3.102.7 \(\int \frac {e^{\frac {3 x-9 e x-3 x^3+(3 e x+x^3) \log (x^2)}{-9 e+3 e \log (x^2)}} (-20 x+36 x^3+e (36+36 x)+(e (-24-24 x)+4 x-24 x^3) \log (x^2)+(4 x^3+e (4+4 x)) \log ^2(x^2))}{9 e-6 e \log (x^2)+e \log ^2(x^2)} \, dx\)

Optimal. Leaf size=32 \[ 4 e^{x-\frac {-\frac {x^3}{3}+\frac {x}{3-\log \left (x^2\right )}}{e}} x \]

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Rubi [F]  time = 4.43, 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 {\exp \left (\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}\right ) \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((3*x - 9*E*x - 3*x^3 + (3*E*x + x^3)*Log[x^2])/(-9*E + 3*E*Log[x^2]))*(-20*x + 36*x^3 + E*(36 + 36*x)
+ (E*(-24 - 24*x) + 4*x - 24*x^3)*Log[x^2] + (4*x^3 + E*(4 + 4*x))*Log[x^2]^2))/(9*E - 6*E*Log[x^2] + E*Log[x^
2]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{\left (3-\log \left (x^2\right )\right )^2} \, dx\\ &=\int \left (4 \exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) \left (e+e x+x^3\right )-\frac {8 \exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x}{\left (-3+\log \left (x^2\right )\right )^2}+\frac {4 \exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x}{-3+\log \left (x^2\right )}\right ) \, dx\\ &=4 \int \exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) \left (e+e x+x^3\right ) \, dx+4 \int \frac {\exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x}{-3+\log \left (x^2\right )} \, dx-8 \int \frac {\exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x}{\left (-3+\log \left (x^2\right )\right )^2} \, dx\\ &=4 \int \left (\exp \left (\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right )+\exp \left (\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x+\exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x^3\right ) \, dx+4 \int \frac {\exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x}{-3+\log \left (x^2\right )} \, dx-8 \int \frac {\exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x}{\left (-3+\log \left (x^2\right )\right )^2} \, dx\\ &=4 \int \exp \left (\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) \, dx+4 \int \exp \left (\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x \, dx+4 \int \exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x^3 \, dx+4 \int \frac {\exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x}{-3+\log \left (x^2\right )} \, dx-8 \int \frac {\exp \left (-1+\frac {3 (1-3 e) x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{3 e \left (-3+\log \left (x^2\right )\right )}\right ) x}{\left (-3+\log \left (x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 44, normalized size = 1.38 \begin {gather*} 4 e^{\frac {x \left (-3 \left (-1+3 e+x^2\right )+\left (3 e+x^2\right ) \log \left (x^2\right )\right )}{3 e \left (-3+\log \left (x^2\right )\right )}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((3*x - 9*E*x - 3*x^3 + (3*E*x + x^3)*Log[x^2])/(-9*E + 3*E*Log[x^2]))*(-20*x + 36*x^3 + E*(36 +
36*x) + (E*(-24 - 24*x) + 4*x - 24*x^3)*Log[x^2] + (4*x^3 + E*(4 + 4*x))*Log[x^2]^2))/(9*E - 6*E*Log[x^2] + E*
Log[x^2]^2),x]

[Out]

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

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fricas [A]  time = 0.76, size = 49, normalized size = 1.53 \begin {gather*} 4 \, x e^{\left (-\frac {3 \, x^{3} + 9 \, x e - {\left (x^{3} + 3 \, x e\right )} \log \left (x^{2}\right ) - 3 \, x}{3 \, {\left (e \log \left (x^{2}\right ) - 3 \, e\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+4)*exp(1)+4*x^3)*log(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x)*log(x^2)+(36*x+36)*exp(1)+36*x^3-2
0*x)*exp(((3*x*exp(1)+x^3)*log(x^2)-9*x*exp(1)-3*x^3+3*x)/(3*exp(1)*log(x^2)-9*exp(1)))/(exp(1)*log(x^2)^2-6*e
xp(1)*log(x^2)+9*exp(1)),x, algorithm="fricas")

[Out]

4*x*e^(-1/3*(3*x^3 + 9*x*e - (x^3 + 3*x*e)*log(x^2) - 3*x)/(e*log(x^2) - 3*e))

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giac [B]  time = 2.69, size = 51, normalized size = 1.59 \begin {gather*} 4 \, x e^{\left (\frac {x^{3} \log \left (x^{2}\right ) - 3 \, x^{3} + 3 \, x e \log \left (x^{2}\right ) - 9 \, x e + 3 \, x}{3 \, {\left (e \log \left (x^{2}\right ) - 3 \, e\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+4)*exp(1)+4*x^3)*log(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x)*log(x^2)+(36*x+36)*exp(1)+36*x^3-2
0*x)*exp(((3*x*exp(1)+x^3)*log(x^2)-9*x*exp(1)-3*x^3+3*x)/(3*exp(1)*log(x^2)-9*exp(1)))/(exp(1)*log(x^2)^2-6*e
xp(1)*log(x^2)+9*exp(1)),x, algorithm="giac")

[Out]

4*x*e^(1/3*(x^3*log(x^2) - 3*x^3 + 3*x*e*log(x^2) - 9*x*e + 3*x)/(e*log(x^2) - 3*e))

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maple [A]  time = 0.50, size = 45, normalized size = 1.41

method	result	size
risch	\(4 x \,{\mathrm e}^{\frac {x \left (x^{2} \ln \left (x^{2}\right )+3 \,{\mathrm e} \ln \left (x^{2}\right )-3 x^{2}-9 \,{\mathrm e}+3\right ) {\mathrm e}^{-1}}{3 \ln \left (x^{2}\right )-9}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x+4)*exp(1)+4*x^3)*ln(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x)*ln(x^2)+(36*x+36)*exp(1)+36*x^3-20*x)*exp
(((3*x*exp(1)+x^3)*ln(x^2)-9*x*exp(1)-3*x^3+3*x)/(3*exp(1)*ln(x^2)-9*exp(1)))/(exp(1)*ln(x^2)^2-6*exp(1)*ln(x^
2)+9*exp(1)),x,method=_RETURNVERBOSE)

[Out]

4*x*exp(1/3*x*(x^2*ln(x^2)+3*exp(1)*ln(x^2)-3*x^2-9*exp(1)+3)*exp(-1)/(ln(x^2)-3))

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maxima [B]  time = 0.54, size = 82, normalized size = 2.56 \begin {gather*} 4 \, x e^{\left (\frac {2 \, x^{3} \log \relax (x)}{3 \, {\left (2 \, e \log \relax (x) - 3 \, e\right )}} - \frac {x^{3}}{2 \, e \log \relax (x) - 3 \, e} + \frac {2 \, x \log \relax (x)}{2 \, \log \relax (x) - 3} + \frac {x}{2 \, e \log \relax (x) - 3 \, e} - \frac {3 \, x}{2 \, \log \relax (x) - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+4)*exp(1)+4*x^3)*log(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x)*log(x^2)+(36*x+36)*exp(1)+36*x^3-2
0*x)*exp(((3*x*exp(1)+x^3)*log(x^2)-9*x*exp(1)-3*x^3+3*x)/(3*exp(1)*log(x^2)-9*exp(1)))/(exp(1)*log(x^2)^2-6*e
xp(1)*log(x^2)+9*exp(1)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 7.44, size = 95, normalized size = 2.97 \begin {gather*} \frac {4\,x\,{\mathrm {e}}^{\frac {3\,x^3}{9\,\mathrm {e}-3\,\ln \left (x^2\right )\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {9\,x\,\mathrm {e}}{9\,\mathrm {e}-3\,\ln \left (x^2\right )\,\mathrm {e}}}\,{\mathrm {e}}^{-\frac {3\,x}{9\,\mathrm {e}-3\,\ln \left (x^2\right )\,\mathrm {e}}}}{{\left (x^2\right )}^{\frac {x^3+3\,\mathrm {e}\,x}{3\,\left (3\,\mathrm {e}-\ln \left (x^2\right )\,\mathrm {e}\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(3*x + log(x^2)*(3*x*exp(1) + x^3) - 9*x*exp(1) - 3*x^3)/(9*exp(1) - 3*log(x^2)*exp(1)))*(log(x^2)^2
*(4*x^3 + exp(1)*(4*x + 4)) - log(x^2)*(24*x^3 - 4*x + exp(1)*(24*x + 24)) - 20*x + 36*x^3 + exp(1)*(36*x + 36
)))/(9*exp(1) - 6*log(x^2)*exp(1) + log(x^2)^2*exp(1)),x)

[Out]

(4*x*exp((3*x^3)/(9*exp(1) - 3*log(x^2)*exp(1)))*exp((9*x*exp(1))/(9*exp(1) - 3*log(x^2)*exp(1)))*exp(-(3*x)/(
9*exp(1) - 3*log(x^2)*exp(1))))/(x^2)^((3*x*exp(1) + x^3)/(3*(3*exp(1) - log(x^2)*exp(1))))

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sympy [B]  time = 8.72, size = 49, normalized size = 1.53 \begin {gather*} 4 x e^{\frac {- 3 x^{3} - 9 e x + 3 x + \left (x^{3} + 3 e x\right ) \log {\left (x^{2} \right )}}{3 e \log {\left (x^{2} \right )} - 9 e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+4)*exp(1)+4*x**3)*ln(x**2)**2+((-24*x-24)*exp(1)-24*x**3+4*x)*ln(x**2)+(36*x+36)*exp(1)+36*x*
*3-20*x)*exp(((3*x*exp(1)+x**3)*ln(x**2)-9*x*exp(1)-3*x**3+3*x)/(3*exp(1)*ln(x**2)-9*exp(1)))/(exp(1)*ln(x**2)
**2-6*exp(1)*ln(x**2)+9*exp(1)),x)

[Out]

4*x*exp((-3*x**3 - 9*E*x + 3*x + (x**3 + 3*E*x)*log(x**2))/(3*E*log(x**2) - 9*E))

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