3.94.53 \(\int \frac {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 (9+6 x+x^2))}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 (27+18 x+3 x^2)+e^4 (-54 x+180 x^2+138 x^3+24 x^4)+e^x (-18 x+66 x^2+24 x^3+e^4 (18+6 x))} \, dx\)

Optimal. Leaf size=33 \[ \frac {2+\frac {e^x}{3}}{e^4-x+4 x^2+\frac {e^x}{3+x}} \]

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Rubi [F]  time = 2.73, 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 {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 \left (9+6 x+x^2\right )\right )}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 \left (27+18 x+3 x^2\right )+e^4 \left (-54 x+180 x^2+138 x^3+24 x^4\right )+e^x \left (-18 x+66 x^2+24 x^3+e^4 (18+6 x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(54 + E^(2*x) - 396*x - 282*x^2 - 48*x^3 + E^x*(-3 - 81*x - 17*x^2 + 15*x^3 + 4*x^4 + E^4*(9 + 6*x + x^2))
)/(3*E^(2*x) + 27*x^2 - 198*x^3 + 291*x^4 + 264*x^5 + 48*x^6 + E^8*(27 + 18*x + 3*x^2) + E^4*(-54*x + 180*x^2
+ 138*x^3 + 24*x^4) + E^x*(-18*x + 66*x^2 + 24*x^3 + E^4*(18 + 6*x))),x]

[Out]

x/3 + 3*(6 + E^4 - 2*E^8)*Defer[Int][(3*E^4 + E^x - 3*(1 - E^4/3)*x + 11*x^2 + 4*x^3)^(-2), x] - (135 - 85*E^4
+ 7*E^8)*Defer[Int][x/(3*E^4 + E^x - 3*(1 - E^4/3)*x + 11*x^2 + 4*x^3)^2, x] - ((483 - 111*E^4 + 8*E^8)*Defer
[Int][x^2/(3*E^4 + E^x - 3*(1 - E^4/3)*x + 11*x^2 + 4*x^3)^2, x])/3 + ((738 - 81*E^4 - E^8)*Defer[Int][x^3/(3*
E^4 + E^x - 3*(1 - E^4/3)*x + 11*x^2 + 4*x^3)^2, x])/3 + ((653 - 54*E^4)*Defer[Int][x^4/(3*E^4 + E^x - 3*(1 -
E^4/3)*x + 11*x^2 + 4*x^3)^2, x])/3 + ((3 - 8*E^4)*Defer[Int][x^5/(3*E^4 + E^x - 3*(1 - E^4/3)*x + 11*x^2 + 4*
x^3)^2, x])/3 - (88*Defer[Int][x^6/(3*E^4 + E^x - 3*(1 - E^4/3)*x + 11*x^2 + 4*x^3)^2, x])/3 - (16*Defer[Int][
x^7/(3*E^4 + E^x - 3*(1 - E^4/3)*x + 11*x^2 + 4*x^3)^2, x])/3 - (1 - E^4)*Defer[Int][(3*E^4 + E^x - 3*(1 - E^4
/3)*x + 11*x^2 + 4*x^3)^(-1), x] - ((75 - 4*E^4)*Defer[Int][x/(3*E^4 + E^x - 3*(1 - E^4/3)*x + 11*x^2 + 4*x^3)
, x])/3 - ((39 - E^4)*Defer[Int][x^2/(3*E^4 + E^x - 3*(1 - E^4/3)*x + 11*x^2 + 4*x^3), x])/3 + (7*Defer[Int][x
^3/(3*E^4 + E^x - 3*(1 - E^4/3)*x + 11*x^2 + 4*x^3), x])/3 + (4*Defer[Int][x^4/(3*E^4 + E^x - 3*(1 - E^4/3)*x
+ 11*x^2 + 4*x^3), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x}+e^{4+x} (3+x)^2-6 (3+x)^2 (-1+8 x)+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4\right )}{3 \left (e^x+e^4 (3+x)+x \left (-3+11 x+4 x^2\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{2 x}+e^{4+x} (3+x)^2-6 (3+x)^2 (-1+8 x)+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4\right )}{\left (e^x+e^4 (3+x)+x \left (-3+11 x+4 x^2\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \left (1+\frac {-3 \left (1-e^4\right )-\left (75-4 e^4\right ) x-\left (39-e^4\right ) x^2+7 x^3+4 x^4}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3}+\frac {54 \left (1+\frac {1}{6} \left (e^4-2 e^8\right )\right )-405 \left (1+\frac {1}{135} e^4 \left (-85+7 e^4\right )\right ) x-483 \left (1+\frac {1}{483} e^4 \left (-111+8 e^4\right )\right ) x^2+738 \left (1-\frac {1}{738} e^4 \left (81+e^4\right )\right ) x^3+653 \left (1-\frac {54 e^4}{653}\right ) x^4+3 \left (1-\frac {8 e^4}{3}\right ) x^5-88 x^6-16 x^7}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2}\right ) \, dx\\ &=\frac {x}{3}+\frac {1}{3} \int \frac {-3 \left (1-e^4\right )-\left (75-4 e^4\right ) x-\left (39-e^4\right ) x^2+7 x^3+4 x^4}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx+\frac {1}{3} \int \frac {54 \left (1+\frac {1}{6} \left (e^4-2 e^8\right )\right )-405 \left (1+\frac {1}{135} e^4 \left (-85+7 e^4\right )\right ) x-483 \left (1+\frac {1}{483} e^4 \left (-111+8 e^4\right )\right ) x^2+738 \left (1-\frac {1}{738} e^4 \left (81+e^4\right )\right ) x^3+653 \left (1-\frac {54 e^4}{653}\right ) x^4+3 \left (1-\frac {8 e^4}{3}\right ) x^5-88 x^6-16 x^7}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2} \, dx\\ &=\frac {x}{3}+\frac {1}{3} \int \left (\frac {3 \left (-1+e^4\right )}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3}+\frac {\left (-75+4 e^4\right ) x}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3}+\frac {\left (-39+e^4\right ) x^2}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3}+\frac {7 x^3}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3}+\frac {4 x^4}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3}\right ) \, dx+\frac {1}{3} \int \frac {54-405 x-483 x^2+738 x^3+653 x^4+3 x^5-88 x^6-16 x^7-e^8 (2+x) (3+x)^2+e^4 \left (9+255 x+111 x^2-81 x^3-54 x^4-8 x^5\right )}{\left (e^x+e^4 (3+x)+x \left (-3+11 x+4 x^2\right )\right )^2} \, dx\\ &=\frac {x}{3}+\frac {1}{3} \int \left (\frac {9 \left (6+e^4-2 e^8\right )}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2}+\frac {3 \left (-135+85 e^4-7 e^8\right ) x}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2}+\frac {\left (-483+111 e^4-8 e^8\right ) x^2}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2}+\frac {\left (738-81 e^4-e^8\right ) x^3}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2}+\frac {\left (653-54 e^4\right ) x^4}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2}+\frac {\left (3-8 e^4\right ) x^5}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2}-\frac {88 x^6}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2}-\frac {16 x^7}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2}\right ) \, dx+\frac {4}{3} \int \frac {x^4}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx+\frac {7}{3} \int \frac {x^3}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx+\frac {1}{3} \left (-39+e^4\right ) \int \frac {x^2}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx+\left (-1+e^4\right ) \int \frac {1}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx+\frac {1}{3} \left (-75+4 e^4\right ) \int \frac {x}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx\\ &=\frac {x}{3}+\frac {4}{3} \int \frac {x^4}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx+\frac {7}{3} \int \frac {x^3}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx-\frac {16}{3} \int \frac {x^7}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2} \, dx-\frac {88}{3} \int \frac {x^6}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2} \, dx+\frac {1}{3} \left (653-54 e^4\right ) \int \frac {x^4}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2} \, dx+\frac {1}{3} \left (3-8 e^4\right ) \int \frac {x^5}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2} \, dx+\frac {1}{3} \left (-39+e^4\right ) \int \frac {x^2}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx+\left (-1+e^4\right ) \int \frac {1}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx+\frac {1}{3} \left (-75+4 e^4\right ) \int \frac {x}{3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3} \, dx+\frac {1}{3} \left (-483+111 e^4-8 e^8\right ) \int \frac {x^2}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2} \, dx+\left (-135+85 e^4-7 e^8\right ) \int \frac {x}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2} \, dx+\left (3 \left (6+e^4-2 e^8\right )\right ) \int \frac {1}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2} \, dx+\frac {1}{3} \left (738-81 e^4-e^8\right ) \int \frac {x^3}{\left (3 e^4+e^x-3 \left (1-\frac {e^4}{3}\right ) x+11 x^2+4 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.94, size = 60, normalized size = 1.82 \begin {gather*} \frac {e^x x-3 e^4 (3+x)+3 \left (6+5 x-11 x^2-4 x^3\right )}{3 \left (e^x+e^4 (3+x)+x \left (-3+11 x+4 x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(54 + E^(2*x) - 396*x - 282*x^2 - 48*x^3 + E^x*(-3 - 81*x - 17*x^2 + 15*x^3 + 4*x^4 + E^4*(9 + 6*x +
x^2)))/(3*E^(2*x) + 27*x^2 - 198*x^3 + 291*x^4 + 264*x^5 + 48*x^6 + E^8*(27 + 18*x + 3*x^2) + E^4*(-54*x + 18
0*x^2 + 138*x^3 + 24*x^4) + E^x*(-18*x + 66*x^2 + 24*x^3 + E^4*(18 + 6*x))),x]

[Out]

(E^x*x - 3*E^4*(3 + x) + 3*(6 + 5*x - 11*x^2 - 4*x^3))/(3*(E^x + E^4*(3 + x) + x*(-3 + 11*x + 4*x^2)))

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fricas [A]  time = 0.86, size = 53, normalized size = 1.61 \begin {gather*} -\frac {12 \, x^{3} + 33 \, x^{2} + 3 \, {\left (x + 3\right )} e^{4} - x e^{x} - 15 \, x - 18}{3 \, {\left (4 \, x^{3} + 11 \, x^{2} + {\left (x + 3\right )} e^{4} - 3 \, x + e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+((x^2+6*x+9)*exp(4)+4*x^4+15*x^3-17*x^2-81*x-3)*exp(x)-48*x^3-282*x^2-396*x+54)/(3*exp(x)^
2+((18+6*x)*exp(4)+24*x^3+66*x^2-18*x)*exp(x)+(3*x^2+18*x+27)*exp(4)^2+(24*x^4+138*x^3+180*x^2-54*x)*exp(4)+48
*x^6+264*x^5+291*x^4-198*x^3+27*x^2),x, algorithm="fricas")

[Out]

-1/3*(12*x^3 + 33*x^2 + 3*(x + 3)*e^4 - x*e^x - 15*x - 18)/(4*x^3 + 11*x^2 + (x + 3)*e^4 - 3*x + e^x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+((x^2+6*x+9)*exp(4)+4*x^4+15*x^3-17*x^2-81*x-3)*exp(x)-48*x^3-282*x^2-396*x+54)/(3*exp(x)^
2+((18+6*x)*exp(4)+24*x^3+66*x^2-18*x)*exp(x)+(3*x^2+18*x+27)*exp(4)^2+(24*x^4+138*x^3+180*x^2-54*x)*exp(4)+48
*x^6+264*x^5+291*x^4-198*x^3+27*x^2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.43, size = 57, normalized size = 1.73

method	result	size
norman	\(\frac {-11 x^{2}-4 x^{3}+\left (5-{\mathrm e}^{4}\right ) x +\frac {{\mathrm e}^{x} x}{3}-3 \,{\mathrm e}^{4}+6}{4 x^{3}+x \,{\mathrm e}^{4}+11 x^{2}+3 \,{\mathrm e}^{4}+{\mathrm e}^{x}-3 x}\)	\(57\)
risch	\(\frac {x}{3}-\frac {4 x^{4}+x^{2} {\mathrm e}^{4}+23 x^{3}+6 x \,{\mathrm e}^{4}+30 x^{2}+9 \,{\mathrm e}^{4}-15 x -18}{3 \left (4 x^{3}+x \,{\mathrm e}^{4}+11 x^{2}+3 \,{\mathrm e}^{4}+{\mathrm e}^{x}-3 x \right )}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)^2+((x^2+6*x+9)*exp(4)+4*x^4+15*x^3-17*x^2-81*x-3)*exp(x)-48*x^3-282*x^2-396*x+54)/(3*exp(x)^2+((18
+6*x)*exp(4)+24*x^3+66*x^2-18*x)*exp(x)+(3*x^2+18*x+27)*exp(4)^2+(24*x^4+138*x^3+180*x^2-54*x)*exp(4)+48*x^6+2
64*x^5+291*x^4-198*x^3+27*x^2),x,method=_RETURNVERBOSE)

[Out]

(-11*x^2-4*x^3+(5-exp(4))*x+1/3*exp(x)*x-3*exp(4)+6)/(4*x^3+x*exp(4)+11*x^2+3*exp(4)+exp(x)-3*x)

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maxima [B]  time = 0.41, size = 55, normalized size = 1.67 \begin {gather*} -\frac {12 \, x^{3} + 33 \, x^{2} + 3 \, x {\left (e^{4} - 5\right )} - x e^{x} + 9 \, e^{4} - 18}{3 \, {\left (4 \, x^{3} + 11 \, x^{2} + x {\left (e^{4} - 3\right )} + 3 \, e^{4} + e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+((x^2+6*x+9)*exp(4)+4*x^4+15*x^3-17*x^2-81*x-3)*exp(x)-48*x^3-282*x^2-396*x+54)/(3*exp(x)^
2+((18+6*x)*exp(4)+24*x^3+66*x^2-18*x)*exp(x)+(3*x^2+18*x+27)*exp(4)^2+(24*x^4+138*x^3+180*x^2-54*x)*exp(4)+48
*x^6+264*x^5+291*x^4-198*x^3+27*x^2),x, algorithm="maxima")

[Out]

-1/3*(12*x^3 + 33*x^2 + 3*x*(e^4 - 5) - x*e^x + 9*e^4 - 18)/(4*x^3 + 11*x^2 + x*(e^4 - 3) + 3*e^4 + e^x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {396\,x-{\mathrm {e}}^{2\,x}+282\,x^2+48\,x^3+{\mathrm {e}}^x\,\left (81\,x-{\mathrm {e}}^4\,\left (x^2+6\,x+9\right )+17\,x^2-15\,x^3-4\,x^4+3\right )-54}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^8\,\left (3\,x^2+18\,x+27\right )+{\mathrm {e}}^4\,\left (24\,x^4+138\,x^3+180\,x^2-54\,x\right )+27\,x^2-198\,x^3+291\,x^4+264\,x^5+48\,x^6+{\mathrm {e}}^x\,\left (66\,x^2-18\,x+24\,x^3+{\mathrm {e}}^4\,\left (6\,x+18\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(396*x - exp(2*x) + 282*x^2 + 48*x^3 + exp(x)*(81*x - exp(4)*(6*x + x^2 + 9) + 17*x^2 - 15*x^3 - 4*x^4 +
3) - 54)/(3*exp(2*x) + exp(8)*(18*x + 3*x^2 + 27) + exp(4)*(180*x^2 - 54*x + 138*x^3 + 24*x^4) + 27*x^2 - 198*
x^3 + 291*x^4 + 264*x^5 + 48*x^6 + exp(x)*(66*x^2 - 18*x + 24*x^3 + exp(4)*(6*x + 18))),x)

[Out]

-int((396*x - exp(2*x) + 282*x^2 + 48*x^3 + exp(x)*(81*x - exp(4)*(6*x + x^2 + 9) + 17*x^2 - 15*x^3 - 4*x^4 +
3) - 54)/(3*exp(2*x) + exp(8)*(18*x + 3*x^2 + 27) + exp(4)*(180*x^2 - 54*x + 138*x^3 + 24*x^4) + 27*x^2 - 198*
x^3 + 291*x^4 + 264*x^5 + 48*x^6 + exp(x)*(66*x^2 - 18*x + 24*x^3 + exp(4)*(6*x + 18))), x)

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sympy [B]  time = 0.23, size = 71, normalized size = 2.15 \begin {gather*} \frac {x}{3} + \frac {- 4 x^{4} - 23 x^{3} - x^{2} e^{4} - 30 x^{2} - 6 x e^{4} + 15 x - 9 e^{4} + 18}{12 x^{3} + 33 x^{2} - 9 x + 3 x e^{4} + 3 e^{x} + 9 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)**2+((x**2+6*x+9)*exp(4)+4*x**4+15*x**3-17*x**2-81*x-3)*exp(x)-48*x**3-282*x**2-396*x+54)/(3*
exp(x)**2+((18+6*x)*exp(4)+24*x**3+66*x**2-18*x)*exp(x)+(3*x**2+18*x+27)*exp(4)**2+(24*x**4+138*x**3+180*x**2-
54*x)*exp(4)+48*x**6+264*x**5+291*x**4-198*x**3+27*x**2),x)

[Out]

x/3 + (-4*x**4 - 23*x**3 - x**2*exp(4) - 30*x**2 - 6*x*exp(4) + 15*x - 9*exp(4) + 18)/(12*x**3 + 33*x**2 - 9*x
+ 3*x*exp(4) + 3*exp(x) + 9*exp(4))

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