3.84.16 \(\int \frac {4 e^{x^2+\frac {3+x^3}{x}} x^3+e^{\frac {3+x^3}{x}} (9-3 x-6 x^3+e^{x^2} (-6+2 x+4 x^3)) \log (\frac {1}{2} (-3+2 e^{x^2}))}{-3 x+2 e^{x^2} x} \, dx\)

Optimal. Leaf size=23 \[ e^{\frac {3}{x}+x^2} x \log \left (-\frac {3}{2}+e^{x^2}\right ) \]

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Rubi [F]  time = 2.74, 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 {4 e^{x^2+\frac {3+x^3}{x}} x^3+e^{\frac {3+x^3}{x}} \left (9-3 x-6 x^3+e^{x^2} \left (-6+2 x+4 x^3\right )\right ) \log \left (\frac {1}{2} \left (-3+2 e^{x^2}\right )\right )}{-3 x+2 e^{x^2} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.67, size = 23, normalized size = 1.00 \begin {gather*} e^{\frac {3}{x}+x^2} x \log \left (-\frac {3}{2}+e^{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(3/x + x^2)*x*Log[-3/2 + E^x^2]

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fricas [B]  time = 0.93, size = 53, normalized size = 2.30 \begin {gather*} x e^{\left (\frac {x^{3} + 3}{x}\right )} \log \left (\frac {1}{2} \, {\left (2 \, e^{\left (\frac {2 \, x^{3} + 3}{x}\right )} - 3 \, e^{\left (\frac {x^{3} + 3}{x}\right )}\right )} e^{\left (-\frac {x^{3} + 3}{x}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3+2*x-6)*exp(x^2)-6*x^3-3*x+9)*exp((x^3+3)/x)*log(exp(x^2)-3/2)+4*x^3*exp(x^2)*exp((x^3+3)/x)
)/(2*exp(x^2)*x-3*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 37, normalized size = 1.61 \begin {gather*} -x e^{\left (\frac {x^{3} + 3}{x}\right )} \log \relax (2) + x e^{\left (\frac {x^{3} + 3}{x}\right )} \log \left (2 \, e^{\left (x^{2}\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3+2*x-6)*exp(x^2)-6*x^3-3*x+9)*exp((x^3+3)/x)*log(exp(x^2)-3/2)+4*x^3*exp(x^2)*exp((x^3+3)/x)
)/(2*exp(x^2)*x-3*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 20, normalized size = 0.87

method	result	size
risch	\({\mathrm e}^{\frac {x^{3}+3}{x}} x \ln \left ({\mathrm e}^{x^{2}}-\frac {3}{2}\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x^3+2*x-6)*exp(x^2)-6*x^3-3*x+9)*exp((x^3+3)/x)*ln(exp(x^2)-3/2)+4*x^3*exp(x^2)*exp((x^3+3)/x))/(2*ex
p(x^2)*x-3*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.52, size = 37, normalized size = 1.61 \begin {gather*} -x e^{\left (x^{2} + \frac {3}{x}\right )} \log \relax (2) + x e^{\left (x^{2} + \frac {3}{x}\right )} \log \left (2 \, e^{\left (x^{2}\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3+2*x-6)*exp(x^2)-6*x^3-3*x+9)*exp((x^3+3)/x)*log(exp(x^2)-3/2)+4*x^3*exp(x^2)*exp((x^3+3)/x)
)/(2*exp(x^2)*x-3*x),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {4\,x^3\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{\frac {x^3+3}{x}}-\ln \left ({\mathrm {e}}^{x^2}-\frac {3}{2}\right )\,{\mathrm {e}}^{\frac {x^3+3}{x}}\,\left (3\,x-{\mathrm {e}}^{x^2}\,\left (4\,x^3+2\,x-6\right )+6\,x^3-9\right )}{3\,x-2\,x\,{\mathrm {e}}^{x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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