3.92.39 \(\int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log (\frac {1-x^2}{x})-\log ^2(\frac {1-x^2}{x})} (2 x+2 x^2+2 x^3-2 x^4+(-2-2 x-2 x^2+2 x^3) \log (\frac {1-x^2}{x}))}{-x+x^3} \, dx\)

Optimal. Leaf size=35 \[ e^{e^2-\left (x-\log \left (\frac {1}{x}-x\right )\right )^2}-4 (4-x)^2 x^4 \]

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Rubi [F]  time = 13.79, 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 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(256*x^4 - 160*x^5 - 232*x^6 + 160*x^7 - 24*x^8 + E^(E^2 - x^2 + 2*x*Log[(1 - x^2)/x] - Log[(1 - x^2)/x]^2
)*(2*x + 2*x^2 + 2*x^3 - 2*x^4 + (-2 - 2*x - 2*x^2 + 2*x^3)*Log[(1 - x^2)/x]))/(-x + x^3),x]

[Out]

-64*x^4 + 32*x^5 - 4*x^6 + 2*Defer[Int][E^(E^2 - x^2 - Log[x^(-1) - x]^2)*((1 - x^2)/x)^(2*x), x] - 2*Defer[In
t][(E^(E^2 - x^2 - Log[x^(-1) - x]^2)*((1 - x^2)/x)^(2*x))/(1 - x), x] - 2*Defer[Int][E^(E^2 - x^2 - Log[x^(-1
) - x]^2)*x*((1 - x^2)/x)^(2*x), x] - 2*Defer[Int][(E^(E^2 - x^2 - Log[x^(-1) - x]^2)*((1 - x^2)/x)^(2*x))/(1
+ x), x] + 2*Defer[Int][E^(E^2 - x^2 - Log[x^(-1) - x]^2)*((1 - x^2)/x)^(2*x)*Log[(1 - x^2)/x], x] + 2*Defer[I
nt][(E^(E^2 - x^2 - Log[x^(-1) - x]^2)*((1 - x^2)/x)^(2*x)*Log[(1 - x^2)/x])/(-1 - x), x] + 2*Defer[Int][(E^(E
^2 - x^2 - Log[x^(-1) - x]^2)*((1 - x^2)/x)^(2*x)*Log[(1 - x^2)/x])/(1 - x), x] + 2*Defer[Int][(E^(E^2 - x^2 -
Log[x^(-1) - x]^2)*((1 - x^2)/x)^(2*x)*Log[(1 - x^2)/x])/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{x \left (-1+x^2\right )} \, dx\\ &=\int \left (\frac {256 x^3}{-1+x^2}-\frac {160 x^4}{-1+x^2}-\frac {232 x^5}{-1+x^2}+\frac {160 x^6}{-1+x^2}-\frac {24 x^7}{-1+x^2}+\frac {2 e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right ) \left (-x+\log \left (\frac {1}{x}-x\right )\right )}{x \left (1-x^2\right )}\right ) \, dx\\ &=2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right ) \left (-x+\log \left (\frac {1}{x}-x\right )\right )}{x \left (1-x^2\right )} \, dx-24 \int \frac {x^7}{-1+x^2} \, dx-160 \int \frac {x^4}{-1+x^2} \, dx+160 \int \frac {x^6}{-1+x^2} \, dx-232 \int \frac {x^5}{-1+x^2} \, dx+256 \int \frac {x^3}{-1+x^2} \, dx\\ &=2 \int \left (\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right )}{-1+x^2}+\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right ) \log \left (\frac {1-x^2}{x}\right )}{x \left (1-x^2\right )}\right ) \, dx-12 \operatorname {Subst}\left (\int \frac {x^3}{-1+x} \, dx,x,x^2\right )-116 \operatorname {Subst}\left (\int \frac {x^2}{-1+x} \, dx,x,x^2\right )+128 \operatorname {Subst}\left (\int \frac {x}{-1+x} \, dx,x,x^2\right )-160 \int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx+160 \int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx\\ &=32 x^5+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right )}{-1+x^2} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right ) \log \left (\frac {1-x^2}{x}\right )}{x \left (1-x^2\right )} \, dx-12 \operatorname {Subst}\left (\int \left (1+\frac {1}{-1+x}+x+x^2\right ) \, dx,x,x^2\right )-116 \operatorname {Subst}\left (\int \left (1+\frac {1}{-1+x}+x\right ) \, dx,x,x^2\right )+128 \operatorname {Subst}\left (\int \left (1+\frac {1}{-1+x}\right ) \, dx,x,x^2\right )\\ &=-64 x^4+32 x^5-4 x^6+2 \int \left (e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}-e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} x \left (\frac {1-x^2}{x}\right )^{2 x}+\frac {2 e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{-1+x^2}\right ) \, dx+2 \int \left (e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )+\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{-1-x}+\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{1-x}+\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{x}\right ) \, dx\\ &=-64 x^4+32 x^5-4 x^6+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \, dx-2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} x \left (\frac {1-x^2}{x}\right )^{2 x} \, dx+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right ) \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{-1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{x} \, dx+4 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{-1+x^2} \, dx\\ &=-64 x^4+32 x^5-4 x^6+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \, dx-2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} x \left (\frac {1-x^2}{x}\right )^{2 x} \, dx+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right ) \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{-1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{x} \, dx+4 \int \left (-\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{2 (1-x)}-\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{2 (1+x)}\right ) \, dx\\ &=-64 x^4+32 x^5-4 x^6+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \, dx-2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{1-x} \, dx-2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} x \left (\frac {1-x^2}{x}\right )^{2 x} \, dx-2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{1+x} \, dx+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right ) \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{-1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 46, normalized size = 1.31 \begin {gather*} e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1}{x}-x\right )^{2 x}-4 (-4+x)^2 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(256*x^4 - 160*x^5 - 232*x^6 + 160*x^7 - 24*x^8 + E^(E^2 - x^2 + 2*x*Log[(1 - x^2)/x] - Log[(1 - x^2
)/x]^2)*(2*x + 2*x^2 + 2*x^3 - 2*x^4 + (-2 - 2*x - 2*x^2 + 2*x^3)*Log[(1 - x^2)/x]))/(-x + x^3),x]

[Out]

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

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fricas [A]  time = 0.60, size = 54, normalized size = 1.54 \begin {gather*} -4 \, x^{6} + 32 \, x^{5} - 64 \, x^{4} + e^{\left (-x^{2} + 2 \, x \log \left (-\frac {x^{2} - 1}{x}\right ) - \log \left (-\frac {x^{2} - 1}{x}\right )^{2} + e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-2*x^2-2*x-2)*log((-x^2+1)/x)-2*x^4+2*x^3+2*x^2+2*x)*exp(-log((-x^2+1)/x)^2+2*x*log((-x^2+1)
/x)+exp(2)-x^2)-24*x^8+160*x^7-232*x^6-160*x^5+256*x^4)/(x^3-x),x, algorithm="fricas")

[Out]

-4*x^6 + 32*x^5 - 64*x^4 + e^(-x^2 + 2*x*log(-(x^2 - 1)/x) - log(-(x^2 - 1)/x)^2 + e^2)

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giac [A]  time = 0.92, size = 48, normalized size = 1.37 \begin {gather*} -4 \, x^{6} + 32 \, x^{5} - 64 \, x^{4} + e^{\left (-x^{2} + 2 \, x \log \left (-x + \frac {1}{x}\right ) - \log \left (-x + \frac {1}{x}\right )^{2} + e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-2*x^2-2*x-2)*log((-x^2+1)/x)-2*x^4+2*x^3+2*x^2+2*x)*exp(-log((-x^2+1)/x)^2+2*x*log((-x^2+1)
/x)+exp(2)-x^2)-24*x^8+160*x^7-232*x^6-160*x^5+256*x^4)/(x^3-x),x, algorithm="giac")

[Out]

-4*x^6 + 32*x^5 - 64*x^4 + e^(-x^2 + 2*x*log(-x + 1/x) - log(-x + 1/x)^2 + e^2)

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maple [A]  time = 0.51, size = 57, normalized size = 1.63

method	result	size
default	\({\mathrm e}^{-\ln \left (\frac {-x^{2}+1}{x}\right )^{2}+2 x \ln \left (\frac {-x^{2}+1}{x}\right )+{\mathrm e}^{2}-x^{2}}-4 x^{6}+32 x^{5}-64 x^{4}\)	\(57\)
risch	\(\left (\frac {-x^{2}+1}{x}\right )^{2 x} {\mathrm e}^{-\ln \left (\frac {-x^{2}+1}{x}\right )^{2}+{\mathrm e}^{2}-x^{2}}-4 x^{6}+32 x^{5}-64 x^{4}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^3-2*x^2-2*x-2)*ln((-x^2+1)/x)-2*x^4+2*x^3+2*x^2+2*x)*exp(-ln((-x^2+1)/x)^2+2*x*ln((-x^2+1)/x)+exp(2
)-x^2)-24*x^8+160*x^7-232*x^6-160*x^5+256*x^4)/(x^3-x),x,method=_RETURNVERBOSE)

[Out]

exp(-ln((-x^2+1)/x)^2+2*x*ln((-x^2+1)/x)+exp(2)-x^2)-4*x^6+32*x^5-64*x^4

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maxima [B]  time = 0.77, size = 100, normalized size = 2.86 \begin {gather*} -4 \, x^{6} + 32 \, x^{5} - 64 \, x^{4} + e^{\left (-x^{2} + 2 \, x \log \left (x + 1\right ) - \log \left (x + 1\right )^{2} - 2 \, x \log \relax (x) + 2 \, \log \left (x + 1\right ) \log \relax (x) - \log \relax (x)^{2} + 2 \, x \log \left (-x + 1\right ) - 2 \, \log \left (x + 1\right ) \log \left (-x + 1\right ) + 2 \, \log \relax (x) \log \left (-x + 1\right ) - \log \left (-x + 1\right )^{2} + e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-2*x^2-2*x-2)*log((-x^2+1)/x)-2*x^4+2*x^3+2*x^2+2*x)*exp(-log((-x^2+1)/x)^2+2*x*log((-x^2+1)
/x)+exp(2)-x^2)-24*x^8+160*x^7-232*x^6-160*x^5+256*x^4)/(x^3-x),x, algorithm="maxima")

[Out]

-4*x^6 + 32*x^5 - 64*x^4 + e^(-x^2 + 2*x*log(x + 1) - log(x + 1)^2 - 2*x*log(x) + 2*log(x + 1)*log(x) - log(x)
^2 + 2*x*log(-x + 1) - 2*log(x + 1)*log(-x + 1) + 2*log(x)*log(-x + 1) - log(-x + 1)^2 + e^2)

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mupad [B]  time = 7.41, size = 55, normalized size = 1.57 \begin {gather*} 32\,x^5-64\,x^4-4\,x^6+{\mathrm {e}}^{-x^2-{\ln \left (-\frac {x^2-1}{x}\right )}^2+{\mathrm {e}}^2}\,{\left (-\frac {x^2-1}{x}\right )}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(2) - log(-(x^2 - 1)/x)^2 + 2*x*log(-(x^2 - 1)/x) - x^2)*(2*x + 2*x^2 + 2*x^3 - 2*x^4 - log(-(x^2
- 1)/x)*(2*x + 2*x^2 - 2*x^3 + 2)) + 256*x^4 - 160*x^5 - 232*x^6 + 160*x^7 - 24*x^8)/(x - x^3),x)

[Out]

32*x^5 - 64*x^4 - 4*x^6 + exp(exp(2) - log(-(x^2 - 1)/x)^2 - x^2)*(-(x^2 - 1)/x)^(2*x)

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sympy [A]  time = 0.55, size = 44, normalized size = 1.26 \begin {gather*} - 4 x^{6} + 32 x^{5} - 64 x^{4} + e^{- x^{2} + 2 x \log {\left (\frac {1 - x^{2}}{x} \right )} - \log {\left (\frac {1 - x^{2}}{x} \right )}^{2} + e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**3-2*x**2-2*x-2)*ln((-x**2+1)/x)-2*x**4+2*x**3+2*x**2+2*x)*exp(-ln((-x**2+1)/x)**2+2*x*ln((-x
**2+1)/x)+exp(2)-x**2)-24*x**8+160*x**7-232*x**6-160*x**5+256*x**4)/(x**3-x),x)

[Out]

-4*x**6 + 32*x**5 - 64*x**4 + exp(-x**2 + 2*x*log((1 - x**2)/x) - log((1 - x**2)/x)**2 + exp(2))

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