3.16.25 \(\int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} (72-864 x+432 x^2)+e^x (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6)}{1+18 x^2+108 x^4+216 x^6} \, dx\)

Optimal. Leaf size=28 \[ x^2 \left (-2+x+\frac {x+\frac {e^x}{\frac {1}{6}+x^2}}{x}\right )^2 \]

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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 {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x - 6*x^2 + 40*x^3 - 108*x^4 + 288*x^5 - 648*x^6 + 864*x^7 - 1296*x^8 + 864*x^9 + E^(2*x)*(72 - 864*x +
432*x^2) + E^x*(-12 + 12*x + 12*x^2 + 576*x^4 - 432*x^5 + 432*x^6))/(1 + 18*x^2 + 108*x^4 + 216*x^6),x]

[Out]

2*E^x + (6*E^x)/(I*Sqrt[6] - 6*x) + (1 - x)^2*x^2 - (Sqrt[6]*E^x)/(I + Sqrt[6]*x) + (36*E^(2*x))/(1 + 6*x^2)^2
+ E^(I/Sqrt[6])*ExpIntegralEi[-((I - Sqrt[6]*x)/Sqrt[6])] + I*Sqrt[6]*E^(I/Sqrt[6])*ExpIntegralEi[-((I - Sqrt
[6]*x)/Sqrt[6])] - ((6 + (5*I)*Sqrt[6])*E^(I/Sqrt[6])*ExpIntegralEi[-((I - Sqrt[6]*x)/Sqrt[6])])/6 + ExpIntegr
alEi[(I + Sqrt[6]*x)/Sqrt[6]]/E^(I/Sqrt[6]) - (I*Sqrt[6]*ExpIntegralEi[(I + Sqrt[6]*x)/Sqrt[6]])/E^(I/Sqrt[6])
- ((6 - (5*I)*Sqrt[6])*ExpIntegralEi[(I + Sqrt[6]*x)/Sqrt[6]])/(6*E^(I/Sqrt[6])) + 24*Defer[Int][(E^x*x)/(1 +
6*x^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (x \left (1-3 x+2 x^2\right ) \left (1+6 x^2\right )^3+36 e^{2 x} \left (1-12 x+6 x^2\right )+6 e^x \left (-1+x+x^2+48 x^4-36 x^5+36 x^6\right )\right )}{\left (1+6 x^2\right )^3} \, dx\\ &=2 \int \frac {x \left (1-3 x+2 x^2\right ) \left (1+6 x^2\right )^3+36 e^{2 x} \left (1-12 x+6 x^2\right )+6 e^x \left (-1+x+x^2+48 x^4-36 x^5+36 x^6\right )}{\left (1+6 x^2\right )^3} \, dx\\ &=2 \int \left ((-1+x) x (-1+2 x)+\frac {36 e^{2 x} \left (1-12 x+6 x^2\right )}{\left (1+6 x^2\right )^3}+\frac {6 e^x \left (-1+x+7 x^2-6 x^3+6 x^4\right )}{\left (1+6 x^2\right )^2}\right ) \, dx\\ &=2 \int (-1+x) x (-1+2 x) \, dx+12 \int \frac {e^x \left (-1+x+7 x^2-6 x^3+6 x^4\right )}{\left (1+6 x^2\right )^2} \, dx+72 \int \frac {e^{2 x} \left (1-12 x+6 x^2\right )}{\left (1+6 x^2\right )^3} \, dx\\ &=(1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}+12 \int \left (\frac {e^x}{6}+\frac {2 e^x (-1+x)}{\left (1+6 x^2\right )^2}+\frac {e^x (5-6 x)}{6 \left (1+6 x^2\right )}\right ) \, dx\\ &=(1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}+2 \int e^x \, dx+2 \int \frac {e^x (5-6 x)}{1+6 x^2} \, dx+24 \int \frac {e^x (-1+x)}{\left (1+6 x^2\right )^2} \, dx\\ &=2 e^x+(1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}+2 \int \left (\frac {\left (5 i+\sqrt {6}\right ) e^x}{2 \left (i-\sqrt {6} x\right )}+\frac {\left (5 i-\sqrt {6}\right ) e^x}{2 \left (i+\sqrt {6} x\right )}\right ) \, dx+24 \int \left (-\frac {e^x}{\left (1+6 x^2\right )^2}+\frac {e^x x}{\left (1+6 x^2\right )^2}\right ) \, dx\\ &=2 e^x+(1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}-24 \int \frac {e^x}{\left (1+6 x^2\right )^2} \, dx+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx+\left (5 i-\sqrt {6}\right ) \int \frac {e^x}{i+\sqrt {6} x} \, dx+\left (5 i+\sqrt {6}\right ) \int \frac {e^x}{i-\sqrt {6} x} \, dx\\ &=2 e^x+(1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \text {Ei}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \text {Ei}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx-24 \int \left (-\frac {3 e^x}{2 \left (i \sqrt {6}-6 x\right )^2}-\frac {3 e^x}{2 \left (i \sqrt {6}+6 x\right )^2}-\frac {3 e^x}{-6-36 x^2}\right ) \, dx\\ &=2 e^x+(1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \text {Ei}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \text {Ei}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx+36 \int \frac {e^x}{\left (i \sqrt {6}-6 x\right )^2} \, dx+36 \int \frac {e^x}{\left (i \sqrt {6}+6 x\right )^2} \, dx+72 \int \frac {e^x}{-6-36 x^2} \, dx\\ &=2 e^x+\frac {6 e^x}{i \sqrt {6}-6 x}+(1-x)^2 x^2-\frac {\sqrt {6} e^x}{i+\sqrt {6} x}+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \text {Ei}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \text {Ei}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )-6 \int \frac {e^x}{i \sqrt {6}-6 x} \, dx+6 \int \frac {e^x}{i \sqrt {6}+6 x} \, dx+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx+72 \int \left (-\frac {i e^x}{12 \left (i-\sqrt {6} x\right )}-\frac {i e^x}{12 \left (i+\sqrt {6} x\right )}\right ) \, dx\\ &=2 e^x+\frac {6 e^x}{i \sqrt {6}-6 x}+(1-x)^2 x^2-\frac {\sqrt {6} e^x}{i+\sqrt {6} x}+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}+e^{\frac {i}{\sqrt {6}}} \text {Ei}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \text {Ei}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )+e^{-\frac {i}{\sqrt {6}}} \text {Ei}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \text {Ei}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )-6 i \int \frac {e^x}{i-\sqrt {6} x} \, dx-6 i \int \frac {e^x}{i+\sqrt {6} x} \, dx+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx\\ &=2 e^x+\frac {6 e^x}{i \sqrt {6}-6 x}+(1-x)^2 x^2-\frac {\sqrt {6} e^x}{i+\sqrt {6} x}+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}+e^{\frac {i}{\sqrt {6}}} \text {Ei}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )+i \sqrt {6} e^{\frac {i}{\sqrt {6}}} \text {Ei}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \text {Ei}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )+e^{-\frac {i}{\sqrt {6}}} \text {Ei}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )-i \sqrt {6} e^{-\frac {i}{\sqrt {6}}} \text {Ei}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \text {Ei}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.38, size = 33, normalized size = 1.18 \begin {gather*} \frac {\left (6 e^x+x \left (-1+x-6 x^2+6 x^3\right )\right )^2}{\left (1+6 x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x - 6*x^2 + 40*x^3 - 108*x^4 + 288*x^5 - 648*x^6 + 864*x^7 - 1296*x^8 + 864*x^9 + E^(2*x)*(72 - 8
64*x + 432*x^2) + E^x*(-12 + 12*x + 12*x^2 + 576*x^4 - 432*x^5 + 432*x^6))/(1 + 18*x^2 + 108*x^4 + 216*x^6),x]

[Out]

(6*E^x + x*(-1 + x - 6*x^2 + 6*x^3))^2/(1 + 6*x^2)^2

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fricas [B]  time = 0.60, size = 76, normalized size = 2.71 \begin {gather*} \frac {36 \, x^{8} - 72 \, x^{7} + 48 \, x^{6} - 24 \, x^{5} + 13 \, x^{4} - 2 \, x^{3} + x^{2} + 12 \, {\left (6 \, x^{4} - 6 \, x^{3} + x^{2} - x\right )} e^{x} + 36 \, e^{\left (2 \, x\right )}}{36 \, x^{4} + 12 \, x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((432*x^2-864*x+72)*exp(x)^2+(432*x^6-432*x^5+576*x^4+12*x^2+12*x-12)*exp(x)+864*x^9-1296*x^8+864*x^
7-648*x^6+288*x^5-108*x^4+40*x^3-6*x^2+2*x)/(216*x^6+108*x^4+18*x^2+1),x, algorithm="fricas")

[Out]

(36*x^8 - 72*x^7 + 48*x^6 - 24*x^5 + 13*x^4 - 2*x^3 + x^2 + 12*(6*x^4 - 6*x^3 + x^2 - x)*e^x + 36*e^(2*x))/(36
*x^4 + 12*x^2 + 1)

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giac [B]  time = 0.31, size = 81, normalized size = 2.89 \begin {gather*} \frac {36 \, x^{8} - 72 \, x^{7} + 48 \, x^{6} - 24 \, x^{5} + 72 \, x^{4} e^{x} + 13 \, x^{4} - 72 \, x^{3} e^{x} - 2 \, x^{3} + 12 \, x^{2} e^{x} + x^{2} - 12 \, x e^{x} + 36 \, e^{\left (2 \, x\right )}}{36 \, x^{4} + 12 \, x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((432*x^2-864*x+72)*exp(x)^2+(432*x^6-432*x^5+576*x^4+12*x^2+12*x-12)*exp(x)+864*x^9-1296*x^8+864*x^
7-648*x^6+288*x^5-108*x^4+40*x^3-6*x^2+2*x)/(216*x^6+108*x^4+18*x^2+1),x, algorithm="giac")

[Out]

(36*x^8 - 72*x^7 + 48*x^6 - 24*x^5 + 72*x^4*e^x + 13*x^4 - 72*x^3*e^x - 2*x^3 + 12*x^2*e^x + x^2 - 12*x*e^x +
36*e^(2*x))/(36*x^4 + 12*x^2 + 1)

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maple [A]  time = 0.17, size = 45, normalized size = 1.61

method	result	size
risch	\(x^{4}-2 x^{3}+x^{2}+\frac {36 \,{\mathrm e}^{2 x}}{\left (6 x^{2}+1\right )^{2}}+\frac {12 x \left (x -1\right ) {\mathrm e}^{x}}{6 x^{2}+1}\)	\(45\)
norman	\(\frac {x^{2}+13 x^{4}-2 x^{3}-24 x^{5}+48 x^{6}-72 x^{7}+36 x^{8}+36 \,{\mathrm e}^{2 x}-12 \,{\mathrm e}^{x} x +12 \,{\mathrm e}^{x} x^{2}-72 \,{\mathrm e}^{x} x^{3}+72 \,{\mathrm e}^{x} x^{4}}{\left (6 x^{2}+1\right )^{2}}\)	\(77\)
default	\(x^{4}-2 x^{3}+x^{2}+2 \,{\mathrm e}^{x}-\frac {3888 \left (-\frac {5}{1728} x^{3}-\frac {1}{3456} x \right )}{\left (6 x^{2}+1\right )^{2}}+\frac {{\mathrm e}^{x} \left (36 x^{3}-6 x^{2}-6 x -1\right )}{864 x^{4}+288 x^{2}+24}-\frac {36 \,{\mathrm e}^{2 x} \left (6 x^{3}+x -1\right )}{36 x^{4}+12 x^{2}+1}-\frac {216 \left (\frac {1}{288} x^{3}-\frac {1}{1728} x \right )}{\left (6 x^{2}+1\right )^{2}}+\frac {-\frac {81}{4} x^{3}-\frac {21}{8} x}{\left (6 x^{2}+1\right )^{2}}+\frac {{\mathrm e}^{x} \left (6 x^{3}+x -2\right )}{144 x^{4}+48 x^{2}+4}-\frac {108 \left (-\frac {13}{144} x^{3}-\frac {11}{864} x \right )}{\left (6 x^{2}+1\right )^{2}}-\frac {{\mathrm e}^{x} \left (6 x^{3}+48 x^{2}+x +6\right )}{4 \left (36 x^{4}+12 x^{2}+1\right )}+\frac {{\mathrm e}^{x} \left (324 x^{3}-6 x^{2}+42 x -1\right )}{864 x^{4}+288 x^{2}+24}-\frac {{\mathrm e}^{x} \left (108 x^{3}+6 x^{2}+30 x +1\right )}{4 \left (36 x^{4}+12 x^{2}+1\right )}+\frac {3 \,{\mathrm e}^{2 x} \left (18 x^{3}-6 x^{2}-3 x -1\right )}{36 x^{4}+12 x^{2}+1}-\frac {{\mathrm e}^{x} \left (180 x^{3}-6 x^{2}+18 x -1\right )}{3 \left (36 x^{4}+12 x^{2}+1\right )}+\frac {3 \,{\mathrm e}^{2 x} \left (54 x^{3}+6 x^{2}+15 x +1\right )}{36 x^{4}+12 x^{2}+1}\)	\(384\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((432*x^2-864*x+72)*exp(x)^2+(432*x^6-432*x^5+576*x^4+12*x^2+12*x-12)*exp(x)+864*x^9-1296*x^8+864*x^7-648*
x^6+288*x^5-108*x^4+40*x^3-6*x^2+2*x)/(216*x^6+108*x^4+18*x^2+1),x,method=_RETURNVERBOSE)

[Out]

x^4-2*x^3+x^2+36/(6*x^2+1)^2*exp(2*x)+12*x*(x-1)/(6*x^2+1)*exp(x)

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maxima [B]  time = 0.59, size = 260, normalized size = 9.29 \begin {gather*} x^{4} - 2 \, x^{3} + x^{2} + \frac {78 \, x^{3} + 11 \, x}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {3 \, {\left (54 \, x^{3} + 7 \, x\right )}}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} + \frac {9 \, {\left (10 \, x^{3} + x\right )}}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {6 \, x^{3} - x}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} + \frac {48 \, x^{2} + 7}{36 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {36 \, x^{2} + 5}{6 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {5 \, {\left (12 \, x^{2} + 1\right )}}{18 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} + \frac {8 \, x^{2} + 1}{36 \, x^{4} + 12 \, x^{2} + 1} + \frac {12 \, {\left ({\left (6 \, x^{4} - 6 \, x^{3} + x^{2} - x\right )} e^{x} + 3 \, e^{\left (2 \, x\right )}\right )}}{36 \, x^{4} + 12 \, x^{2} + 1} - \frac {1}{12 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((432*x^2-864*x+72)*exp(x)^2+(432*x^6-432*x^5+576*x^4+12*x^2+12*x-12)*exp(x)+864*x^9-1296*x^8+864*x^
7-648*x^6+288*x^5-108*x^4+40*x^3-6*x^2+2*x)/(216*x^6+108*x^4+18*x^2+1),x, algorithm="maxima")

[Out]

x^4 - 2*x^3 + x^2 + 1/8*(78*x^3 + 11*x)/(36*x^4 + 12*x^2 + 1) - 3/8*(54*x^3 + 7*x)/(36*x^4 + 12*x^2 + 1) + 9/8
*(10*x^3 + x)/(36*x^4 + 12*x^2 + 1) - 1/8*(6*x^3 - x)/(36*x^4 + 12*x^2 + 1) + 1/36*(48*x^2 + 7)/(36*x^4 + 12*x
^2 + 1) - 1/6*(36*x^2 + 5)/(36*x^4 + 12*x^2 + 1) - 5/18*(12*x^2 + 1)/(36*x^4 + 12*x^2 + 1) + (8*x^2 + 1)/(36*x
^4 + 12*x^2 + 1) + 12*((6*x^4 - 6*x^3 + x^2 - x)*e^x + 3*e^(2*x))/(36*x^4 + 12*x^2 + 1) - 1/12/(36*x^4 + 12*x^
2 + 1)

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mupad [B]  time = 1.19, size = 49, normalized size = 1.75 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{x^4+\frac {x^2}{3}+\frac {1}{36}}+x^2-2\,x^3+x^4-\frac {{\mathrm {e}}^x\,\left (2\,x-2\,x^2\right )}{x^2+\frac {1}{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + exp(2*x)*(432*x^2 - 864*x + 72) + exp(x)*(12*x + 12*x^2 + 576*x^4 - 432*x^5 + 432*x^6 - 12) - 6*x^2
+ 40*x^3 - 108*x^4 + 288*x^5 - 648*x^6 + 864*x^7 - 1296*x^8 + 864*x^9)/(18*x^2 + 108*x^4 + 216*x^6 + 1),x)

[Out]

exp(2*x)/(x^2/3 + x^4 + 1/36) + x^2 - 2*x^3 + x^4 - (exp(x)*(2*x - 2*x^2))/(x^2 + 1/6)

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sympy [B]  time = 0.17, size = 71, normalized size = 2.54 \begin {gather*} x^{4} - 2 x^{3} + x^{2} + \frac {\left (216 x^{2} + 36\right ) e^{2 x} + \left (432 x^{6} - 432 x^{5} + 144 x^{4} - 144 x^{3} + 12 x^{2} - 12 x\right ) e^{x}}{216 x^{6} + 108 x^{4} + 18 x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((432*x**2-864*x+72)*exp(x)**2+(432*x**6-432*x**5+576*x**4+12*x**2+12*x-12)*exp(x)+864*x**9-1296*x**
8+864*x**7-648*x**6+288*x**5-108*x**4+40*x**3-6*x**2+2*x)/(216*x**6+108*x**4+18*x**2+1),x)

[Out]

x**4 - 2*x**3 + x**2 + ((216*x**2 + 36)*exp(2*x) + (432*x**6 - 432*x**5 + 144*x**4 - 144*x**3 + 12*x**2 - 12*x
)*exp(x))/(216*x**6 + 108*x**4 + 18*x**2 + 1)

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