3.48.31 \(\int \frac {-37 x^2+e^{x^2} (-1+37 x+2 x^2)}{e^{x^2} x-x^2+(e^{x^2} x-x^2) \log (\frac {e^{37 x+x^2}-e^{37 x} x}{x})} \, dx\)

Optimal. Leaf size=22 \[ \log \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right ) \]

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Rubi [F]  time = 1.27, 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 {-37 x^2+e^{x^2} \left (-1+37 x+2 x^2\right )}{e^{x^2} x-x^2+\left (e^{x^2} x-x^2\right ) \log \left (\frac {e^{37 x+x^2}-e^{37 x} x}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-37*x^2 + E^x^2*(-1 + 37*x + 2*x^2))/(E^x^2*x - x^2 + (E^x^2*x - x^2)*Log[(E^(37*x + x^2) - E^(37*x)*x)/x
]),x]

[Out]

37*Defer[Int][(1 + Log[(E^(37*x)*(E^x^2 - x))/x])^(-1), x] - Defer[Int][1/((E^x^2 - x)*(1 + Log[(E^(37*x)*(E^x
^2 - x))/x])), x] - Defer[Int][1/(x*(1 + Log[(E^(37*x)*(E^x^2 - x))/x])), x] + 2*Defer[Int][x/(1 + Log[(E^(37*
x)*(E^x^2 - x))/x]), x] + 2*Defer[Int][x^2/((E^x^2 - x)*(1 + Log[(E^(37*x)*(E^x^2 - x))/x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-37 x^2+e^{x^2} \left (-1+37 x+2 x^2\right )}{\left (e^{x^2}-x\right ) x \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )} \, dx\\ &=\int \left (\frac {-1+2 x^2}{\left (e^{x^2}-x\right ) \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )}+\frac {-1+37 x+2 x^2}{x \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )}\right ) \, dx\\ &=\int \frac {-1+2 x^2}{\left (e^{x^2}-x\right ) \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )} \, dx+\int \frac {-1+37 x+2 x^2}{x \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )} \, dx\\ &=\int \left (\frac {37}{1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )}-\frac {1}{x \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )}+\frac {2 x}{1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )}\right ) \, dx+\int \left (-\frac {1}{\left (e^{x^2}-x\right ) \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )}+\frac {2 x^2}{\left (e^{x^2}-x\right ) \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )}\right ) \, dx\\ &=2 \int \frac {x}{1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )} \, dx+2 \int \frac {x^2}{\left (e^{x^2}-x\right ) \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )} \, dx+37 \int \frac {1}{1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )} \, dx-\int \frac {1}{\left (e^{x^2}-x\right ) \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )} \, dx-\int \frac {1}{x \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 22, normalized size = 1.00 \begin {gather*} \log \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-37*x^2 + E^x^2*(-1 + 37*x + 2*x^2))/(E^x^2*x - x^2 + (E^x^2*x - x^2)*Log[(E^(37*x + x^2) - E^(37*x
)*x)/x]),x]

[Out]

Log[1 + Log[(E^(37*x)*(E^x^2 - x))/x]]

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fricas [A]  time = 0.98, size = 21, normalized size = 0.95 \begin {gather*} \log \left (\log \left (-\frac {{\left (x - e^{\left (x^{2}\right )}\right )} e^{\left (37 \, x\right )}}{x}\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+37*x-1)*exp(x^2)-37*x^2)/((exp(x^2)*x-x^2)*log((exp(37*x)*exp(x^2)-x*exp(37*x))/x)+exp(x^2)*
x-x^2),x, algorithm="fricas")

[Out]

log(log(-(x - e^(x^2))*e^(37*x)/x) + 1)

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giac [A]  time = 0.26, size = 19, normalized size = 0.86 \begin {gather*} \log \left (37 \, x - \log \relax (x) + \log \left (-x + e^{\left (x^{2}\right )}\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+37*x-1)*exp(x^2)-37*x^2)/((exp(x^2)*x-x^2)*log((exp(37*x)*exp(x^2)-x*exp(37*x))/x)+exp(x^2)*
x-x^2),x, algorithm="giac")

[Out]

log(37*x - log(x) + log(-x + e^(x^2)) + 1)

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maple [C]  time = 0.13, size = 317, normalized size = 14.41

method	result	size
risch	\(\ln \left (\ln \left ({\mathrm e}^{37 x}\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )}{x}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (-{\mathrm e}^{x^{2}}+x \right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{37 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right )-\pi \,\mathrm {csgn}\left (i \left (-{\mathrm e}^{x^{2}}+x \right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{37 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )}{x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )}{x}\right )^{3}-2 i \ln \relax (x )+2 i \ln \left (-{\mathrm e}^{x^{2}}+x \right )-2 \pi +2 i\right )}{2}\right )\)	\(317\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+37*x-1)*exp(x^2)-37*x^2)/((exp(x^2)*x-x^2)*ln((exp(37*x)*exp(x^2)-x*exp(37*x))/x)+exp(x^2)*x-x^2),
x,method=_RETURNVERBOSE)

[Out]

ln(ln(exp(37*x))-1/2*I*(Pi*csgn(I/x)*csgn(I*exp(37*x)*(-exp(x^2)+x))*csgn(I/x*exp(37*x)*(-exp(x^2)+x))-Pi*csgn
(I/x)*csgn(I/x*exp(37*x)*(-exp(x^2)+x))^2+2*Pi*csgn(I/x*exp(37*x)*(-exp(x^2)+x))^2+Pi*csgn(I*(-exp(x^2)+x))*cs
gn(I*exp(37*x))*csgn(I*exp(37*x)*(-exp(x^2)+x))-Pi*csgn(I*(-exp(x^2)+x))*csgn(I*exp(37*x)*(-exp(x^2)+x))^2-Pi*
csgn(I*exp(37*x))*csgn(I*exp(37*x)*(-exp(x^2)+x))^2+Pi*csgn(I*exp(37*x)*(-exp(x^2)+x))^3-Pi*csgn(I*exp(37*x)*(
-exp(x^2)+x))*csgn(I/x*exp(37*x)*(-exp(x^2)+x))^2-Pi*csgn(I/x*exp(37*x)*(-exp(x^2)+x))^3-2*I*ln(x)+2*I*ln(-exp
(x^2)+x)-2*Pi+2*I))

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maxima [A]  time = 0.42, size = 19, normalized size = 0.86 \begin {gather*} \log \left (37 \, x - \log \relax (x) + \log \left (-x + e^{\left (x^{2}\right )}\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+37*x-1)*exp(x^2)-37*x^2)/((exp(x^2)*x-x^2)*log((exp(37*x)*exp(x^2)-x*exp(37*x))/x)+exp(x^2)*
x-x^2),x, algorithm="maxima")

[Out]

log(37*x - log(x) + log(-x + e^(x^2)) + 1)

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mupad [B]  time = 3.42, size = 23, normalized size = 1.05 \begin {gather*} \ln \left (\ln \left (\frac {{\mathrm {e}}^{37\,x}\,{\mathrm {e}}^{x^2}}{x}-{\mathrm {e}}^{37\,x}\right )+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2)*(37*x + 2*x^2 - 1) - 37*x^2)/(x*exp(x^2) - x^2 + log(-(x*exp(37*x) - exp(37*x)*exp(x^2))/x)*(x*e
xp(x^2) - x^2)),x)

[Out]

log(log((exp(37*x)*exp(x^2))/x - exp(37*x)) + 1)

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sympy [A]  time = 0.52, size = 22, normalized size = 1.00 \begin {gather*} \log {\left (\log {\left (\frac {- x e^{37 x} + e^{37 x} e^{x^{2}}}{x} \right )} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+37*x-1)*exp(x**2)-37*x**2)/((exp(x**2)*x-x**2)*ln((exp(37*x)*exp(x**2)-x*exp(37*x))/x)+exp(
x**2)*x-x**2),x)

[Out]

log(log((-x*exp(37*x) + exp(37*x)*exp(x**2))/x) + 1)

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