3.82.47 \(\int \frac {2 e^{5 x^6} x-10 e^{4 x^6} x^2-242 x^6+220 e^2 x^6-50 e^4 x^6+e^{3 x^6} (100 x^3-240 x^9+e^2 (-40 x^3+120 x^9))+e^{2 x^6} (-220 x^4+480 x^{10}+e^2 (100 x^4-240 x^{10}))+e^{x^6} (770 x^5-2640 x^{11}+e^4 (150 x^5-600 x^{11})+e^2 (-680 x^5+2520 x^{11}))}{e^{5 x^6}-5 e^{4 x^6} x+10 e^{3 x^6} x^2-10 e^{2 x^6} x^3+5 e^{x^6} x^4-x^5} \, dx\) [8147]

Optimal. Leaf size=27 \[ \left (x+\frac {5 \left (2-e^2\right ) x^3}{\left (-e^{x^6}+x\right )^2}\right )^2 \]

[Out]

(5*x^3/(x-exp(x^6))^2*(2-exp(2))+x)^2

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Rubi [F]
time = 2.54, 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 {2 e^{5 x^6} x-10 e^{4 x^6} x^2-242 x^6+220 e^2 x^6-50 e^4 x^6+e^{3 x^6} \left (100 x^3-240 x^9+e^2 \left (-40 x^3+120 x^9\right )\right )+e^{2 x^6} \left (-220 x^4+480 x^{10}+e^2 \left (100 x^4-240 x^{10}\right )\right )+e^{x^6} \left (770 x^5-2640 x^{11}+e^4 \left (150 x^5-600 x^{11}\right )+e^2 \left (-680 x^5+2520 x^{11}\right )\right )}{e^{5 x^6}-5 e^{4 x^6} x+10 e^{3 x^6} x^2-10 e^{2 x^6} x^3+5 e^{x^6} x^4-x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*E^(5*x^6)*x - 10*E^(4*x^6)*x^2 - 242*x^6 + 220*E^2*x^6 - 50*E^4*x^6 + E^(3*x^6)*(100*x^3 - 240*x^9 + E^
2*(-40*x^3 + 120*x^9)) + E^(2*x^6)*(-220*x^4 + 480*x^10 + E^2*(100*x^4 - 240*x^10)) + E^x^6*(770*x^5 - 2640*x^
11 + E^4*(150*x^5 - 600*x^11) + E^2*(-680*x^5 + 2520*x^11)))/(E^(5*x^6) - 5*E^(4*x^6)*x + 10*E^(3*x^6)*x^2 - 1
0*E^(2*x^6)*x^3 + 5*E^x^6*x^4 - x^5),x]

[Out]

x^2 + 40*(2 - E^2)*Defer[Int][x^3/(E^x^6 - x)^2, x] + 20*(2 - E^2)*Defer[Int][x^4/(E^x^6 - x)^3, x] + 150*(2 -
E^2)^2*Defer[Int][x^5/(E^x^6 - x)^4, x] + 100*(2 - E^2)^2*Defer[Int][x^6/(E^x^6 - x)^5, x] - 120*(2 - E^2)*De
fer[Int][x^9/(E^x^6 - x)^2, x] - 120*(2 - E^2)*Defer[Int][x^10/(E^x^6 - x)^3, x] - 600*(2 - E^2)^2*Defer[Int][
x^11/(E^x^6 - x)^4, x] - 600*(2 - E^2)^2*Defer[Int][x^12/(E^x^6 - x)^5, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{5 x^6} x-10 e^{4 x^6} x^2-50 e^4 x^6+\left (-242+220 e^2\right ) x^6+e^{3 x^6} \left (100 x^3-240 x^9+e^2 \left (-40 x^3+120 x^9\right )\right )+e^{2 x^6} \left (-220 x^4+480 x^{10}+e^2 \left (100 x^4-240 x^{10}\right )\right )+e^{x^6} \left (770 x^5-2640 x^{11}+e^4 \left (150 x^5-600 x^{11}\right )+e^2 \left (-680 x^5+2520 x^{11}\right )\right )}{e^{5 x^6}-5 e^{4 x^6} x+10 e^{3 x^6} x^2-10 e^{2 x^6} x^3+5 e^{x^6} x^4-x^5} \, dx\\ &=\int \frac {2 e^{5 x^6} x-10 e^{4 x^6} x^2+\left (-242+220 e^2-50 e^4\right ) x^6+e^{3 x^6} \left (100 x^3-240 x^9+e^2 \left (-40 x^3+120 x^9\right )\right )+e^{2 x^6} \left (-220 x^4+480 x^{10}+e^2 \left (100 x^4-240 x^{10}\right )\right )+e^{x^6} \left (770 x^5-2640 x^{11}+e^4 \left (150 x^5-600 x^{11}\right )+e^2 \left (-680 x^5+2520 x^{11}\right )\right )}{e^{5 x^6}-5 e^{4 x^6} x+10 e^{3 x^6} x^2-10 e^{2 x^6} x^3+5 e^{x^6} x^4-x^5} \, dx\\ &=\int \frac {2 e^{5 x^6} x-10 e^{4 x^6} x^2+\left (-242+220 e^2-50 e^4\right ) x^6+e^{3 x^6} \left (100 x^3-240 x^9+e^2 \left (-40 x^3+120 x^9\right )\right )+e^{2 x^6} \left (-220 x^4+480 x^{10}+e^2 \left (100 x^4-240 x^{10}\right )\right )+e^{x^6} \left (770 x^5-2640 x^{11}+e^4 \left (150 x^5-600 x^{11}\right )+e^2 \left (-680 x^5+2520 x^{11}\right )\right )}{\left (e^{x^6}-x\right )^5} \, dx\\ &=\int \left (2 x+\frac {40 \left (-2+e^2\right ) x^3 \left (-1+3 x^6\right )}{\left (e^{x^6}-x\right )^2}-\frac {150 \left (-2+e^2\right )^2 x^5 \left (-1+4 x^6\right )}{\left (e^{x^6}-x\right )^4}+\frac {20 \left (-2+e^2\right ) x^4 \left (-1+6 x^6\right )}{\left (e^{x^6}-x\right )^3}-\frac {100 \left (-2+e^2\right )^2 x^6 \left (-1+6 x^6\right )}{\left (e^{x^6}-x\right )^5}\right ) \, dx\\ &=x^2-\left (20 \left (2-e^2\right )\right ) \int \frac {x^4 \left (-1+6 x^6\right )}{\left (e^{x^6}-x\right )^3} \, dx-\left (40 \left (2-e^2\right )\right ) \int \frac {x^3 \left (-1+3 x^6\right )}{\left (e^{x^6}-x\right )^2} \, dx-\left (100 \left (2-e^2\right )^2\right ) \int \frac {x^6 \left (-1+6 x^6\right )}{\left (e^{x^6}-x\right )^5} \, dx-\left (150 \left (2-e^2\right )^2\right ) \int \frac {x^5 \left (-1+4 x^6\right )}{\left (e^{x^6}-x\right )^4} \, dx\\ &=x^2-\left (20 \left (2-e^2\right )\right ) \int \left (-\frac {x^4}{\left (e^{x^6}-x\right )^3}+\frac {6 x^{10}}{\left (e^{x^6}-x\right )^3}\right ) \, dx-\left (40 \left (2-e^2\right )\right ) \int \left (-\frac {x^3}{\left (e^{x^6}-x\right )^2}+\frac {3 x^9}{\left (e^{x^6}-x\right )^2}\right ) \, dx-\left (100 \left (2-e^2\right )^2\right ) \int \left (-\frac {x^6}{\left (e^{x^6}-x\right )^5}+\frac {6 x^{12}}{\left (e^{x^6}-x\right )^5}\right ) \, dx-\left (150 \left (2-e^2\right )^2\right ) \int \left (-\frac {x^5}{\left (e^{x^6}-x\right )^4}+\frac {4 x^{11}}{\left (e^{x^6}-x\right )^4}\right ) \, dx\\ &=x^2+\left (20 \left (2-e^2\right )\right ) \int \frac {x^4}{\left (e^{x^6}-x\right )^3} \, dx+\left (40 \left (2-e^2\right )\right ) \int \frac {x^3}{\left (e^{x^6}-x\right )^2} \, dx-\left (120 \left (2-e^2\right )\right ) \int \frac {x^9}{\left (e^{x^6}-x\right )^2} \, dx-\left (120 \left (2-e^2\right )\right ) \int \frac {x^{10}}{\left (e^{x^6}-x\right )^3} \, dx+\left (100 \left (2-e^2\right )^2\right ) \int \frac {x^6}{\left (e^{x^6}-x\right )^5} \, dx+\left (150 \left (2-e^2\right )^2\right ) \int \frac {x^5}{\left (e^{x^6}-x\right )^4} \, dx-\left (600 \left (2-e^2\right )^2\right ) \int \frac {x^{11}}{\left (e^{x^6}-x\right )^4} \, dx-\left (600 \left (2-e^2\right )^2\right ) \int \frac {x^{12}}{\left (e^{x^6}-x\right )^5} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 10.11, size = 46, normalized size = 1.70 \begin {gather*} \frac {x^2 \left (e^{2 x^6}-2 e^{x^6} x+11 x^2-5 e^2 x^2\right )^2}{\left (e^{x^6}-x\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*E^(5*x^6)*x - 10*E^(4*x^6)*x^2 - 242*x^6 + 220*E^2*x^6 - 50*E^4*x^6 + E^(3*x^6)*(100*x^3 - 240*x^
9 + E^2*(-40*x^3 + 120*x^9)) + E^(2*x^6)*(-220*x^4 + 480*x^10 + E^2*(100*x^4 - 240*x^10)) + E^x^6*(770*x^5 - 2
640*x^11 + E^4*(150*x^5 - 600*x^11) + E^2*(-680*x^5 + 2520*x^11)))/(E^(5*x^6) - 5*E^(4*x^6)*x + 10*E^(3*x^6)*x
^2 - 10*E^(2*x^6)*x^3 + 5*E^x^6*x^4 - x^5),x]

[Out]

(x^2*(E^(2*x^6) - 2*E^x^6*x + 11*x^2 - 5*E^2*x^2)^2)/(E^x^6 - x)^4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(25)=50\).
time = 0.30, size = 84, normalized size = 3.11

method	result	size
risch	\(x^{2}+\frac {5 x^{4} \left (5 x^{2} {\mathrm e}^{4}-22 x^{2} {\mathrm e}^{2}+4 x \,{\mathrm e}^{x^{6}+2}-2 \,{\mathrm e}^{2 \left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}+24 x^{2}-8 x \,{\mathrm e}^{x^{6}}+4 \,{\mathrm e}^{2 x^{6}}\right )}{\left (x -{\mathrm e}^{x^{6}}\right )^{4}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(x^6)^5-10*x^2*exp(x^6)^4+((120*x^9-40*x^3)*exp(2)-240*x^9+100*x^3)*exp(x^6)^3+((-240*x^10+100*x^4
)*exp(2)+480*x^10-220*x^4)*exp(x^6)^2+((-600*x^11+150*x^5)*exp(2)^2+(2520*x^11-680*x^5)*exp(2)-2640*x^11+770*x
^5)*exp(x^6)-50*x^6*exp(2)^2+220*x^6*exp(2)-242*x^6)/(exp(x^6)^5-5*x*exp(x^6)^4+10*x^2*exp(x^6)^3-10*x^3*exp(x
^6)^2+5*x^4*exp(x^6)-x^5),x,method=_RETURNVERBOSE)

[Out]

x^2+5*x^4*(5*x^2*exp(4)-22*x^2*exp(2)+4*x*exp(x^6+2)-2*exp(2*(x^2+1)*(x^4-x^2+1))+24*x^2-8*x*exp(x^6)+4*exp(2*
x^6))/(x-exp(x^6))^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (25) = 50\).
time = 0.35, size = 110, normalized size = 4.07 \begin {gather*} \frac {x^{6} {\left (25 \, e^{4} - 110 \, e^{2} + 121\right )} + 4 \, x^{5} {\left (5 \, e^{2} - 11\right )} e^{\left (x^{6}\right )} - 2 \, x^{4} {\left (5 \, e^{2} - 13\right )} e^{\left (2 \, x^{6}\right )} - 4 \, x^{3} e^{\left (3 \, x^{6}\right )} + x^{2} e^{\left (4 \, x^{6}\right )}}{x^{4} - 4 \, x^{3} e^{\left (x^{6}\right )} + 6 \, x^{2} e^{\left (2 \, x^{6}\right )} - 4 \, x e^{\left (3 \, x^{6}\right )} + e^{\left (4 \, x^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x^6)^5-10*x^2*exp(x^6)^4+((120*x^9-40*x^3)*exp(2)-240*x^9+100*x^3)*exp(x^6)^3+((-240*x^10+1
00*x^4)*exp(2)+480*x^10-220*x^4)*exp(x^6)^2+((-600*x^11+150*x^5)*exp(2)^2+(2520*x^11-680*x^5)*exp(2)-2640*x^11
+770*x^5)*exp(x^6)-50*x^6*exp(2)^2+220*x^6*exp(2)-242*x^6)/(exp(x^6)^5-5*x*exp(x^6)^4+10*x^2*exp(x^6)^3-10*x^3
*exp(x^6)^2+5*x^4*exp(x^6)-x^5),x, algorithm="maxima")

[Out]

(x^6*(25*e^4 - 110*e^2 + 121) + 4*x^5*(5*e^2 - 11)*e^(x^6) - 2*x^4*(5*e^2 - 13)*e^(2*x^6) - 4*x^3*e^(3*x^6) +
x^2*e^(4*x^6))/(x^4 - 4*x^3*e^(x^6) + 6*x^2*e^(2*x^6) - 4*x*e^(3*x^6) + e^(4*x^6))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (25) = 50\).
time = 0.38, size = 123, normalized size = 4.56 \begin {gather*} \frac {25 \, x^{6} e^{4} - 110 \, x^{6} e^{2} + 121 \, x^{6} - 4 \, x^{3} e^{\left (3 \, x^{6}\right )} + x^{2} e^{\left (4 \, x^{6}\right )} - 2 \, {\left (5 \, x^{4} e^{2} - 13 \, x^{4}\right )} e^{\left (2 \, x^{6}\right )} + 4 \, {\left (5 \, x^{5} e^{2} - 11 \, x^{5}\right )} e^{\left (x^{6}\right )}}{x^{4} - 4 \, x^{3} e^{\left (x^{6}\right )} + 6 \, x^{2} e^{\left (2 \, x^{6}\right )} - 4 \, x e^{\left (3 \, x^{6}\right )} + e^{\left (4 \, x^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x^6)^5-10*x^2*exp(x^6)^4+((120*x^9-40*x^3)*exp(2)-240*x^9+100*x^3)*exp(x^6)^3+((-240*x^10+1
00*x^4)*exp(2)+480*x^10-220*x^4)*exp(x^6)^2+((-600*x^11+150*x^5)*exp(2)^2+(2520*x^11-680*x^5)*exp(2)-2640*x^11
+770*x^5)*exp(x^6)-50*x^6*exp(2)^2+220*x^6*exp(2)-242*x^6)/(exp(x^6)^5-5*x*exp(x^6)^4+10*x^2*exp(x^6)^3-10*x^3
*exp(x^6)^2+5*x^4*exp(x^6)-x^5),x, algorithm="fricas")

[Out]

(25*x^6*e^4 - 110*x^6*e^2 + 121*x^6 - 4*x^3*e^(3*x^6) + x^2*e^(4*x^6) - 2*(5*x^4*e^2 - 13*x^4)*e^(2*x^6) + 4*(
5*x^5*e^2 - 11*x^5)*e^(x^6))/(x^4 - 4*x^3*e^(x^6) + 6*x^2*e^(2*x^6) - 4*x*e^(3*x^6) + e^(4*x^6))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (20) = 40\).
time = 0.16, size = 105, normalized size = 3.89 \begin {gather*} x^{2} + \frac {- 110 x^{6} e^{2} + 120 x^{6} + 25 x^{6} e^{4} + \left (- 40 x^{5} + 20 x^{5} e^{2}\right ) e^{x^{6}} + \left (- 10 x^{4} e^{2} + 20 x^{4}\right ) e^{2 x^{6}}}{x^{4} - 4 x^{3} e^{x^{6}} + 6 x^{2} e^{2 x^{6}} - 4 x e^{3 x^{6}} + e^{4 x^{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x**6)**5-10*x**2*exp(x**6)**4+((120*x**9-40*x**3)*exp(2)-240*x**9+100*x**3)*exp(x**6)**3+((
-240*x**10+100*x**4)*exp(2)+480*x**10-220*x**4)*exp(x**6)**2+((-600*x**11+150*x**5)*exp(2)**2+(2520*x**11-680*
x**5)*exp(2)-2640*x**11+770*x**5)*exp(x**6)-50*x**6*exp(2)**2+220*x**6*exp(2)-242*x**6)/(exp(x**6)**5-5*x*exp(
x**6)**4+10*x**2*exp(x**6)**3-10*x**3*exp(x**6)**2+5*x**4*exp(x**6)-x**5),x)

[Out]

x**2 + (-110*x**6*exp(2) + 120*x**6 + 25*x**6*exp(4) + (-40*x**5 + 20*x**5*exp(2))*exp(x**6) + (-10*x**4*exp(2
) + 20*x**4)*exp(2*x**6))/(x**4 - 4*x**3*exp(x**6) + 6*x**2*exp(2*x**6) - 4*x*exp(3*x**6) + exp(4*x**6))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (25) = 50\).
time = 0.47, size = 127, normalized size = 4.70 \begin {gather*} \frac {25 \, x^{6} e^{4} - 110 \, x^{6} e^{2} + 121 \, x^{6} + 20 \, x^{5} e^{\left (x^{6} + 2\right )} - 44 \, x^{5} e^{\left (x^{6}\right )} + 26 \, x^{4} e^{\left (2 \, x^{6}\right )} - 10 \, x^{4} e^{\left (2 \, x^{6} + 2\right )} - 4 \, x^{3} e^{\left (3 \, x^{6}\right )} + x^{2} e^{\left (4 \, x^{6}\right )}}{x^{4} - 4 \, x^{3} e^{\left (x^{6}\right )} + 6 \, x^{2} e^{\left (2 \, x^{6}\right )} - 4 \, x e^{\left (3 \, x^{6}\right )} + e^{\left (4 \, x^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x^6)^5-10*x^2*exp(x^6)^4+((120*x^9-40*x^3)*exp(2)-240*x^9+100*x^3)*exp(x^6)^3+((-240*x^10+1
00*x^4)*exp(2)+480*x^10-220*x^4)*exp(x^6)^2+((-600*x^11+150*x^5)*exp(2)^2+(2520*x^11-680*x^5)*exp(2)-2640*x^11
+770*x^5)*exp(x^6)-50*x^6*exp(2)^2+220*x^6*exp(2)-242*x^6)/(exp(x^6)^5-5*x*exp(x^6)^4+10*x^2*exp(x^6)^3-10*x^3
*exp(x^6)^2+5*x^4*exp(x^6)-x^5),x, algorithm="giac")

[Out]

(25*x^6*e^4 - 110*x^6*e^2 + 121*x^6 + 20*x^5*e^(x^6 + 2) - 44*x^5*e^(x^6) + 26*x^4*e^(2*x^6) - 10*x^4*e^(2*x^6
+ 2) - 4*x^3*e^(3*x^6) + x^2*e^(4*x^6))/(x^4 - 4*x^3*e^(x^6) + 6*x^2*e^(2*x^6) - 4*x*e^(3*x^6) + e^(4*x^6))

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Mupad [B]
time = 5.72, size = 42, normalized size = 1.56 \begin {gather*} \frac {x^2\,{\left ({\mathrm {e}}^{2\,x^6}-2\,x\,{\mathrm {e}}^{x^6}-5\,x^2\,{\mathrm {e}}^2+11\,x^2\right )}^2}{{\left (x-{\mathrm {e}}^{x^6}\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3*x^6)*(exp(2)*(40*x^3 - 120*x^9) - 100*x^3 + 240*x^9) - exp(2*x^6)*(exp(2)*(100*x^4 - 240*x^10) - 2
20*x^4 + 480*x^10) - 2*x*exp(5*x^6) - exp(x^6)*(exp(4)*(150*x^5 - 600*x^11) - exp(2)*(680*x^5 - 2520*x^11) + 7
70*x^5 - 2640*x^11) - 220*x^6*exp(2) + 50*x^6*exp(4) + 10*x^2*exp(4*x^6) + 242*x^6)/(exp(5*x^6) - 5*x*exp(4*x^
6) + 5*x^4*exp(x^6) + 10*x^2*exp(3*x^6) - 10*x^3*exp(2*x^6) - x^5),x)

[Out]

(x^2*(exp(2*x^6) - 2*x*exp(x^6) - 5*x^2*exp(2) + 11*x^2)^2)/(x - exp(x^6))^4

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