3.70.89 \(\int \frac {-72+36 x+e^{x^2} (36-20 x-72 x^2+18 x^3)}{4-4 e^{x^2}+e^{2 x^2}} \, dx\)

Optimal. Leaf size=30 \[ \frac {x+9 x (-x+(5-x) x)}{-2 x+e^{x^2} x} \]

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Rubi [F]  time = 0.92, 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 {-72+36 x+e^{x^2} \left (36-20 x-72 x^2+18 x^3\right )}{4-4 e^{x^2}+e^{2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-72 + 36*x + E^x^2*(36 - 20*x - 72*x^2 + 18*x^3))/(4 - 4*E^x^2 + E^(2*x^2)),x]

[Out]

-(2 - E^x^2)^(-1) + (9*x^2)/(2 - E^x^2) + 36*Defer[Int][(-2 + E^x^2)^(-1), x] - 144*Defer[Int][x^2/(-2 + E^x^2
)^2, x] - 72*Defer[Int][x^2/(-2 + E^x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-72+36 x+e^{x^2} \left (36-20 x-72 x^2+18 x^3\right )}{\left (2-e^{x^2}\right )^2} \, dx\\ &=\int \left (\frac {4 x \left (-1-36 x+9 x^2\right )}{\left (-2+e^{x^2}\right )^2}+\frac {2 \left (18-10 x-36 x^2+9 x^3\right )}{-2+e^{x^2}}\right ) \, dx\\ &=2 \int \frac {18-10 x-36 x^2+9 x^3}{-2+e^{x^2}} \, dx+4 \int \frac {x \left (-1-36 x+9 x^2\right )}{\left (-2+e^{x^2}\right )^2} \, dx\\ &=2 \int \left (\frac {18}{-2+e^{x^2}}-\frac {10 x}{-2+e^{x^2}}-\frac {36 x^2}{-2+e^{x^2}}+\frac {9 x^3}{-2+e^{x^2}}\right ) \, dx+4 \int \left (-\frac {x}{\left (-2+e^{x^2}\right )^2}-\frac {36 x^2}{\left (-2+e^{x^2}\right )^2}+\frac {9 x^3}{\left (-2+e^{x^2}\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {x}{\left (-2+e^{x^2}\right )^2} \, dx\right )+18 \int \frac {x^3}{-2+e^{x^2}} \, dx-20 \int \frac {x}{-2+e^{x^2}} \, dx+36 \int \frac {1}{-2+e^{x^2}} \, dx+36 \int \frac {x^3}{\left (-2+e^{x^2}\right )^2} \, dx-72 \int \frac {x^2}{-2+e^{x^2}} \, dx-144 \int \frac {x^2}{\left (-2+e^{x^2}\right )^2} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{\left (-2+e^x\right )^2} \, dx,x,x^2\right )\right )+9 \operatorname {Subst}\left (\int \frac {x}{-2+e^x} \, dx,x,x^2\right )-10 \operatorname {Subst}\left (\int \frac {1}{-2+e^x} \, dx,x,x^2\right )+18 \operatorname {Subst}\left (\int \frac {x}{\left (-2+e^x\right )^2} \, dx,x,x^2\right )+36 \int \frac {1}{-2+e^{x^2}} \, dx-72 \int \frac {x^2}{-2+e^{x^2}} \, dx-144 \int \frac {x^2}{\left (-2+e^{x^2}\right )^2} \, dx\\ &=-\frac {9 x^4}{4}-2 \operatorname {Subst}\left (\int \frac {1}{(-2+x)^2 x} \, dx,x,e^{x^2}\right )+\frac {9}{2} \operatorname {Subst}\left (\int \frac {e^x x}{-2+e^x} \, dx,x,x^2\right )+9 \operatorname {Subst}\left (\int \frac {e^x x}{\left (-2+e^x\right )^2} \, dx,x,x^2\right )-9 \operatorname {Subst}\left (\int \frac {x}{-2+e^x} \, dx,x,x^2\right )-10 \operatorname {Subst}\left (\int \frac {1}{(-2+x) x} \, dx,x,e^{x^2}\right )+36 \int \frac {1}{-2+e^{x^2}} \, dx-72 \int \frac {x^2}{-2+e^{x^2}} \, dx-144 \int \frac {x^2}{\left (-2+e^{x^2}\right )^2} \, dx\\ &=\frac {9 x^2}{2-e^{x^2}}+\frac {9}{2} x^2 \log \left (1-\frac {e^{x^2}}{2}\right )-2 \operatorname {Subst}\left (\int \left (\frac {1}{2 (-2+x)^2}-\frac {1}{4 (-2+x)}+\frac {1}{4 x}\right ) \, dx,x,e^{x^2}\right )-\frac {9}{2} \operatorname {Subst}\left (\int \frac {e^x x}{-2+e^x} \, dx,x,x^2\right )-\frac {9}{2} \operatorname {Subst}\left (\int \log \left (1-\frac {e^x}{2}\right ) \, dx,x,x^2\right )-5 \operatorname {Subst}\left (\int \frac {1}{-2+x} \, dx,x,e^{x^2}\right )+5 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x^2}\right )+9 \operatorname {Subst}\left (\int \frac {1}{-2+e^x} \, dx,x,x^2\right )+36 \int \frac {1}{-2+e^{x^2}} \, dx-72 \int \frac {x^2}{-2+e^{x^2}} \, dx-144 \int \frac {x^2}{\left (-2+e^{x^2}\right )^2} \, dx\\ &=-\frac {1}{2-e^{x^2}}+\frac {9 x^2}{2}+\frac {9 x^2}{2-e^{x^2}}-\frac {9}{2} \log \left (2-e^{x^2}\right )+\frac {9}{2} \operatorname {Subst}\left (\int \log \left (1-\frac {e^x}{2}\right ) \, dx,x,x^2\right )-\frac {9}{2} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,e^{x^2}\right )+9 \operatorname {Subst}\left (\int \frac {1}{(-2+x) x} \, dx,x,e^{x^2}\right )+36 \int \frac {1}{-2+e^{x^2}} \, dx-72 \int \frac {x^2}{-2+e^{x^2}} \, dx-144 \int \frac {x^2}{\left (-2+e^{x^2}\right )^2} \, dx\\ &=-\frac {1}{2-e^{x^2}}+\frac {9 x^2}{2}+\frac {9 x^2}{2-e^{x^2}}-\frac {9}{2} \log \left (2-e^{x^2}\right )+\frac {9}{2} \text {Li}_2\left (\frac {e^{x^2}}{2}\right )+\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{-2+x} \, dx,x,e^{x^2}\right )-\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x^2}\right )+\frac {9}{2} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,e^{x^2}\right )+36 \int \frac {1}{-2+e^{x^2}} \, dx-72 \int \frac {x^2}{-2+e^{x^2}} \, dx-144 \int \frac {x^2}{\left (-2+e^{x^2}\right )^2} \, dx\\ &=-\frac {1}{2-e^{x^2}}+\frac {9 x^2}{2-e^{x^2}}+36 \int \frac {1}{-2+e^{x^2}} \, dx-72 \int \frac {x^2}{-2+e^{x^2}} \, dx-144 \int \frac {x^2}{\left (-2+e^{x^2}\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 20, normalized size = 0.67 \begin {gather*} \frac {1+36 x-9 x^2}{-2+e^{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-72 + 36*x + E^x^2*(36 - 20*x - 72*x^2 + 18*x^3))/(4 - 4*E^x^2 + E^(2*x^2)),x]

[Out]

(1 + 36*x - 9*x^2)/(-2 + E^x^2)

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fricas [A]  time = 0.69, size = 20, normalized size = 0.67 \begin {gather*} -\frac {9 \, x^{2} - 36 \, x - 1}{e^{\left (x^{2}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3-72*x^2-20*x+36)*exp(x^2)+36*x-72)/(exp(x^2)^2-4*exp(x^2)+4),x, algorithm="fricas")

[Out]

-(9*x^2 - 36*x - 1)/(e^(x^2) - 2)

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giac [A]  time = 0.15, size = 20, normalized size = 0.67 \begin {gather*} -\frac {9 \, x^{2} - 36 \, x - 1}{e^{\left (x^{2}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3-72*x^2-20*x+36)*exp(x^2)+36*x-72)/(exp(x^2)^2-4*exp(x^2)+4),x, algorithm="giac")

[Out]

-(9*x^2 - 36*x - 1)/(e^(x^2) - 2)

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maple [A]  time = 0.03, size = 20, normalized size = 0.67

method	result	size
norman	\(\frac {-9 x^{2}+36 x +1}{{\mathrm e}^{x^{2}}-2}\)	\(20\)
risch	\(-\frac {9 x^{2}-36 x -1}{{\mathrm e}^{x^{2}}-2}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((18*x^3-72*x^2-20*x+36)*exp(x^2)+36*x-72)/(exp(x^2)^2-4*exp(x^2)+4),x,method=_RETURNVERBOSE)

[Out]

(-9*x^2+36*x+1)/(exp(x^2)-2)

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maxima [A]  time = 0.41, size = 20, normalized size = 0.67 \begin {gather*} -\frac {9 \, x^{2} - 36 \, x - 1}{e^{\left (x^{2}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3-72*x^2-20*x+36)*exp(x^2)+36*x-72)/(exp(x^2)^2-4*exp(x^2)+4),x, algorithm="maxima")

[Out]

-(9*x^2 - 36*x - 1)/(e^(x^2) - 2)

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mupad [B]  time = 4.12, size = 19, normalized size = 0.63 \begin {gather*} \frac {-9\,x^2+36\,x+1}{{\mathrm {e}}^{x^2}-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2)*(20*x + 72*x^2 - 18*x^3 - 36) - 36*x + 72)/(exp(2*x^2) - 4*exp(x^2) + 4),x)

[Out]

(36*x - 9*x^2 + 1)/(exp(x^2) - 2)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.50 \begin {gather*} \frac {- 9 x^{2} + 36 x + 1}{e^{x^{2}} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x**3-72*x**2-20*x+36)*exp(x**2)+36*x-72)/(exp(x**2)**2-4*exp(x**2)+4),x)

[Out]

(-9*x**2 + 36*x + 1)/(exp(x**2) - 2)

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