3.42.53 \(\int \frac {8+14 x^2+2 e^{2 x} x^2-12 x^3+2 x^4+e^x (-8+16 x^2-4 x^3)+(-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} (-2+x^2)+e^x (-12+4 x+6 x^2-2 x^3)) \log (-2+x^2)}{-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} (-2+x^2)+e^x (-12+4 x+6 x^2-2 x^3)} \, dx\)

Optimal. Leaf size=28 \[ 5+x \left (\frac {4}{x-\left (4+e^x-x\right ) x}+\log \left (-2+x^2\right )\right ) \]

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Rubi [F]  time = 0.72, 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 {8+14 x^2+2 e^{2 x} x^2-12 x^3+2 x^4+e^x \left (-8+16 x^2-4 x^3\right )+\left (-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )\right ) \log \left (-2+x^2\right )}{-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8 + 14*x^2 + 2*E^(2*x)*x^2 - 12*x^3 + 2*x^4 + E^x*(-8 + 16*x^2 - 4*x^3) + (-18 + 12*x + 7*x^2 - 6*x^3 + x
^4 + E^(2*x)*(-2 + x^2) + E^x*(-12 + 4*x + 6*x^2 - 2*x^3))*Log[-2 + x^2])/(-18 + 12*x + 7*x^2 - 6*x^3 + x^4 +
E^(2*x)*(-2 + x^2) + E^x*(-12 + 4*x + 6*x^2 - 2*x^3)),x]

[Out]

x*Log[-2 + x^2] - 16*Defer[Int][(3 + E^x - x)^(-2), x] + 4*Defer[Int][(3 + E^x - x)^(-1), x] + 4*Defer[Int][x/
(3 + E^x - x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (4+7 x^2+e^{2 x} x^2-6 x^3+x^4-2 e^x \left (2-4 x^2+x^3\right )\right )}{\left (3+e^x-x\right )^2 \left (-2+x^2\right )}+\log \left (-2+x^2\right )\right ) \, dx\\ &=2 \int \frac {4+7 x^2+e^{2 x} x^2-6 x^3+x^4-2 e^x \left (2-4 x^2+x^3\right )}{\left (3+e^x-x\right )^2 \left (-2+x^2\right )} \, dx+\int \log \left (-2+x^2\right ) \, dx\\ &=x \log \left (-2+x^2\right )-2 \int \frac {x^2}{-2+x^2} \, dx+2 \int \left (\frac {2}{3+e^x-x}+\frac {2 (-4+x)}{\left (3+e^x-x\right )^2}+\frac {x^2}{-2+x^2}\right ) \, dx\\ &=-2 x+x \log \left (-2+x^2\right )+2 \int \frac {x^2}{-2+x^2} \, dx+4 \int \frac {1}{3+e^x-x} \, dx+4 \int \frac {-4+x}{\left (3+e^x-x\right )^2} \, dx-4 \int \frac {1}{-2+x^2} \, dx\\ &=2 \sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )+x \log \left (-2+x^2\right )+4 \int \frac {1}{3+e^x-x} \, dx+4 \int \left (-\frac {4}{\left (3+e^x-x\right )^2}+\frac {x}{\left (3+e^x-x\right )^2}\right ) \, dx+4 \int \frac {1}{-2+x^2} \, dx\\ &=x \log \left (-2+x^2\right )+4 \int \frac {1}{3+e^x-x} \, dx+4 \int \frac {x}{\left (3+e^x-x\right )^2} \, dx-16 \int \frac {1}{\left (3+e^x-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.20, size = 67, normalized size = 2.39 \begin {gather*} -\frac {4}{3+e^x-x}+2 \sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )+\sqrt {2} \log \left (\sqrt {2}-x\right )-\sqrt {2} \log \left (\sqrt {2}+x\right )+x \log \left (-2+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + 14*x^2 + 2*E^(2*x)*x^2 - 12*x^3 + 2*x^4 + E^x*(-8 + 16*x^2 - 4*x^3) + (-18 + 12*x + 7*x^2 - 6*x
^3 + x^4 + E^(2*x)*(-2 + x^2) + E^x*(-12 + 4*x + 6*x^2 - 2*x^3))*Log[-2 + x^2])/(-18 + 12*x + 7*x^2 - 6*x^3 +
x^4 + E^(2*x)*(-2 + x^2) + E^x*(-12 + 4*x + 6*x^2 - 2*x^3)),x]

[Out]

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

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fricas [A]  time = 0.59, size = 31, normalized size = 1.11 \begin {gather*} \frac {{\left (x^{2} - x e^{x} - 3 \, x\right )} \log \left (x^{2} - 2\right ) + 4}{x - e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+12*x-18)*log(x^2-2)+2*exp(x)^2*x^2+(
-4*x^3+16*x^2-8)*exp(x)+2*x^4-12*x^3+14*x^2+8)/((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+
12*x-18),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.20, size = 42, normalized size = 1.50 \begin {gather*} \frac {x^{2} \log \left (x^{2} - 2\right ) - x e^{x} \log \left (x^{2} - 2\right ) - 3 \, x \log \left (x^{2} - 2\right ) + 4}{x - e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+12*x-18)*log(x^2-2)+2*exp(x)^2*x^2+(
-4*x^3+16*x^2-8)*exp(x)+2*x^4-12*x^3+14*x^2+8)/((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+
12*x-18),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 21, normalized size = 0.75

method	result	size
risch	\(\ln \left (x^{2}-2\right ) x +\frac {4}{-{\mathrm e}^{x}+x -3}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+12*x-18)*ln(x^2-2)+2*exp(x)^2*x^2+(-4*x^3+
16*x^2-8)*exp(x)+2*x^4-12*x^3+14*x^2+8)/((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+12*x-18
),x,method=_RETURNVERBOSE)

[Out]

ln(x^2-2)*x+4/(-exp(x)+x-3)

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maxima [A]  time = 0.83, size = 31, normalized size = 1.11 \begin {gather*} \frac {{\left (x^{2} - x e^{x} - 3 \, x\right )} \log \left (x^{2} - 2\right ) + 4}{x - e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+12*x-18)*log(x^2-2)+2*exp(x)^2*x^2+(
-4*x^3+16*x^2-8)*exp(x)+2*x^4-12*x^3+14*x^2+8)/((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+
12*x-18),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.06, size = 20, normalized size = 0.71 \begin {gather*} x\,\ln \left (x^2-2\right )-\frac {4}{{\mathrm {e}}^x-x+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2 - 2)*(12*x + exp(2*x)*(x^2 - 2) + 7*x^2 - 6*x^3 + x^4 + exp(x)*(4*x + 6*x^2 - 2*x^3 - 12) - 18) -
exp(x)*(4*x^3 - 16*x^2 + 8) + 2*x^2*exp(2*x) + 14*x^2 - 12*x^3 + 2*x^4 + 8)/(12*x + exp(2*x)*(x^2 - 2) + 7*x^
2 - 6*x^3 + x^4 + exp(x)*(4*x + 6*x^2 - 2*x^3 - 12) - 18),x)

[Out]

x*log(x^2 - 2) - 4/(exp(x) - x + 3)

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sympy [A]  time = 0.36, size = 15, normalized size = 0.54 \begin {gather*} x \log {\left (x^{2} - 2 \right )} - \frac {4}{- x + e^{x} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-2)*exp(x)**2+(-2*x**3+6*x**2+4*x-12)*exp(x)+x**4-6*x**3+7*x**2+12*x-18)*ln(x**2-2)+2*exp(x)*
*2*x**2+(-4*x**3+16*x**2-8)*exp(x)+2*x**4-12*x**3+14*x**2+8)/((x**2-2)*exp(x)**2+(-2*x**3+6*x**2+4*x-12)*exp(x
)+x**4-6*x**3+7*x**2+12*x-18),x)

[Out]

x*log(x**2 - 2) - 4/(-x + exp(x) + 3)

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