3.101.76 \(\int \frac {e^{-4+\frac {25+115 x-49 x^2+5 x^3+(25-10 x+x^2) \log (x^2)}{e^4}} (100+190 x-192 x^2+30 x^3+(-20 x+4 x^2) \log (x^2))}{x} \, dx\)

Optimal. Leaf size=24 \[ 2 e^{\frac {(5-x)^2 \left (1+5 x+\log \left (x^2\right )\right )}{e^4}} \]

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Rubi [F]  time = 4.48, 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 (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) \left (100+190 x-192 x^2+30 x^3+\left (-20 x+4 x^2\right ) \log \left (x^2\right )\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-4 + (25 + 115*x - 49*x^2 + 5*x^3 + (25 - 10*x + x^2)*Log[x^2])/E^4)*(100 + 190*x - 192*x^2 + 30*x^3 +
(-20*x + 4*x^2)*Log[x^2]))/x,x]

[Out]

190*Defer[Int][E^(-4 + (25 + 115*x - 49*x^2 + 5*x^3 + (25 - 10*x + x^2)*Log[x^2])/E^4), x] + 100*Defer[Int][E^
(-4 + (25 + 115*x - 49*x^2 + 5*x^3 + (25 - 10*x + x^2)*Log[x^2])/E^4)/x, x] - 192*Defer[Int][E^(-4 + (25 + 115
*x - 49*x^2 + 5*x^3 + (25 - 10*x + x^2)*Log[x^2])/E^4)*x, x] + 30*Defer[Int][E^(-4 + (25 + 115*x - 49*x^2 + 5*
x^3 + (25 - 10*x + x^2)*Log[x^2])/E^4)*x^2, x] - 20*Defer[Int][E^(-4 + (25 + 115*x - 49*x^2 + 5*x^3 + (25 - 10
*x + x^2)*Log[x^2])/E^4)*Log[x^2], x] + 4*Defer[Int][E^(-4 + (25 + 115*x - 49*x^2 + 5*x^3 + (25 - 10*x + x^2)*
Log[x^2])/E^4)*x*Log[x^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) (5-x) \left (10+21 x-15 x^2-2 x \log \left (x^2\right )\right )}{x} \, dx\\ &=2 \int \frac {\exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) (5-x) \left (10+21 x-15 x^2-2 x \log \left (x^2\right )\right )}{x} \, dx\\ &=2 \int \left (\frac {\exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) \left (50+95 x-96 x^2+15 x^3\right )}{x}+2 \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) (-5+x) \log \left (x^2\right )\right ) \, dx\\ &=2 \int \frac {\exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) \left (50+95 x-96 x^2+15 x^3\right )}{x} \, dx+4 \int \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) (-5+x) \log \left (x^2\right ) \, dx\\ &=2 \int \left (95 \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right )+\frac {50 \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right )}{x}-96 \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) x+15 \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) x^2\right ) \, dx+4 \int \left (-5 \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) \log \left (x^2\right )+\exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) x \log \left (x^2\right )\right ) \, dx\\ &=4 \int \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) x \log \left (x^2\right ) \, dx-20 \int \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) \log \left (x^2\right ) \, dx+30 \int \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) x^2 \, dx+100 \int \frac {\exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right )}{x} \, dx+190 \int \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) \, dx-192 \int \exp \left (-4+\frac {25+115 x-49 x^2+5 x^3+\left (25-10 x+x^2\right ) \log \left (x^2\right )}{e^4}\right ) x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.68, size = 31, normalized size = 1.29 \begin {gather*} 2 e^{\frac {(-5+x)^2 (1+5 x)}{e^4}} \left (x^2\right )^{\frac {(-5+x)^2}{e^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-4 + (25 + 115*x - 49*x^2 + 5*x^3 + (25 - 10*x + x^2)*Log[x^2])/E^4)*(100 + 190*x - 192*x^2 + 30
*x^3 + (-20*x + 4*x^2)*Log[x^2]))/x,x]

[Out]

2*E^(((-5 + x)^2*(1 + 5*x))/E^4)*(x^2)^((-5 + x)^2/E^4)

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fricas [A]  time = 0.80, size = 40, normalized size = 1.67 \begin {gather*} 2 \, e^{\left ({\left (5 \, x^{3} - 49 \, x^{2} + {\left (x^{2} - 10 \, x + 25\right )} \log \left (x^{2}\right ) + 115 \, x - 4 \, e^{4} + 25\right )} e^{\left (-4\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2-20*x)*log(x^2)+30*x^3-192*x^2+190*x+100)*exp(((x^2-10*x+25)*log(x^2)+5*x^3-49*x^2+115*x+25)/
exp(4))/x/exp(4),x, algorithm="fricas")

[Out]

2*e^((5*x^3 - 49*x^2 + (x^2 - 10*x + 25)*log(x^2) + 115*x - 4*e^4 + 25)*e^(-4) + 4)

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giac [B]  time = 2.99, size = 54, normalized size = 2.25 \begin {gather*} 2 \, e^{\left (5 \, x^{3} e^{\left (-4\right )} + x^{2} e^{\left (-4\right )} \log \left (x^{2}\right ) - 49 \, x^{2} e^{\left (-4\right )} - 10 \, x e^{\left (-4\right )} \log \left (x^{2}\right ) + 115 \, x e^{\left (-4\right )} + 25 \, e^{\left (-4\right )} \log \left (x^{2}\right ) + 25 \, e^{\left (-4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2-20*x)*log(x^2)+30*x^3-192*x^2+190*x+100)*exp(((x^2-10*x+25)*log(x^2)+5*x^3-49*x^2+115*x+25)/
exp(4))/x/exp(4),x, algorithm="giac")

[Out]

2*e^(5*x^3*e^(-4) + x^2*e^(-4)*log(x^2) - 49*x^2*e^(-4) - 10*x*e^(-4)*log(x^2) + 115*x*e^(-4) + 25*e^(-4)*log(
x^2) + 25*e^(-4))

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maple [A]  time = 0.17, size = 21, normalized size = 0.88

method	result	size
risch	\(2 \,{\mathrm e}^{\left (x -5\right )^{2} \left (1+\ln \left (x^{2}\right )+5 x \right ) {\mathrm e}^{-4}}\)	\(21\)
default	\(2 \,{\mathrm e}^{\left (\left (x^{2}-10 x +25\right ) \ln \left (x^{2}\right )+5 x^{3}-49 x^{2}+115 x +25\right ) {\mathrm e}^{-4}}\)	\(37\)
norman	\(2 \,{\mathrm e}^{\left (\left (x^{2}-10 x +25\right ) \ln \left (x^{2}\right )+5 x^{3}-49 x^{2}+115 x +25\right ) {\mathrm e}^{-4}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2-20*x)*ln(x^2)+30*x^3-192*x^2+190*x+100)*exp(((x^2-10*x+25)*ln(x^2)+5*x^3-49*x^2+115*x+25)/exp(4))/
x/exp(4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.70, size = 49, normalized size = 2.04 \begin {gather*} 2 \, e^{\left (5 \, x^{3} e^{\left (-4\right )} + 2 \, x^{2} e^{\left (-4\right )} \log \relax (x) - 49 \, x^{2} e^{\left (-4\right )} - 20 \, x e^{\left (-4\right )} \log \relax (x) + 115 \, x e^{\left (-4\right )} + 50 \, e^{\left (-4\right )} \log \relax (x) + 25 \, e^{\left (-4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2-20*x)*log(x^2)+30*x^3-192*x^2+190*x+100)*exp(((x^2-10*x+25)*log(x^2)+5*x^3-49*x^2+115*x+25)/
exp(4))/x/exp(4),x, algorithm="maxima")

[Out]

2*e^(5*x^3*e^(-4) + 2*x^2*e^(-4)*log(x) - 49*x^2*e^(-4) - 20*x*e^(-4)*log(x) + 115*x*e^(-4) + 50*e^(-4)*log(x)
+ 25*e^(-4))

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mupad [B]  time = 7.74, size = 44, normalized size = 1.83 \begin {gather*} 2\,{\mathrm {e}}^{5\,x^3\,{\mathrm {e}}^{-4}}\,{\mathrm {e}}^{-49\,x^2\,{\mathrm {e}}^{-4}}\,{\mathrm {e}}^{25\,{\mathrm {e}}^{-4}}\,{\mathrm {e}}^{115\,x\,{\mathrm {e}}^{-4}}\,{\left (x^2\right )}^{{\mathrm {e}}^{-4}\,\left (x^2-10\,x+25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-4)*exp(exp(-4)*(115*x + log(x^2)*(x^2 - 10*x + 25) - 49*x^2 + 5*x^3 + 25))*(190*x - log(x^2)*(20*x -
4*x^2) - 192*x^2 + 30*x^3 + 100))/x,x)

[Out]

2*exp(5*x^3*exp(-4))*exp(-49*x^2*exp(-4))*exp(25*exp(-4))*exp(115*x*exp(-4))*(x^2)^(exp(-4)*(x^2 - 10*x + 25))

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sympy [A]  time = 0.48, size = 34, normalized size = 1.42 \begin {gather*} 2 e^{\frac {5 x^{3} - 49 x^{2} + 115 x + \left (x^{2} - 10 x + 25\right ) \log {\left (x^{2} \right )} + 25}{e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2-20*x)*ln(x**2)+30*x**3-192*x**2+190*x+100)*exp(((x**2-10*x+25)*ln(x**2)+5*x**3-49*x**2+115*
x+25)/exp(4))/x/exp(4),x)

[Out]

2*exp((5*x**3 - 49*x**2 + 115*x + (x**2 - 10*x + 25)*log(x**2) + 25)*exp(-4))

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