3.42.99 \(\int \frac {4 x+e^{x+\frac {2 (10+2 x^2)}{x}} x^2+e^x (4-4 x^2+x^4)+e^{\frac {10+2 x^2}{x}} (-10 x-x^2+2 x^3+e^x (-4 x+2 x^3))}{4-4 x^2+e^{\frac {2 (10+2 x^2)}{x}} x^2+x^4+e^{\frac {10+2 x^2}{x}} (-4 x+2 x^3)} \, dx\)

Optimal. Leaf size=28 \[ 1+e^x-\frac {x}{e^{\frac {10}{x}+2 x}-\frac {2}{x}+x} \]

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Rubi [F]  time = 1.29, 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 x+e^{x+\frac {2 \left (10+2 x^2\right )}{x}} x^2+e^x \left (4-4 x^2+x^4\right )+e^{\frac {10+2 x^2}{x}} \left (-10 x-x^2+2 x^3+e^x \left (-4 x+2 x^3\right )\right )}{4-4 x^2+e^{\frac {2 \left (10+2 x^2\right )}{x}} x^2+x^4+e^{\frac {10+2 x^2}{x}} \left (-4 x+2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x + E^(x + (2*(10 + 2*x^2))/x)*x^2 + E^x*(4 - 4*x^2 + x^4) + E^((10 + 2*x^2)/x)*(-10*x - x^2 + 2*x^3 +
E^x*(-4*x + 2*x^3)))/(4 - 4*x^2 + E^((2*(10 + 2*x^2))/x)*x^2 + x^4 + E^((10 + 2*x^2)/x)*(-4*x + 2*x^3)),x]

[Out]

E^x - 20*Defer[Int][(-2 + E^(10/x + 2*x)*x + x^2)^(-2), x] + 2*Defer[Int][x/(-2 + E^(10/x + 2*x)*x + x^2)^2, x
] + 14*Defer[Int][x^2/(-2 + E^(10/x + 2*x)*x + x^2)^2, x] + Defer[Int][x^3/(-2 + E^(10/x + 2*x)*x + x^2)^2, x]
- 2*Defer[Int][x^4/(-2 + E^(10/x + 2*x)*x + x^2)^2, x] - 10*Defer[Int][(-2 + E^(10/x + 2*x)*x + x^2)^(-1), x]
- Defer[Int][x/(-2 + E^(10/x + 2*x)*x + x^2), x] + 2*Defer[Int][x^2/(-2 + E^(10/x + 2*x)*x + x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x+e^{\frac {20}{x}+5 x} x^2+2 e^{\frac {10}{x}+3 x} x \left (-2+x^2\right )+e^x \left (-2+x^2\right )^2+e^{\frac {10}{x}+2 x} x \left (-10-x+2 x^2\right )}{\left (2-e^{\frac {10}{x}+2 x} x-x^2\right )^2} \, dx\\ &=\int \left (e^x+\frac {-10-x+2 x^2}{-2+e^{\frac {10}{x}+2 x} x+x^2}-\frac {20-2 x-14 x^2-x^3+2 x^4}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2}\right ) \, dx\\ &=\int e^x \, dx+\int \frac {-10-x+2 x^2}{-2+e^{\frac {10}{x}+2 x} x+x^2} \, dx-\int \frac {20-2 x-14 x^2-x^3+2 x^4}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2} \, dx\\ &=e^x-\int \left (\frac {20}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2}-\frac {2 x}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2}-\frac {14 x^2}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2}-\frac {x^3}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2}+\frac {2 x^4}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2}\right ) \, dx+\int \left (-\frac {10}{-2+e^{\frac {10}{x}+2 x} x+x^2}-\frac {x}{-2+e^{\frac {10}{x}+2 x} x+x^2}+\frac {2 x^2}{-2+e^{\frac {10}{x}+2 x} x+x^2}\right ) \, dx\\ &=e^x+2 \int \frac {x}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2} \, dx-2 \int \frac {x^4}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2} \, dx+2 \int \frac {x^2}{-2+e^{\frac {10}{x}+2 x} x+x^2} \, dx-10 \int \frac {1}{-2+e^{\frac {10}{x}+2 x} x+x^2} \, dx+14 \int \frac {x^2}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2} \, dx-20 \int \frac {1}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2} \, dx+\int \frac {x^3}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right )^2} \, dx-\int \frac {x}{-2+e^{\frac {10}{x}+2 x} x+x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 1.49, size = 901, normalized size = 32.18 \begin {gather*} \frac {\frac {2095 e^x \left (-1300-1030 x+946 x^2+661 x^3\right )}{4 \left (-20+2 x+14 x^2+x^3-2 x^4\right )^2}-\frac {16760 e^{\frac {10}{x}+3 x} x^3}{20-2 x-14 x^2-x^3+2 x^4}+\frac {e^x \left (390015+82526 x-110828 x^2+2612 x^3\right )}{80-8 x-56 x^2-4 x^3+8 x^4}-60060 \text {RootSum}\left [20-2 \text {$\#$1}-14 \text {$\#$1}^2-\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {e^{\text {$\#$1}} \text {Ei}(x-\text {$\#$1})}{-2-28 \text {$\#$1}-3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ]-52674 \text {RootSum}\left [20-2 \text {$\#$1}-14 \text {$\#$1}^2-\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {e^{\text {$\#$1}} \text {Ei}(x-\text {$\#$1}) \text {$\#$1}}{-2-28 \text {$\#$1}-3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ]+32550 \text {RootSum}\left [20-2 \text {$\#$1}-14 \text {$\#$1}^2-\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {e^{\text {$\#$1}} \text {Ei}(x-\text {$\#$1}) \text {$\#$1}^2}{-2-28 \text {$\#$1}-3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ]-9033 \text {RootSum}\left [20-2 \text {$\#$1}-14 \text {$\#$1}^2-\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {e^{\text {$\#$1}} \text {Ei}(x-\text {$\#$1}) \text {$\#$1}^3}{-2-28 \text {$\#$1}-3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ]}{16760}+\frac {\frac {16760 e^{\frac {20}{x}+5 x} x^4}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right ) \left (20-2 x-14 x^2-x^3+2 x^4\right )}+\frac {33520 e^{\frac {10}{x}+3 x} x^3 \left (-2+x^2\right )}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right ) \left (20-2 x-14 x^2-x^3+2 x^4\right )}+\frac {e^x \left (5434800-1018960 x-7183172 x^2+172792 x^3+3327352 x^4+275372 x^5-564053 x^6-51586 x^7+33520 x^8\right )}{\left (-20+2 x+14 x^2+x^3-2 x^4\right )^2}+\frac {16760 x^2 \left (-20+2 x+14 x^2+x^3-2 x^4+e^x \left (-2+x^2\right )^2\right )}{\left (-2+e^{\frac {10}{x}+2 x} x+x^2\right ) \left (20-2 x-14 x^2-x^3+2 x^4\right )}+60060 \text {RootSum}\left [20-2 \text {$\#$1}-14 \text {$\#$1}^2-\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {e^{\text {$\#$1}} \text {Ei}(x-\text {$\#$1})}{-2-28 \text {$\#$1}-3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ]+52674 \text {RootSum}\left [20-2 \text {$\#$1}-14 \text {$\#$1}^2-\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {e^{\text {$\#$1}} \text {Ei}(x-\text {$\#$1}) \text {$\#$1}}{-2-28 \text {$\#$1}-3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ]-32550 \text {RootSum}\left [20-2 \text {$\#$1}-14 \text {$\#$1}^2-\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {e^{\text {$\#$1}} \text {Ei}(x-\text {$\#$1}) \text {$\#$1}^2}{-2-28 \text {$\#$1}-3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ]+9033 \text {RootSum}\left [20-2 \text {$\#$1}-14 \text {$\#$1}^2-\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {e^{\text {$\#$1}} \text {Ei}(x-\text {$\#$1}) \text {$\#$1}^3}{-2-28 \text {$\#$1}-3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ]}{16760} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x + E^(x + (2*(10 + 2*x^2))/x)*x^2 + E^x*(4 - 4*x^2 + x^4) + E^((10 + 2*x^2)/x)*(-10*x - x^2 + 2*
x^3 + E^x*(-4*x + 2*x^3)))/(4 - 4*x^2 + E^((2*(10 + 2*x^2))/x)*x^2 + x^4 + E^((10 + 2*x^2)/x)*(-4*x + 2*x^3)),
x]

[Out]

((2095*E^x*(-1300 - 1030*x + 946*x^2 + 661*x^3))/(4*(-20 + 2*x + 14*x^2 + x^3 - 2*x^4)^2) - (16760*E^(10/x + 3
*x)*x^3)/(20 - 2*x - 14*x^2 - x^3 + 2*x^4) + (E^x*(390015 + 82526*x - 110828*x^2 + 2612*x^3))/(80 - 8*x - 56*x
^2 - 4*x^3 + 8*x^4) - 60060*RootSum[20 - 2*#1 - 14*#1^2 - #1^3 + 2*#1^4 & , (E^#1*ExpIntegralEi[x - #1])/(-2 -
28*#1 - 3*#1^2 + 8*#1^3) & ] - 52674*RootSum[20 - 2*#1 - 14*#1^2 - #1^3 + 2*#1^4 & , (E^#1*ExpIntegralEi[x -
#1]*#1)/(-2 - 28*#1 - 3*#1^2 + 8*#1^3) & ] + 32550*RootSum[20 - 2*#1 - 14*#1^2 - #1^3 + 2*#1^4 & , (E^#1*ExpIn
tegralEi[x - #1]*#1^2)/(-2 - 28*#1 - 3*#1^2 + 8*#1^3) & ] - 9033*RootSum[20 - 2*#1 - 14*#1^2 - #1^3 + 2*#1^4 &
, (E^#1*ExpIntegralEi[x - #1]*#1^3)/(-2 - 28*#1 - 3*#1^2 + 8*#1^3) & ])/16760 + ((16760*E^(20/x + 5*x)*x^4)/(
(-2 + E^(10/x + 2*x)*x + x^2)*(20 - 2*x - 14*x^2 - x^3 + 2*x^4)) + (33520*E^(10/x + 3*x)*x^3*(-2 + x^2))/((-2
+ E^(10/x + 2*x)*x + x^2)*(20 - 2*x - 14*x^2 - x^3 + 2*x^4)) + (E^x*(5434800 - 1018960*x - 7183172*x^2 + 17279
2*x^3 + 3327352*x^4 + 275372*x^5 - 564053*x^6 - 51586*x^7 + 33520*x^8))/(-20 + 2*x + 14*x^2 + x^3 - 2*x^4)^2 +
(16760*x^2*(-20 + 2*x + 14*x^2 + x^3 - 2*x^4 + E^x*(-2 + x^2)^2))/((-2 + E^(10/x + 2*x)*x + x^2)*(20 - 2*x -
14*x^2 - x^3 + 2*x^4)) + 60060*RootSum[20 - 2*#1 - 14*#1^2 - #1^3 + 2*#1^4 & , (E^#1*ExpIntegralEi[x - #1])/(-
2 - 28*#1 - 3*#1^2 + 8*#1^3) & ] + 52674*RootSum[20 - 2*#1 - 14*#1^2 - #1^3 + 2*#1^4 & , (E^#1*ExpIntegralEi[x
- #1]*#1)/(-2 - 28*#1 - 3*#1^2 + 8*#1^3) & ] - 32550*RootSum[20 - 2*#1 - 14*#1^2 - #1^3 + 2*#1^4 & , (E^#1*Ex
pIntegralEi[x - #1]*#1^2)/(-2 - 28*#1 - 3*#1^2 + 8*#1^3) & ] + 9033*RootSum[20 - 2*#1 - 14*#1^2 - #1^3 + 2*#1^
4 & , (E^#1*ExpIntegralEi[x - #1]*#1^3)/(-2 - 28*#1 - 3*#1^2 + 8*#1^3) & ])/16760

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fricas [B]  time = 0.95, size = 82, normalized size = 2.93 \begin {gather*} -\frac {x^{2} e^{\left (\frac {4 \, {\left (x^{2} + 5\right )}}{x}\right )} - {\left (x^{2} + x e^{\left (\frac {2 \, {\left (x^{2} + 5\right )}}{x}\right )} - 2\right )} e^{\left (\frac {5 \, {\left (x^{2} + 4\right )}}{x}\right )}}{x e^{\left (\frac {6 \, {\left (x^{2} + 5\right )}}{x}\right )} + {\left (x^{2} - 2\right )} e^{\left (\frac {4 \, {\left (x^{2} + 5\right )}}{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(x)*exp((2*x^2+10)/x)^2+((2*x^3-4*x)*exp(x)+2*x^3-x^2-10*x)*exp((2*x^2+10)/x)+(x^4-4*x^2+4)*
exp(x)+4*x)/(x^2*exp((2*x^2+10)/x)^2+(2*x^3-4*x)*exp((2*x^2+10)/x)+x^4-4*x^2+4),x, algorithm="fricas")

[Out]

-(x^2*e^(4*(x^2 + 5)/x) - (x^2 + x*e^(2*(x^2 + 5)/x) - 2)*e^(5*(x^2 + 4)/x))/(x*e^(6*(x^2 + 5)/x) + (x^2 - 2)*
e^(4*(x^2 + 5)/x))

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giac [B]  time = 0.28, size = 304, normalized size = 10.86 \begin {gather*} \frac {2 \, x^{6} e^{x} - 2 \, x^{6} - x^{5} e^{x} + 2 \, x^{5} e^{\left (\frac {3 \, x^{2} + 10}{x}\right )} + x^{5} - x^{4} e^{\left (x + \frac {4 \, {\left (x^{2} + 5\right )}}{x}\right )} - x^{4} e^{\left (x + \frac {2 \, {\left (x^{2} + 5\right )}}{x}\right )} - 18 \, x^{4} e^{x} + x^{4} e^{\left (\frac {5 \, {\left (x^{2} + 4\right )}}{x}\right )} + 14 \, x^{4} - 10 \, x^{3} e^{\left (x + \frac {2 \, {\left (x^{2} + 5\right )}}{x}\right )} - 4 \, x^{3} e^{\left (\frac {3 \, x^{2} + 10}{x}\right )} + 2 \, x^{3} - 2 \, x^{2} e^{\left (x + \frac {2 \, {\left (x^{2} + 5\right )}}{x}\right )} + 48 \, x^{2} e^{x} - 20 \, x^{2} + 20 \, x e^{\left (x + \frac {2 \, {\left (x^{2} + 5\right )}}{x}\right )} + 4 \, x e^{x} - 40 \, e^{x}}{2 \, x^{6} + 2 \, x^{5} e^{\left (\frac {2 \, {\left (x^{2} + 5\right )}}{x}\right )} - x^{5} - x^{4} e^{\left (\frac {2 \, {\left (x^{2} + 5\right )}}{x}\right )} - 18 \, x^{4} - 14 \, x^{3} e^{\left (\frac {2 \, {\left (x^{2} + 5\right )}}{x}\right )} - 2 \, x^{2} e^{\left (\frac {2 \, {\left (x^{2} + 5\right )}}{x}\right )} + 48 \, x^{2} + 20 \, x e^{\left (\frac {2 \, {\left (x^{2} + 5\right )}}{x}\right )} + 4 \, x - 40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(x)*exp((2*x^2+10)/x)^2+((2*x^3-4*x)*exp(x)+2*x^3-x^2-10*x)*exp((2*x^2+10)/x)+(x^4-4*x^2+4)*
exp(x)+4*x)/(x^2*exp((2*x^2+10)/x)^2+(2*x^3-4*x)*exp((2*x^2+10)/x)+x^4-4*x^2+4),x, algorithm="giac")

[Out]

(2*x^6*e^x - 2*x^6 - x^5*e^x + 2*x^5*e^((3*x^2 + 10)/x) + x^5 - x^4*e^(x + 4*(x^2 + 5)/x) - x^4*e^(x + 2*(x^2
+ 5)/x) - 18*x^4*e^x + x^4*e^(5*(x^2 + 4)/x) + 14*x^4 - 10*x^3*e^(x + 2*(x^2 + 5)/x) - 4*x^3*e^((3*x^2 + 10)/x
) + 2*x^3 - 2*x^2*e^(x + 2*(x^2 + 5)/x) + 48*x^2*e^x - 20*x^2 + 20*x*e^(x + 2*(x^2 + 5)/x) + 4*x*e^x - 40*e^x)
/(2*x^6 + 2*x^5*e^(2*(x^2 + 5)/x) - x^5 - x^4*e^(2*(x^2 + 5)/x) - 18*x^4 - 14*x^3*e^(2*(x^2 + 5)/x) - 2*x^2*e^
(2*(x^2 + 5)/x) + 48*x^2 + 20*x*e^(2*(x^2 + 5)/x) + 4*x - 40)

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maple [A]  time = 0.16, size = 29, normalized size = 1.04

method	result	size
risch	\({\mathrm e}^{x}-\frac {x^{2}}{x^{2}+x \,{\mathrm e}^{\frac {2 x^{2}+10}{x}}-2}\)	\(29\)
norman	\(\frac {{\mathrm e}^{x} x^{2}+x \,{\mathrm e}^{\frac {2 x^{2}+10}{x}}+{\mathrm e}^{x} x \,{\mathrm e}^{\frac {2 x^{2}+10}{x}}-2 \,{\mathrm e}^{x}-2}{x^{2}+x \,{\mathrm e}^{\frac {2 x^{2}+10}{x}}-2}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(x)*exp((2*x^2+10)/x)^2+((2*x^3-4*x)*exp(x)+2*x^3-x^2-10*x)*exp((2*x^2+10)/x)+(x^4-4*x^2+4)*exp(x)
+4*x)/(x^2*exp((2*x^2+10)/x)^2+(2*x^3-4*x)*exp((2*x^2+10)/x)+x^4-4*x^2+4),x,method=_RETURNVERBOSE)

[Out]

exp(x)-x^2/(exp(2*(x^2+5)/x)*x+x^2-2)

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maxima [A]  time = 0.40, size = 47, normalized size = 1.68 \begin {gather*} -\frac {x^{2} - x e^{\left (3 \, x + \frac {10}{x}\right )} - {\left (x^{2} - 2\right )} e^{x}}{x^{2} + x e^{\left (2 \, x + \frac {10}{x}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(x)*exp((2*x^2+10)/x)^2+((2*x^3-4*x)*exp(x)+2*x^3-x^2-10*x)*exp((2*x^2+10)/x)+(x^4-4*x^2+4)*
exp(x)+4*x)/(x^2*exp((2*x^2+10)/x)^2+(2*x^3-4*x)*exp((2*x^2+10)/x)+x^4-4*x^2+4),x, algorithm="maxima")

[Out]

-(x^2 - x*e^(3*x + 10/x) - (x^2 - 2)*e^x)/(x^2 + x*e^(2*x + 10/x) - 2)

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mupad [B]  time = 3.45, size = 27, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^x-\frac {x^2}{x^2+x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{10/x}-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - exp((2*x^2 + 10)/x)*(10*x + exp(x)*(4*x - 2*x^3) + x^2 - 2*x^3) + exp(x)*(x^4 - 4*x^2 + 4) + x^2*ex
p((2*(2*x^2 + 10))/x)*exp(x))/(x^2*exp((2*(2*x^2 + 10))/x) - 4*x^2 + x^4 - exp((2*x^2 + 10)/x)*(4*x - 2*x^3) +
4),x)

[Out]

exp(x) - x^2/(x^2 + x*exp(2*x)*exp(10/x) - 2)

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sympy [A]  time = 0.23, size = 22, normalized size = 0.79 \begin {gather*} - \frac {x^{2}}{x^{2} + x e^{\frac {2 x^{2} + 10}{x}} - 2} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*exp(x)*exp((2*x**2+10)/x)**2+((2*x**3-4*x)*exp(x)+2*x**3-x**2-10*x)*exp((2*x**2+10)/x)+(x**4-4
*x**2+4)*exp(x)+4*x)/(x**2*exp((2*x**2+10)/x)**2+(2*x**3-4*x)*exp((2*x**2+10)/x)+x**4-4*x**2+4),x)

[Out]

-x**2/(x**2 + x*exp((2*x**2 + 10)/x) - 2) + exp(x)

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