)*(1 - 7*x + 3*x^2 + 11*x^3 - 8*x^4))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{\frac {1}{\log ^2\left (4-40 x+36 x^2+320 x^3+256 x^4\right )}} \left (5+11 x-16 x^2\right ) \left (1-10 x+9 x^2+80 x^3+64 x^4\right )}{\left (5-9 x-120 x^2-128 x^3\right ) \left (1-7 x+3 x^2+11 x^3-8 x^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 26, normalized size = 0.90 \begin {gather*} -\frac {e^{\frac {1}{\log ^2\left (4 \left (-1+5 x+8 x^2\right )^2\right )}}}{-1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^Log[4 - 40*x + 36*x^2 + 320*x^3 + 256*x^4]^(-2)*(-20 - 44*x + 64*x^2 + (-1 + 5*x + 8*x^2)*Log[4 -
40*x + 36*x^2 + 320*x^3 + 256*x^4]^3))/((-1 + 7*x - 3*x^2 - 11*x^3 + 8*x^4)*Log[4 - 40*x + 36*x^2 + 320*x^3 +
256*x^4]^3),x]

[Out]

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

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fricas [A]  time = 0.61, size = 31, normalized size = 1.07 \begin {gather*} -\frac {e^{\left (\frac {1}{\log \left (256 \, x^{4} + 320 \, x^{3} + 36 \, x^{2} - 40 \, x + 4\right )^{2}}\right )}}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+5*x-1)*log(256*x^4+320*x^3+36*x^2-40*x+4)^3+64*x^2-44*x-20)*exp(1/log(256*x^4+320*x^3+36*x^2
-40*x+4)^2)/(8*x^4-11*x^3-3*x^2+7*x-1)/log(256*x^4+320*x^3+36*x^2-40*x+4)^3,x, algorithm="fricas")

[Out]

-e^(log(256*x^4 + 320*x^3 + 36*x^2 - 40*x + 4)^(-2))/(x - 1)

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giac [A]  time = 3.11, size = 31, normalized size = 1.07 \begin {gather*} -\frac {e^{\left (\frac {1}{\log \left (256 \, x^{4} + 320 \, x^{3} + 36 \, x^{2} - 40 \, x + 4\right )^{2}}\right )}}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+5*x-1)*log(256*x^4+320*x^3+36*x^2-40*x+4)^3+64*x^2-44*x-20)*exp(1/log(256*x^4+320*x^3+36*x^2
-40*x+4)^2)/(8*x^4-11*x^3-3*x^2+7*x-1)/log(256*x^4+320*x^3+36*x^2-40*x+4)^3,x, algorithm="giac")

[Out]

-e^(log(256*x^4 + 320*x^3 + 36*x^2 - 40*x + 4)^(-2))/(x - 1)

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maple [A]  time = 0.08, size = 32, normalized size = 1.10

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {1}{\ln \left (256 x^{4}+320 x^{3}+36 x^{2}-40 x +4\right )^{2}}}}{x -1}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^2+5*x-1)*ln(256*x^4+320*x^3+36*x^2-40*x+4)^3+64*x^2-44*x-20)*exp(1/ln(256*x^4+320*x^3+36*x^2-40*x+4)
^2)/(8*x^4-11*x^3-3*x^2+7*x-1)/ln(256*x^4+320*x^3+36*x^2-40*x+4)^3,x,method=_RETURNVERBOSE)

[Out]

-1/(x-1)*exp(1/ln(256*x^4+320*x^3+36*x^2-40*x+4)^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {16 \, x^{2} e^{\left (\frac {1}{4 \, {\left (\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (8 \, x^{2} + 5 \, x - 1\right ) + \log \left (8 \, x^{2} + 5 \, x - 1\right )^{2}\right )}}\right )}}{16 \, x^{3} - 27 \, x^{2} + 6 \, x + 5} + \frac {11 \, x e^{\left (\frac {1}{4 \, {\left (\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (8 \, x^{2} + 5 \, x - 1\right ) + \log \left (8 \, x^{2} + 5 \, x - 1\right )^{2}\right )}}\right )}}{16 \, x^{3} - 27 \, x^{2} + 6 \, x + 5} + \frac {5 \, e^{\left (\frac {1}{4 \, {\left (\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (8 \, x^{2} + 5 \, x - 1\right ) + \log \left (8 \, x^{2} + 5 \, x - 1\right )^{2}\right )}}\right )}}{16 \, x^{3} - 27 \, x^{2} + 6 \, x + 5} + \int \frac {e^{\left (\frac {1}{4 \, {\left (\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (8 \, x^{2} + 5 \, x - 1\right ) + \log \left (8 \, x^{2} + 5 \, x - 1\right )^{2}\right )}}\right )}}{x^{2} - 2 \, x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+5*x-1)*log(256*x^4+320*x^3+36*x^2-40*x+4)^3+64*x^2-44*x-20)*exp(1/log(256*x^4+320*x^3+36*x^2
-40*x+4)^2)/(8*x^4-11*x^3-3*x^2+7*x-1)/log(256*x^4+320*x^3+36*x^2-40*x+4)^3,x, algorithm="maxima")

[Out]

-16*x^2*e^(1/4/(log(2)^2 + 2*log(2)*log(8*x^2 + 5*x - 1) + log(8*x^2 + 5*x - 1)^2))/(16*x^3 - 27*x^2 + 6*x + 5
) + 11*x*e^(1/4/(log(2)^2 + 2*log(2)*log(8*x^2 + 5*x - 1) + log(8*x^2 + 5*x - 1)^2))/(16*x^3 - 27*x^2 + 6*x +
5) + 5*e^(1/4/(log(2)^2 + 2*log(2)*log(8*x^2 + 5*x - 1) + log(8*x^2 + 5*x - 1)^2))/(16*x^3 - 27*x^2 + 6*x + 5)
+ integrate(e^(1/4/(log(2)^2 + 2*log(2)*log(8*x^2 + 5*x - 1) + log(8*x^2 + 5*x - 1)^2))/(x^2 - 2*x + 1), x)

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mupad [B]  time = 2.04, size = 31, normalized size = 1.07 \begin {gather*} -\frac {{\mathrm {e}}^{\frac {1}{{\ln \left (256\,x^4+320\,x^3+36\,x^2-40\,x+4\right )}^2}}}{x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1/log(36*x^2 - 40*x + 320*x^3 + 256*x^4 + 4)^2)*(44*x - log(36*x^2 - 40*x + 320*x^3 + 256*x^4 + 4)^3*
(5*x + 8*x^2 - 1) - 64*x^2 + 20))/(log(36*x^2 - 40*x + 320*x^3 + 256*x^4 + 4)^3*(3*x^2 - 7*x + 11*x^3 - 8*x^4
+ 1)),x)

[Out]

-exp(1/log(36*x^2 - 40*x + 320*x^3 + 256*x^4 + 4)^2)/(x - 1)

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sympy [A]  time = 0.40, size = 31, normalized size = 1.07 \begin {gather*} - \frac {e^{\frac {1}{\log {\left (256 x^{4} + 320 x^{3} + 36 x^{2} - 40 x + 4 \right )}^{2}}}}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**2+5*x-1)*ln(256*x**4+320*x**3+36*x**2-40*x+4)**3+64*x**2-44*x-20)*exp(1/ln(256*x**4+320*x**3+
36*x**2-40*x+4)**2)/(8*x**4-11*x**3-3*x**2+7*x-1)/ln(256*x**4+320*x**3+36*x**2-40*x+4)**3,x)

[Out]

-exp(log(256*x**4 + 320*x**3 + 36*x**2 - 40*x + 4)**(-2))/(x - 1)

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