3.38.82 \(\int \frac {-192+64 x-96 x^3+32 x^4+e^x (-128 x+64 x^2+32 x^4-16 x^5)+e^{x^2} (-30 x^3+10 x^4+e^x (10 x^2-5 x^3))+(-64 x+5 e^{x^2} x^2+16 x^4) \log (\frac {64-5 e^{x^2} x-16 x^3}{16 x})}{-64 x+5 e^{x^2} x^2+16 x^4} \, dx\)

Optimal. Leaf size=33 \[ (3-x) \left (e^x-\log \left (-\frac {5 e^{x^2}}{16}+\frac {4}{x}-x^2\right )\right ) \]

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Rubi [F]  time = 1.95, 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 {-192+64 x-96 x^3+32 x^4+e^x \left (-128 x+64 x^2+32 x^4-16 x^5\right )+e^{x^2} \left (-30 x^3+10 x^4+e^x \left (10 x^2-5 x^3\right )\right )+\left (-64 x+5 e^{x^2} x^2+16 x^4\right ) \log \left (\frac {64-5 e^{x^2} x-16 x^3}{16 x}\right )}{-64 x+5 e^{x^2} x^2+16 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-192 + 64*x - 96*x^3 + 32*x^4 + E^x*(-128*x + 64*x^2 + 32*x^4 - 16*x^5) + E^x^2*(-30*x^3 + 10*x^4 + E^x*(
10*x^2 - 5*x^3)) + (-64*x + 5*E^x^2*x^2 + 16*x^4)*Log[(64 - 5*E^x^2*x - 16*x^3)/(16*x)])/(-64*x + 5*E^x^2*x^2
+ 16*x^4),x]

[Out]

3*E^x - E^x*x - 3*x^2 + x*Log[(64 - 5*E^x^2*x - 16*x^3)/(16*x)] - 192*Defer[Int][1/(x*(-64 + 5*E^x^2*x + 16*x^
3)), x] - 384*Defer[Int][x/(-64 + 5*E^x^2*x + 16*x^3), x] - 96*Defer[Int][x^2/(-64 + 5*E^x^2*x + 16*x^3), x] +
96*Defer[Int][x^4/(-64 + 5*E^x^2*x + 16*x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 e^x-6 x-e^x x+2 x^2-\frac {32 \left (6-2 x+12 x^2-x^3-x^4-3 x^5+x^6\right )}{x \left (-64+5 e^{x^2} x+16 x^3\right )}+\log \left (-\frac {-64+5 e^{x^2} x+16 x^3}{16 x}\right )\right ) \, dx\\ &=-3 x^2+\frac {2 x^3}{3}+2 \int e^x \, dx-32 \int \frac {6-2 x+12 x^2-x^3-x^4-3 x^5+x^6}{x \left (-64+5 e^{x^2} x+16 x^3\right )} \, dx-\int e^x x \, dx+\int \log \left (-\frac {-64+5 e^{x^2} x+16 x^3}{16 x}\right ) \, dx\\ &=2 e^x-e^x x-3 x^2+\frac {2 x^3}{3}+x \log \left (\frac {64-5 e^{x^2} x-16 x^3}{16 x}\right )-32 \int \left (-\frac {2}{-64+5 e^{x^2} x+16 x^3}+\frac {6}{x \left (-64+5 e^{x^2} x+16 x^3\right )}+\frac {12 x}{-64+5 e^{x^2} x+16 x^3}-\frac {x^2}{-64+5 e^{x^2} x+16 x^3}-\frac {x^3}{-64+5 e^{x^2} x+16 x^3}-\frac {3 x^4}{-64+5 e^{x^2} x+16 x^3}+\frac {x^5}{-64+5 e^{x^2} x+16 x^3}\right ) \, dx+\int e^x \, dx-\int \frac {64+2 \left (16+5 e^{x^2}\right ) x^3}{5 e^{x^2} x+16 \left (-4+x^3\right )} \, dx\\ &=3 e^x-e^x x-3 x^2+\frac {2 x^3}{3}+x \log \left (\frac {64-5 e^{x^2} x-16 x^3}{16 x}\right )+32 \int \frac {x^2}{-64+5 e^{x^2} x+16 x^3} \, dx+32 \int \frac {x^3}{-64+5 e^{x^2} x+16 x^3} \, dx-32 \int \frac {x^5}{-64+5 e^{x^2} x+16 x^3} \, dx+64 \int \frac {1}{-64+5 e^{x^2} x+16 x^3} \, dx+96 \int \frac {x^4}{-64+5 e^{x^2} x+16 x^3} \, dx-192 \int \frac {1}{x \left (-64+5 e^{x^2} x+16 x^3\right )} \, dx-384 \int \frac {x}{-64+5 e^{x^2} x+16 x^3} \, dx-\int \left (2 x^2-\frac {32 \left (-2-4 x^2-x^3+x^5\right )}{-64+5 e^{x^2} x+16 x^3}\right ) \, dx\\ &=3 e^x-e^x x-3 x^2+x \log \left (\frac {64-5 e^{x^2} x-16 x^3}{16 x}\right )+32 \int \frac {x^2}{-64+5 e^{x^2} x+16 x^3} \, dx+32 \int \frac {x^3}{-64+5 e^{x^2} x+16 x^3} \, dx-32 \int \frac {x^5}{-64+5 e^{x^2} x+16 x^3} \, dx+32 \int \frac {-2-4 x^2-x^3+x^5}{-64+5 e^{x^2} x+16 x^3} \, dx+64 \int \frac {1}{-64+5 e^{x^2} x+16 x^3} \, dx+96 \int \frac {x^4}{-64+5 e^{x^2} x+16 x^3} \, dx-192 \int \frac {1}{x \left (-64+5 e^{x^2} x+16 x^3\right )} \, dx-384 \int \frac {x}{-64+5 e^{x^2} x+16 x^3} \, dx\\ &=3 e^x-e^x x-3 x^2+x \log \left (\frac {64-5 e^{x^2} x-16 x^3}{16 x}\right )+32 \int \frac {x^2}{-64+5 e^{x^2} x+16 x^3} \, dx+32 \int \frac {x^3}{-64+5 e^{x^2} x+16 x^3} \, dx-32 \int \frac {x^5}{-64+5 e^{x^2} x+16 x^3} \, dx+32 \int \left (-\frac {2}{-64+5 e^{x^2} x+16 x^3}-\frac {4 x^2}{-64+5 e^{x^2} x+16 x^3}-\frac {x^3}{-64+5 e^{x^2} x+16 x^3}+\frac {x^5}{-64+5 e^{x^2} x+16 x^3}\right ) \, dx+64 \int \frac {1}{-64+5 e^{x^2} x+16 x^3} \, dx+96 \int \frac {x^4}{-64+5 e^{x^2} x+16 x^3} \, dx-192 \int \frac {1}{x \left (-64+5 e^{x^2} x+16 x^3\right )} \, dx-384 \int \frac {x}{-64+5 e^{x^2} x+16 x^3} \, dx\\ &=3 e^x-e^x x-3 x^2+x \log \left (\frac {64-5 e^{x^2} x-16 x^3}{16 x}\right )+32 \int \frac {x^2}{-64+5 e^{x^2} x+16 x^3} \, dx+96 \int \frac {x^4}{-64+5 e^{x^2} x+16 x^3} \, dx-128 \int \frac {x^2}{-64+5 e^{x^2} x+16 x^3} \, dx-192 \int \frac {1}{x \left (-64+5 e^{x^2} x+16 x^3\right )} \, dx-384 \int \frac {x}{-64+5 e^{x^2} x+16 x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 55, normalized size = 1.67 \begin {gather*} e^x (3-x)+3 \log (x)+x \log \left (-\frac {5 e^{x^2}}{16}+\frac {4}{x}-x^2\right )-3 \log \left (64-5 e^{x^2} x-16 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-192 + 64*x - 96*x^3 + 32*x^4 + E^x*(-128*x + 64*x^2 + 32*x^4 - 16*x^5) + E^x^2*(-30*x^3 + 10*x^4 +
E^x*(10*x^2 - 5*x^3)) + (-64*x + 5*E^x^2*x^2 + 16*x^4)*Log[(64 - 5*E^x^2*x - 16*x^3)/(16*x)])/(-64*x + 5*E^x^
2*x^2 + 16*x^4),x]

[Out]

E^x*(3 - x) + 3*Log[x] + x*Log[(-5*E^x^2)/16 + 4/x - x^2] - 3*Log[64 - 5*E^x^2*x - 16*x^3]

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fricas [A]  time = 1.03, size = 32, normalized size = 0.97 \begin {gather*} -{\left (x - 3\right )} e^{x} + {\left (x - 3\right )} \log \left (-\frac {16 \, x^{3} + 5 \, x e^{\left (x^{2}\right )} - 64}{16 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2*exp(x^2)+16*x^4-64*x)*log(1/16*(-5*exp(x^2)*x-16*x^3+64)/x)+((-5*x^3+10*x^2)*exp(x)+10*x^4-3
0*x^3)*exp(x^2)+(-16*x^5+32*x^4+64*x^2-128*x)*exp(x)+32*x^4-96*x^3+64*x-192)/(5*x^2*exp(x^2)+16*x^4-64*x),x, a
lgorithm="fricas")

[Out]

-(x - 3)*e^x + (x - 3)*log(-1/16*(16*x^3 + 5*x*e^(x^2) - 64)/x)

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giac [A]  time = 0.20, size = 53, normalized size = 1.61 \begin {gather*} -x e^{x} + x \log \left (-\frac {16 \, x^{3} + 5 \, x e^{\left (x^{2}\right )} - 64}{16 \, x}\right ) + 3 \, e^{x} - 3 \, \log \left (16 \, x^{3} + 5 \, x e^{\left (x^{2}\right )} - 64\right ) + 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2*exp(x^2)+16*x^4-64*x)*log(1/16*(-5*exp(x^2)*x-16*x^3+64)/x)+((-5*x^3+10*x^2)*exp(x)+10*x^4-3
0*x^3)*exp(x^2)+(-16*x^5+32*x^4+64*x^2-128*x)*exp(x)+32*x^4-96*x^3+64*x-192)/(5*x^2*exp(x^2)+16*x^4-64*x),x, a
lgorithm="giac")

[Out]

-x*e^x + x*log(-1/16*(16*x^3 + 5*x*e^(x^2) - 64)/x) + 3*e^x - 3*log(16*x^3 + 5*x*e^(x^2) - 64) + 3*log(x)

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maple [A]  time = 0.19, size = 55, normalized size = 1.67

method	result	size
default	\(x \ln \left (\frac {-5 \,{\mathrm e}^{x^{2}} x -16 x^{3}+64}{16 x}\right )-3 \ln \left (\frac {-5 \,{\mathrm e}^{x^{2}} x -16 x^{3}+64}{16 x}\right )-{\mathrm e}^{x} x +3 \,{\mathrm e}^{x}\)	\(55\)
risch	\(x \ln \left (x^{3}+\frac {5 \,{\mathrm e}^{x^{2}} x}{16}-4\right )-x \ln \relax (x )-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x^{3}+\frac {5 \,{\mathrm e}^{x^{2}} x}{16}-4\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{3}+\frac {5 \,{\mathrm e}^{x^{2}} x}{16}-4\right )}{x}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x^{3}+\frac {5 \,{\mathrm e}^{x^{2}} x}{16}-4\right )}{x}\right )^{2}}{2}-i \pi x \mathrm {csgn}\left (\frac {i \left (x^{3}+\frac {5 \,{\mathrm e}^{x^{2}} x}{16}-4\right )}{x}\right )^{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (x^{3}+\frac {5 \,{\mathrm e}^{x^{2}} x}{16}-4\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{3}+\frac {5 \,{\mathrm e}^{x^{2}} x}{16}-4\right )}{x}\right )^{2}}{2}+\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (x^{3}+\frac {5 \,{\mathrm e}^{x^{2}} x}{16}-4\right )}{x}\right )^{3}}{2}+i \pi x -{\mathrm e}^{x} x +3 \,{\mathrm e}^{x}-3 \ln \left ({\mathrm e}^{x^{2}}+\frac {\frac {16 x^{3}}{5}-\frac {64}{5}}{x}\right )\)	\(228\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^2*exp(x^2)+16*x^4-64*x)*ln(1/16*(-5*exp(x^2)*x-16*x^3+64)/x)+((-5*x^3+10*x^2)*exp(x)+10*x^4-30*x^3)*
exp(x^2)+(-16*x^5+32*x^4+64*x^2-128*x)*exp(x)+32*x^4-96*x^3+64*x-192)/(5*x^2*exp(x^2)+16*x^4-64*x),x,method=_R
ETURNVERBOSE)

[Out]

x*ln(1/16*(-5*exp(x^2)*x-16*x^3+64)/x)-3*ln(1/16*(-5*exp(x^2)*x-16*x^3+64)/x)-exp(x)*x+3*exp(x)

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maxima [B]  time = 0.65, size = 57, normalized size = 1.73 \begin {gather*} -{\left (x - 3\right )} e^{x} - 4 \, x \log \relax (2) + x \log \left (-16 \, x^{3} - 5 \, x e^{\left (x^{2}\right )} + 64\right ) - x \log \relax (x) - 3 \, \log \left (\frac {16 \, x^{3} + 5 \, x e^{\left (x^{2}\right )} - 64}{5 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2*exp(x^2)+16*x^4-64*x)*log(1/16*(-5*exp(x^2)*x-16*x^3+64)/x)+((-5*x^3+10*x^2)*exp(x)+10*x^4-3
0*x^3)*exp(x^2)+(-16*x^5+32*x^4+64*x^2-128*x)*exp(x)+32*x^4-96*x^3+64*x-192)/(5*x^2*exp(x^2)+16*x^4-64*x),x, a
lgorithm="maxima")

[Out]

-(x - 3)*e^x - 4*x*log(2) + x*log(-16*x^3 - 5*x*e^(x^2) + 64) - x*log(x) - 3*log(1/5*(16*x^3 + 5*x*e^(x^2) - 6
4)/x)

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mupad [B]  time = 2.63, size = 49, normalized size = 1.48 \begin {gather*} x\,\ln \left (-\frac {\frac {5\,x\,{\mathrm {e}}^{x^2}}{16}+x^3-4}{x}\right )-{\mathrm {e}}^x\,\left (x-3\right )-3\,\ln \left (\frac {5\,x\,{\mathrm {e}}^{x^2}+16\,x^3-64}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((64*x - exp(x)*(128*x - 64*x^2 - 32*x^4 + 16*x^5) + log(-((5*x*exp(x^2))/16 + x^3 - 4)/x)*(5*x^2*exp(x^2)
- 64*x + 16*x^4) + exp(x^2)*(exp(x)*(10*x^2 - 5*x^3) - 30*x^3 + 10*x^4) - 96*x^3 + 32*x^4 - 192)/(5*x^2*exp(x^
2) - 64*x + 16*x^4),x)

[Out]

x*log(-((5*x*exp(x^2))/16 + x^3 - 4)/x) - exp(x)*(x - 3) - 3*log((5*x*exp(x^2) + 16*x^3 - 64)/x)

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sympy [A]  time = 0.95, size = 44, normalized size = 1.33 \begin {gather*} x \log {\left (\frac {- x^{3} - \frac {5 x e^{x^{2}}}{16} + 4}{x} \right )} + \left (3 - x\right ) e^{x} - 3 \log {\left (e^{x^{2}} + \frac {16 x^{3} - 64}{5 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**2*exp(x**2)+16*x**4-64*x)*ln(1/16*(-5*exp(x**2)*x-16*x**3+64)/x)+((-5*x**3+10*x**2)*exp(x)+10
*x**4-30*x**3)*exp(x**2)+(-16*x**5+32*x**4+64*x**2-128*x)*exp(x)+32*x**4-96*x**3+64*x-192)/(5*x**2*exp(x**2)+1
6*x**4-64*x),x)

[Out]

x*log((-x**3 - 5*x*exp(x**2)/16 + 4)/x) + (3 - x)*exp(x) - 3*log(exp(x**2) + (16*x**3 - 64)/(5*x))

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