3.59.83 \(\int \frac {-24+e^x (6-3 x+3 x^3)+(12-12 x^2+e^x (-3+3 x^2)) \log (\frac {x^2}{4-4 x^2+e^x (-1+x^2)})+(-12+12 x^2+e^x (3-3 x^2)) \log ^2(\frac {x^2}{4-4 x^2+e^x (-1+x^2)})}{(4-4 x^2+e^x (-1+x^2)) \log ^2(\frac {x^2}{4-4 x^2+e^x (-1+x^2)})} \, dx\)

Optimal. Leaf size=27 \[ x \left (-3+\frac {3}{\log \left (\frac {x}{\left (-4+e^x\right ) \left (-\frac {1}{x}+x\right )}\right )}\right ) \]

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Rubi [F]  time = 1.26, 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 {-24+e^x \left (6-3 x+3 x^3\right )+\left (12-12 x^2+e^x \left (-3+3 x^2\right )\right ) \log \left (\frac {x^2}{4-4 x^2+e^x \left (-1+x^2\right )}\right )+\left (-12+12 x^2+e^x \left (3-3 x^2\right )\right ) \log ^2\left (\frac {x^2}{4-4 x^2+e^x \left (-1+x^2\right )}\right )}{\left (4-4 x^2+e^x \left (-1+x^2\right )\right ) \log ^2\left (\frac {x^2}{4-4 x^2+e^x \left (-1+x^2\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-24 + E^x*(6 - 3*x + 3*x^3) + (12 - 12*x^2 + E^x*(-3 + 3*x^2))*Log[x^2/(4 - 4*x^2 + E^x*(-1 + x^2))] + (-
12 + 12*x^2 + E^x*(3 - 3*x^2))*Log[x^2/(4 - 4*x^2 + E^x*(-1 + x^2))]^2)/((4 - 4*x^2 + E^x*(-1 + x^2))*Log[x^2/
(4 - 4*x^2 + E^x*(-1 + x^2))]^2),x]

[Out]

-3*x + 3*Defer[Int][1/((-1 + x)*Log[x^2/((-4 + E^x)*(-1 + x^2))]^2), x] + 3*Defer[Int][x/Log[x^2/((-4 + E^x)*(
-1 + x^2))]^2, x] + 12*Defer[Int][x/((-4 + E^x)*Log[x^2/((-4 + E^x)*(-1 + x^2))]^2), x] - 3*Defer[Int][1/((1 +
x)*Log[x^2/((-4 + E^x)*(-1 + x^2))]^2), x] + 3*Defer[Int][Log[x^2/((-4 + E^x)*(-1 + x^2))]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 3 \left (-1+\frac {-8+e^x \left (2-x+x^3\right )}{\left (-4+e^x\right ) \left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}+\frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \, dx\\ &=3 \int \left (-1+\frac {-8+e^x \left (2-x+x^3\right )}{\left (-4+e^x\right ) \left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}+\frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \, dx\\ &=-3 x+3 \int \frac {-8+e^x \left (2-x+x^3\right )}{\left (-4+e^x\right ) \left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \left (\frac {4 x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}+\frac {2-x+x^3}{\left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \frac {2-x+x^3}{\left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+12 \int \frac {x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \left (\frac {x}{\log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}+\frac {2}{\left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+12 \int \frac {x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \frac {x}{\log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+6 \int \frac {1}{\left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+12 \int \frac {x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \frac {x}{\log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+6 \int \left (\frac {1}{2 (-1+x) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}-\frac {1}{2 (1+x) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \, dx+12 \int \frac {x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \frac {1}{(-1+x) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {x}{\log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx-3 \int \frac {1}{(1+x) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+12 \int \frac {x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 29, normalized size = 1.07 \begin {gather*} 3 \left (-x+\frac {x}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-24 + E^x*(6 - 3*x + 3*x^3) + (12 - 12*x^2 + E^x*(-3 + 3*x^2))*Log[x^2/(4 - 4*x^2 + E^x*(-1 + x^2))
] + (-12 + 12*x^2 + E^x*(3 - 3*x^2))*Log[x^2/(4 - 4*x^2 + E^x*(-1 + x^2))]^2)/((4 - 4*x^2 + E^x*(-1 + x^2))*Lo
g[x^2/(4 - 4*x^2 + E^x*(-1 + x^2))]^2),x]

[Out]

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

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fricas [B]  time = 0.61, size = 58, normalized size = 2.15 \begin {gather*} -\frac {3 \, {\left (x \log \left (-\frac {x^{2}}{4 \, x^{2} - {\left (x^{2} - 1\right )} e^{x} - 4}\right ) - x\right )}}{\log \left (-\frac {x^{2}}{4 \, x^{2} - {\left (x^{2} - 1\right )} e^{x} - 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^2+3)*exp(x)+12*x^2-12)*log(x^2/((x^2-1)*exp(x)-4*x^2+4))^2+((3*x^2-3)*exp(x)-12*x^2+12)*log(
x^2/((x^2-1)*exp(x)-4*x^2+4))+(3*x^3-3*x+6)*exp(x)-24)/((x^2-1)*exp(x)-4*x^2+4)/log(x^2/((x^2-1)*exp(x)-4*x^2+
4))^2,x, algorithm="fricas")

[Out]

-3*(x*log(-x^2/(4*x^2 - (x^2 - 1)*e^x - 4)) - x)/log(-x^2/(4*x^2 - (x^2 - 1)*e^x - 4))

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giac [B]  time = 1.29, size = 58, normalized size = 2.15 \begin {gather*} -\frac {3 \, {\left (x \log \left (\frac {x^{2}}{x^{2} e^{x} - 4 \, x^{2} - e^{x} + 4}\right ) - x\right )}}{\log \left (\frac {x^{2}}{x^{2} e^{x} - 4 \, x^{2} - e^{x} + 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^2+3)*exp(x)+12*x^2-12)*log(x^2/((x^2-1)*exp(x)-4*x^2+4))^2+((3*x^2-3)*exp(x)-12*x^2+12)*log(
x^2/((x^2-1)*exp(x)-4*x^2+4))+(3*x^3-3*x+6)*exp(x)-24)/((x^2-1)*exp(x)-4*x^2+4)/log(x^2/((x^2-1)*exp(x)-4*x^2+
4))^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.27, size = 348, normalized size = 12.89

method	result	size
risch	\(-3 x +\frac {6 i x}{\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{x}-4}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{x}-4}\right ) \mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right )^{3}-\pi \,\mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\left (x^{2}-1\right ) \left ({\mathrm e}^{x}-4\right )}\right )^{2}+\pi \,\mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\left (x^{2}-1\right ) \left ({\mathrm e}^{x}-4\right )}\right ) \mathrm {csgn}\left (i x^{2}\right )+\pi \mathrm {csgn}\left (\frac {i x^{2}}{\left (x^{2}-1\right ) \left ({\mathrm e}^{x}-4\right )}\right )^{3}-\pi \mathrm {csgn}\left (\frac {i x^{2}}{\left (x^{2}-1\right ) \left ({\mathrm e}^{x}-4\right )}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )-2 i \ln \left (x^{2}-1\right )-2 i \ln \left ({\mathrm e}^{x}-4\right )}\)	\(348\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-3*x^2+3)*exp(x)+12*x^2-12)*ln(x^2/((x^2-1)*exp(x)-4*x^2+4))^2+((3*x^2-3)*exp(x)-12*x^2+12)*ln(x^2/((x^
2-1)*exp(x)-4*x^2+4))+(3*x^3-3*x+6)*exp(x)-24)/((x^2-1)*exp(x)-4*x^2+4)/ln(x^2/((x^2-1)*exp(x)-4*x^2+4))^2,x,m
ethod=_RETURNVERBOSE)

[Out]

-3*x+6*I*x/(Pi*csgn(I/(exp(x)-4))*csgn(I/(x^2-1))*csgn(I/(exp(x)-4)/(x^2-1))-Pi*csgn(I/(exp(x)-4))*csgn(I/(exp
(x)-4)/(x^2-1))^2-Pi*csgn(I/(x^2-1))*csgn(I/(exp(x)-4)/(x^2-1))^2+Pi*csgn(I/(exp(x)-4)/(x^2-1))^3-Pi*csgn(I/(e
xp(x)-4)/(x^2-1))*csgn(I*x^2/(x^2-1)/(exp(x)-4))^2+Pi*csgn(I/(exp(x)-4)/(x^2-1))*csgn(I*x^2/(x^2-1)/(exp(x)-4)
)*csgn(I*x^2)+Pi*csgn(I*x^2/(x^2-1)/(exp(x)-4))^3-Pi*csgn(I*x^2/(x^2-1)/(exp(x)-4))^2*csgn(I*x^2)+Pi*csgn(I*x)
^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+4*I*ln(x)-2*I*ln(x^2-1)-2*I*ln(exp(x)-4))

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maxima [A]  time = 0.45, size = 48, normalized size = 1.78 \begin {gather*} -\frac {3 \, {\left (x \log \left (x + 1\right ) + x \log \left (x - 1\right ) - 2 \, x \log \relax (x) + x \log \left (e^{x} - 4\right ) + x\right )}}{\log \left (x + 1\right ) + \log \left (x - 1\right ) - 2 \, \log \relax (x) + \log \left (e^{x} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^2+3)*exp(x)+12*x^2-12)*log(x^2/((x^2-1)*exp(x)-4*x^2+4))^2+((3*x^2-3)*exp(x)-12*x^2+12)*log(
x^2/((x^2-1)*exp(x)-4*x^2+4))+(3*x^3-3*x+6)*exp(x)-24)/((x^2-1)*exp(x)-4*x^2+4)/log(x^2/((x^2-1)*exp(x)-4*x^2+
4))^2,x, algorithm="maxima")

[Out]

-3*(x*log(x + 1) + x*log(x - 1) - 2*x*log(x) + x*log(e^x - 4) + x)/(log(x + 1) + log(x - 1) - 2*log(x) + log(e
^x - 4))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\left ({\mathrm {e}}^x\,\left (3\,x^2-3\right )-12\,x^2+12\right )\,{\ln \left (\frac {x^2}{{\mathrm {e}}^x\,\left (x^2-1\right )-4\,x^2+4}\right )}^2+\left (12\,x^2-{\mathrm {e}}^x\,\left (3\,x^2-3\right )-12\right )\,\ln \left (\frac {x^2}{{\mathrm {e}}^x\,\left (x^2-1\right )-4\,x^2+4}\right )-{\mathrm {e}}^x\,\left (3\,x^3-3\,x+6\right )+24}{{\ln \left (\frac {x^2}{{\mathrm {e}}^x\,\left (x^2-1\right )-4\,x^2+4}\right )}^2\,\left ({\mathrm {e}}^x\,\left (x^2-1\right )-4\,x^2+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x^2/(exp(x)*(x^2 - 1) - 4*x^2 + 4))^2*(exp(x)*(3*x^2 - 3) - 12*x^2 + 12) - exp(x)*(3*x^3 - 3*x + 6)
- log(x^2/(exp(x)*(x^2 - 1) - 4*x^2 + 4))*(exp(x)*(3*x^2 - 3) - 12*x^2 + 12) + 24)/(log(x^2/(exp(x)*(x^2 - 1)
- 4*x^2 + 4))^2*(exp(x)*(x^2 - 1) - 4*x^2 + 4)),x)

[Out]

int(-(log(x^2/(exp(x)*(x^2 - 1) - 4*x^2 + 4))^2*(exp(x)*(3*x^2 - 3) - 12*x^2 + 12) - exp(x)*(3*x^3 - 3*x + 6)
- log(x^2/(exp(x)*(x^2 - 1) - 4*x^2 + 4))*(exp(x)*(3*x^2 - 3) - 12*x^2 + 12) + 24)/(log(x^2/(exp(x)*(x^2 - 1)
- 4*x^2 + 4))^2*(exp(x)*(x^2 - 1) - 4*x^2 + 4)), x)

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sympy [A]  time = 0.30, size = 26, normalized size = 0.96 \begin {gather*} - 3 x + \frac {3 x}{\log {\left (\frac {x^{2}}{- 4 x^{2} + \left (x^{2} - 1\right ) e^{x} + 4} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x**2+3)*exp(x)+12*x**2-12)*ln(x**2/((x**2-1)*exp(x)-4*x**2+4))**2+((3*x**2-3)*exp(x)-12*x**2+1
2)*ln(x**2/((x**2-1)*exp(x)-4*x**2+4))+(3*x**3-3*x+6)*exp(x)-24)/((x**2-1)*exp(x)-4*x**2+4)/ln(x**2/((x**2-1)*
exp(x)-4*x**2+4))**2,x)

[Out]

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

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