3.60.16 \(\int \frac {-1+4 x^2-5 x^4+2 x^6+e^{x^2} (-3+2 x-x^4)}{-x+2 x^3-x^5+e^{x^2} (2-3 x-3 x^2+4 x^3+x^4-x^5+e^5 (4-8 x^2+4 x^4))} \, dx\)

Optimal. Leaf size=31 \[ \log \left (-2-4 e^5+x+e^{-x^2} x+\frac {-2+x}{-\frac {1}{x}+x}\right ) \]

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Rubi [F]  time = 5.34, 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 {-1+4 x^2-5 x^4+2 x^6+e^{x^2} \left (-3+2 x-x^4\right )}{-x+2 x^3-x^5+e^{x^2} \left (2-3 x-3 x^2+4 x^3+x^4-x^5+e^5 \left (4-8 x^2+4 x^4\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + 4*x^2 - 5*x^4 + 2*x^6 + E^x^2*(-3 + 2*x - x^4))/(-x + 2*x^3 - x^5 + E^x^2*(2 - 3*x - 3*x^2 + 4*x^3 +
x^4 - x^5 + E^5*(4 - 8*x^2 + 4*x^4))),x]

[Out]

-Log[1 - x^2] + Log[2*(1 + 2*E^5) - 3*x - (1 + 4*E^5)*x^2 + x^3] + 2*Defer[Int][x^4/(-2*E^x^2*(1 + 2*E^5) + x
+ 3*E^x^2*x + E^x^2*(1 + 4*E^5)*x^2 - x^3 - E^x^2*x^3), x] - (1 + 4*E^5)*Defer[Int][x/(2*E^x^2*(1 + 2*E^5) - x
- 3*E^x^2*x - E^x^2*(1 + 4*E^5)*x^2 + x^3 + E^x^2*x^3), x] + 2*Defer[Int][x^2/(2*E^x^2*(1 + 2*E^5) - x - 3*E^
x^2*x - E^x^2*(1 + 4*E^5)*x^2 + x^3 + E^x^2*x^3), x] + 8*(1 + 4*E^5 + 8*E^10 + 8*E^15)*Defer[Int][1/((2*(1 + 2
*E^5) - 3*x - (1 + 4*E^5)*x^2 + x^3)*(2*E^x^2*(1 + 2*E^5) - x - 3*E^x^2*x - E^x^2*(1 + 4*E^5)*x^2 + x^3 + E^x^
2*x^3)), x] - (13 + 12*E^5 + 32*E^10)*Defer[Int][x/((2*(1 + 2*E^5) - 3*x - (1 + 4*E^5)*x^2 + x^3)*(2*E^x^2*(1
+ 2*E^5) - x - 3*E^x^2*x - E^x^2*(1 + 4*E^5)*x^2 + x^3 + E^x^2*x^3)), x] - (3 + 32*E^5 + 48*E^10 + 64*E^15)*De
fer[Int][x^2/((2*(1 + 2*E^5) - 3*x - (1 + 4*E^5)*x^2 + x^3)*(2*E^x^2*(1 + 2*E^5) - x - 3*E^x^2*x - E^x^2*(1 +
4*E^5)*x^2 + x^3 + E^x^2*x^3)), x] + (5 + 8*E^5 + 16*E^10)*Defer[Int][(x - x^3 + 4*E^(5 + x^2)*(-1 + x^2) + E^
x^2*(-2 + 3*x + x^2 - x^3))^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-3+2 x-x^4}{(1-x) (1+x) \left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right )}+\frac {-2 \left (1+2 e^5\right )+9 \left (1+\frac {16 e^5}{9}\right ) x^2-10 x^3-7 \left (1+\frac {20 e^5}{7}\right ) x^4+8 x^5+2 \left (1+4 e^5\right ) x^6-2 x^7}{\left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right ) \left (2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3\right )}\right ) \, dx\\ &=\int \frac {-3+2 x-x^4}{(1-x) (1+x) \left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right )} \, dx+\int \frac {-2 \left (1+2 e^5\right )+9 \left (1+\frac {16 e^5}{9}\right ) x^2-10 x^3-7 \left (1+\frac {20 e^5}{7}\right ) x^4+8 x^5+2 \left (1+4 e^5\right ) x^6-2 x^7}{\left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right ) \left (2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3\right )} \, dx\\ &=\int \left (-\frac {2 x}{-1+x^2}+\frac {-3-2 \left (1+4 e^5\right ) x+3 x^2}{2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3}\right ) \, dx+\int \left (\frac {5 \left (1+\frac {8}{5} e^5 \left (1+2 e^5\right )\right )}{-2 e^{x^2} \left (1+2 e^5\right )+x+3 e^{x^2} x+e^{x^2} \left (1+4 e^5\right ) x^2-x^3-e^{x^2} x^3}+\frac {2 x^4}{-2 e^{x^2} \left (1+2 e^5\right )+x+3 e^{x^2} x+e^{x^2} \left (1+4 e^5\right ) x^2-x^3-e^{x^2} x^3}+\frac {\left (-1-4 e^5\right ) x}{2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3}+\frac {2 x^2}{2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3}+\frac {8 \left (1+4 e^5+8 e^{10}+8 e^{15}\right )-\left (13+12 e^5+32 e^{10}\right ) x-\left (3+32 e^5+48 e^{10}+64 e^{15}\right ) x^2}{\left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right ) \left (2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x}{-1+x^2} \, dx\right )+2 \int \frac {x^4}{-2 e^{x^2} \left (1+2 e^5\right )+x+3 e^{x^2} x+e^{x^2} \left (1+4 e^5\right ) x^2-x^3-e^{x^2} x^3} \, dx+2 \int \frac {x^2}{2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3} \, dx+\left (-1-4 e^5\right ) \int \frac {x}{2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3} \, dx+\left (5+8 e^5+16 e^{10}\right ) \int \frac {1}{-2 e^{x^2} \left (1+2 e^5\right )+x+3 e^{x^2} x+e^{x^2} \left (1+4 e^5\right ) x^2-x^3-e^{x^2} x^3} \, dx+\int \frac {-3-2 \left (1+4 e^5\right ) x+3 x^2}{2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3} \, dx+\int \frac {8 \left (1+4 e^5+8 e^{10}+8 e^{15}\right )-\left (13+12 e^5+32 e^{10}\right ) x-\left (3+32 e^5+48 e^{10}+64 e^{15}\right ) x^2}{\left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right ) \left (2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3\right )} \, dx\\ &=-\log \left (1-x^2\right )+\log \left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right )+2 \int \frac {x^4}{-2 e^{x^2} \left (1+2 e^5\right )+x+3 e^{x^2} x+e^{x^2} \left (1+4 e^5\right ) x^2-x^3-e^{x^2} x^3} \, dx+2 \int \frac {x^2}{2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3} \, dx+\left (-1-4 e^5\right ) \int \frac {x}{2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3} \, dx+\left (5+8 e^5+16 e^{10}\right ) \int \frac {1}{x-x^3+4 e^{5+x^2} \left (-1+x^2\right )+e^{x^2} \left (-2+3 x+x^2-x^3\right )} \, dx+\int \left (\frac {8 \left (1+4 e^5+8 e^{10}+8 e^{15}\right )}{\left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right ) \left (2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3\right )}+\frac {\left (-13-12 e^5-32 e^{10}\right ) x}{\left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right ) \left (2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3\right )}+\frac {\left (-3-32 e^5-48 e^{10}-64 e^{15}\right ) x^2}{\left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right ) \left (2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3\right )}\right ) \, dx\\ &=-\log \left (1-x^2\right )+\log \left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right )+2 \int \frac {x^4}{-2 e^{x^2} \left (1+2 e^5\right )+x+3 e^{x^2} x+e^{x^2} \left (1+4 e^5\right ) x^2-x^3-e^{x^2} x^3} \, dx+2 \int \frac {x^2}{2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3} \, dx+\left (-1-4 e^5\right ) \int \frac {x}{2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3} \, dx+\left (-13-12 e^5-32 e^{10}\right ) \int \frac {x}{\left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right ) \left (2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3\right )} \, dx+\left (5+8 e^5+16 e^{10}\right ) \int \frac {1}{x-x^3+4 e^{5+x^2} \left (-1+x^2\right )+e^{x^2} \left (-2+3 x+x^2-x^3\right )} \, dx+\left (-3-32 e^5-48 e^{10}-64 e^{15}\right ) \int \frac {x^2}{\left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right ) \left (2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3\right )} \, dx+\left (8 \left (1+4 e^5+8 e^{10}+8 e^{15}\right )\right ) \int \frac {1}{\left (2 \left (1+2 e^5\right )-3 x-\left (1+4 e^5\right ) x^2+x^3\right ) \left (2 e^{x^2} \left (1+2 e^5\right )-x-3 e^{x^2} x-e^{x^2} \left (1+4 e^5\right ) x^2+x^3+e^{x^2} x^3\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.10, size = 79, normalized size = 2.55 \begin {gather*} -x^2-\log \left (1-x^2\right )+\log \left (-2 e^{x^2}-4 e^{5+x^2}+x+3 e^{x^2} x+e^{x^2} x^2+4 e^{5+x^2} x^2-x^3-e^{x^2} x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 4*x^2 - 5*x^4 + 2*x^6 + E^x^2*(-3 + 2*x - x^4))/(-x + 2*x^3 - x^5 + E^x^2*(2 - 3*x - 3*x^2 + 4
*x^3 + x^4 - x^5 + E^5*(4 - 8*x^2 + 4*x^4))),x]

[Out]

-x^2 - Log[1 - x^2] + Log[-2*E^x^2 - 4*E^(5 + x^2) + x + 3*E^x^2*x + E^x^2*x^2 + 4*E^(5 + x^2)*x^2 - x^3 - E^x
^2*x^3]

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fricas [B]  time = 0.70, size = 98, normalized size = 3.16 \begin {gather*} -x^{2} + \log \left (x^{3} - x^{2} - 4 \, {\left (x^{2} - 1\right )} e^{5} - 3 \, x + 2\right ) - \log \left (x^{2} - 1\right ) + \log \left (-\frac {x^{3} + {\left (x^{3} - x^{2} - 4 \, {\left (x^{2} - 1\right )} e^{5} - 3 \, x + 2\right )} e^{\left (x^{2}\right )} - x}{x^{3} - x^{2} - 4 \, {\left (x^{2} - 1\right )} e^{5} - 3 \, x + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+2*x-3)*exp(x^2)+2*x^6-5*x^4+4*x^2-1)/(((4*x^4-8*x^2+4)*exp(5)-x^5+x^4+4*x^3-3*x^2-3*x+2)*exp(
x^2)-x^5+2*x^3-x),x, algorithm="fricas")

[Out]

-x^2 + log(x^3 - x^2 - 4*(x^2 - 1)*e^5 - 3*x + 2) - log(x^2 - 1) + log(-(x^3 + (x^3 - x^2 - 4*(x^2 - 1)*e^5 -
3*x + 2)*e^(x^2) - x)/(x^3 - x^2 - 4*(x^2 - 1)*e^5 - 3*x + 2))

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giac [B]  time = 0.18, size = 123, normalized size = 3.97 \begin {gather*} -x^{2} + \log \left (x^{3} e^{\left (x^{2}\right )} + x^{3} - 4 \, x^{2} e^{\left (x^{2} + 5\right )} - x^{2} e^{\left (x^{2}\right )} - 3 \, x e^{\left (x^{2}\right )} - x + 4 \, e^{\left (x^{2} + 5\right )} + 2 \, e^{\left (x^{2}\right )}\right ) - \log \left (x^{3} - 4 \, x^{2} e^{5} - x^{2} - 3 \, x + 4 \, e^{5} + 2\right ) + \log \left (-x^{3} + 4 \, x^{2} e^{5} + x^{2} + 3 \, x - 4 \, e^{5} - 2\right ) - \log \left (x^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+2*x-3)*exp(x^2)+2*x^6-5*x^4+4*x^2-1)/(((4*x^4-8*x^2+4)*exp(5)-x^5+x^4+4*x^3-3*x^2-3*x+2)*exp(
x^2)-x^5+2*x^3-x),x, algorithm="giac")

[Out]

-x^2 + log(x^3*e^(x^2) + x^3 - 4*x^2*e^(x^2 + 5) - x^2*e^(x^2) - 3*x*e^(x^2) - x + 4*e^(x^2 + 5) + 2*e^(x^2))
- log(x^3 - 4*x^2*e^5 - x^2 - 3*x + 4*e^5 + 2) + log(-x^3 + 4*x^2*e^5 + x^2 + 3*x - 4*e^5 - 2) - log(x^2 - 1)

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maple [B]  time = 0.29, size = 76, normalized size = 2.45

method	result	size
norman	\(-x^{2}-\ln \left (x -1\right )-\ln \left (x +1\right )+\ln \left (4 x^{2} {\mathrm e}^{5} {\mathrm e}^{x^{2}}-x^{3} {\mathrm e}^{x^{2}}-x^{3}+x^{2} {\mathrm e}^{x^{2}}-4 \,{\mathrm e}^{5} {\mathrm e}^{x^{2}}+3 \,{\mathrm e}^{x^{2}} x +x -2 \,{\mathrm e}^{x^{2}}\right )\)	\(76\)
risch	\(-\ln \left (-x^{2}+1\right )+\ln \left (x^{3}+\left (-4 \,{\mathrm e}^{5}-1\right ) x^{2}-3 x +4 \,{\mathrm e}^{5}+2\right )-x^{2}+\ln \left ({\mathrm e}^{x^{2}}-\frac {\left (x^{2}-1\right ) x}{4 x^{2} {\mathrm e}^{5}-x^{3}+x^{2}-4 \,{\mathrm e}^{5}+3 x -2}\right )\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^4+2*x-3)*exp(x^2)+2*x^6-5*x^4+4*x^2-1)/(((4*x^4-8*x^2+4)*exp(5)-x^5+x^4+4*x^3-3*x^2-3*x+2)*exp(x^2)-x
^5+2*x^3-x),x,method=_RETURNVERBOSE)

[Out]

-x^2-ln(x-1)-ln(x+1)+ln(4*x^2*exp(5)*exp(x^2)-x^3*exp(x^2)-x^3+x^2*exp(x^2)-4*exp(5)*exp(x^2)+3*exp(x^2)*x+x-2
*exp(x^2))

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maxima [B]  time = 0.41, size = 104, normalized size = 3.35 \begin {gather*} -x^{2} + \log \left (x^{3} - x^{2} {\left (4 \, e^{5} + 1\right )} - 3 \, x + 4 \, e^{5} + 2\right ) - \log \left (x + 1\right ) - \log \left (x - 1\right ) + \log \left (\frac {x^{3} + {\left (x^{3} - x^{2} {\left (4 \, e^{5} + 1\right )} - 3 \, x + 4 \, e^{5} + 2\right )} e^{\left (x^{2}\right )} - x}{x^{3} - x^{2} {\left (4 \, e^{5} + 1\right )} - 3 \, x + 4 \, e^{5} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+2*x-3)*exp(x^2)+2*x^6-5*x^4+4*x^2-1)/(((4*x^4-8*x^2+4)*exp(5)-x^5+x^4+4*x^3-3*x^2-3*x+2)*exp(
x^2)-x^5+2*x^3-x),x, algorithm="maxima")

[Out]

-x^2 + log(x^3 - x^2*(4*e^5 + 1) - 3*x + 4*e^5 + 2) - log(x + 1) - log(x - 1) + log((x^3 + (x^3 - x^2*(4*e^5 +
1) - 3*x + 4*e^5 + 2)*e^(x^2) - x)/(x^3 - x^2*(4*e^5 + 1) - 3*x + 4*e^5 + 2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{x^2}\,\left (x^4-2\,x+3\right )-4\,x^2+5\,x^4-2\,x^6+1}{x-2\,x^3+x^5-{\mathrm {e}}^{x^2}\,\left ({\mathrm {e}}^5\,\left (4\,x^4-8\,x^2+4\right )-3\,x-3\,x^2+4\,x^3+x^4-x^5+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2)*(x^4 - 2*x + 3) - 4*x^2 + 5*x^4 - 2*x^6 + 1)/(x - 2*x^3 + x^5 - exp(x^2)*(exp(5)*(4*x^4 - 8*x^2
+ 4) - 3*x - 3*x^2 + 4*x^3 + x^4 - x^5 + 2)),x)

[Out]

int((exp(x^2)*(x^4 - 2*x + 3) - 4*x^2 + 5*x^4 - 2*x^6 + 1)/(x - 2*x^3 + x^5 - exp(x^2)*(exp(5)*(4*x^4 - 8*x^2
+ 4) - 3*x - 3*x^2 + 4*x^3 + x^4 - x^5 + 2)), x)

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sympy [B]  time = 5.45, size = 73, normalized size = 2.35 \begin {gather*} - x^{2} - \log {\left (x^{2} - 1 \right )} + \log {\left (\frac {x^{3} - x}{x^{3} - 4 x^{2} e^{5} - x^{2} - 3 x + 2 + 4 e^{5}} + e^{x^{2}} \right )} + \log {\left (x^{3} + x^{2} \left (- 4 e^{5} - 1\right ) - 3 x + 2 + 4 e^{5} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**4+2*x-3)*exp(x**2)+2*x**6-5*x**4+4*x**2-1)/(((4*x**4-8*x**2+4)*exp(5)-x**5+x**4+4*x**3-3*x**2-
3*x+2)*exp(x**2)-x**5+2*x**3-x),x)

[Out]

-x**2 - log(x**2 - 1) + log((x**3 - x)/(x**3 - 4*x**2*exp(5) - x**2 - 3*x + 2 + 4*exp(5)) + exp(x**2)) + log(x
**3 + x**2*(-4*exp(5) - 1) - 3*x + 2 + 4*exp(5))

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