3.19.62 \(\int \frac {320-432 x^2+204 x^4-41 x^6+3 x^8+e^{3 x} (5 x^3-30 x^4+57 x^5-22 x^6-36 x^7+24 x^8)+e^{2 x} (60 x^2-240 x^3+189 x^4+204 x^5-195 x^6-36 x^7+36 x^8)+(-4 x^2-3 x^4) \log (4)+e^x (240 x-480 x^2-264 x^3+528 x^4+87 x^5-174 x^6-9 x^7+18 x^8+(x^3-4 x^4-4 x^5) \log (4))}{-64 x^2+48 x^4-12 x^6+x^8+e^x (-48 x^3+96 x^4+24 x^5-48 x^6-3 x^7+6 x^8)+e^{3 x} (-x^5+6 x^6-12 x^7+8 x^8)+e^{2 x} (-12 x^4+48 x^5-45 x^6-12 x^7+12 x^8)} \, dx\)

Optimal. Leaf size=34 \[ \frac {5+3 x^2+\frac {\log (4)}{\left (e^x (1-2 x)+\frac {4}{x}-x\right )^2}}{x} \]

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Rubi [F]  time = 3.27, 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 {320-432 x^2+204 x^4-41 x^6+3 x^8+e^{3 x} \left (5 x^3-30 x^4+57 x^5-22 x^6-36 x^7+24 x^8\right )+e^{2 x} \left (60 x^2-240 x^3+189 x^4+204 x^5-195 x^6-36 x^7+36 x^8\right )+\left (-4 x^2-3 x^4\right ) \log (4)+e^x \left (240 x-480 x^2-264 x^3+528 x^4+87 x^5-174 x^6-9 x^7+18 x^8+\left (x^3-4 x^4-4 x^5\right ) \log (4)\right )}{-64 x^2+48 x^4-12 x^6+x^8+e^x \left (-48 x^3+96 x^4+24 x^5-48 x^6-3 x^7+6 x^8\right )+e^{3 x} \left (-x^5+6 x^6-12 x^7+8 x^8\right )+e^{2 x} \left (-12 x^4+48 x^5-45 x^6-12 x^7+12 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(320 - 432*x^2 + 204*x^4 - 41*x^6 + 3*x^8 + E^(3*x)*(5*x^3 - 30*x^4 + 57*x^5 - 22*x^6 - 36*x^7 + 24*x^8) +
E^(2*x)*(60*x^2 - 240*x^3 + 189*x^4 + 204*x^5 - 195*x^6 - 36*x^7 + 36*x^8) + (-4*x^2 - 3*x^4)*Log[4] + E^x*(2
40*x - 480*x^2 - 264*x^3 + 528*x^4 + 87*x^5 - 174*x^6 - 9*x^7 + 18*x^8 + (x^3 - 4*x^4 - 4*x^5)*Log[4]))/(-64*x
^2 + 48*x^4 - 12*x^6 + x^8 + E^x*(-48*x^3 + 96*x^4 + 24*x^5 - 48*x^6 - 3*x^7 + 6*x^8) + E^(3*x)*(-x^5 + 6*x^6
- 12*x^7 + 8*x^8) + E^(2*x)*(-12*x^4 + 48*x^5 - 45*x^6 - 12*x^7 + 12*x^8)),x]

[Out]

5/x + 3*x - (31*Log[4]*Defer[Int][(-4 - E^x*x + x^2 + 2*E^x*x^2)^(-3), x])/2 - 7*Log[4]*Defer[Int][x/(-4 - E^x
*x + x^2 + 2*E^x*x^2)^3, x] + 2*Log[4]*Defer[Int][x^3/(-4 - E^x*x + x^2 + 2*E^x*x^2)^3, x] - (15*Log[4]*Defer[
Int][1/((-1 + 2*x)*(-4 - E^x*x + x^2 + 2*E^x*x^2)^3), x])/2 - 3*Log[4]*Defer[Int][(-4 - E^x*x + x^2 + 2*E^x*x^
2)^(-2), x] - 2*Log[4]*Defer[Int][x/(-4 - E^x*x + x^2 + 2*E^x*x^2)^2, x] - 2*Log[4]*Defer[Int][1/((-1 + 2*x)*(
-4 - E^x*x + x^2 + 2*E^x*x^2)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-320+41 x^6-3 x^8-e^{3 x} x^3 (-1+2 x)^3 \left (-5+3 x^2\right )-3 e^{2 x} (1-2 x)^2 x^2 \left (20-17 x^2+3 x^4\right )-e^x x \left (240-480 x-174 x^5-9 x^6+18 x^7+x^4 (87-4 \log (4))+x^2 (-264+\log (4))-4 x^3 (-132+\log (4))\right )+3 x^4 (-68+\log (4))+4 x^2 (108+\log (4))}{x^2 \left (4-x^2-e^x x (-1+2 x)\right )^3} \, dx\\ &=\int \left (\frac {-5+3 x^2}{x^2}-\frac {\left (-1+4 x+4 x^2\right ) \log (4)}{(-1+2 x) \left (-4-e^x x+x^2+2 e^x x^2\right )^2}+\frac {2 \left (4-12 x-7 x^2-x^3+2 x^4\right ) \log (4)}{(-1+2 x) \left (-4-e^x x+x^2+2 e^x x^2\right )^3}\right ) \, dx\\ &=-\left (\log (4) \int \frac {-1+4 x+4 x^2}{(-1+2 x) \left (-4-e^x x+x^2+2 e^x x^2\right )^2} \, dx\right )+(2 \log (4)) \int \frac {4-12 x-7 x^2-x^3+2 x^4}{(-1+2 x) \left (-4-e^x x+x^2+2 e^x x^2\right )^3} \, dx+\int \frac {-5+3 x^2}{x^2} \, dx\\ &=-\left (\log (4) \int \left (\frac {3}{\left (-4-e^x x+x^2+2 e^x x^2\right )^2}+\frac {2 x}{\left (-4-e^x x+x^2+2 e^x x^2\right )^2}+\frac {2}{(-1+2 x) \left (-4-e^x x+x^2+2 e^x x^2\right )^2}\right ) \, dx\right )+(2 \log (4)) \int \left (-\frac {31}{4 \left (-4-e^x x+x^2+2 e^x x^2\right )^3}-\frac {7 x}{2 \left (-4-e^x x+x^2+2 e^x x^2\right )^3}+\frac {x^3}{\left (-4-e^x x+x^2+2 e^x x^2\right )^3}-\frac {15}{4 (-1+2 x) \left (-4-e^x x+x^2+2 e^x x^2\right )^3}\right ) \, dx+\int \left (3-\frac {5}{x^2}\right ) \, dx\\ &=\frac {5}{x}+3 x+(2 \log (4)) \int \frac {x^3}{\left (-4-e^x x+x^2+2 e^x x^2\right )^3} \, dx-(2 \log (4)) \int \frac {x}{\left (-4-e^x x+x^2+2 e^x x^2\right )^2} \, dx-(2 \log (4)) \int \frac {1}{(-1+2 x) \left (-4-e^x x+x^2+2 e^x x^2\right )^2} \, dx-(3 \log (4)) \int \frac {1}{\left (-4-e^x x+x^2+2 e^x x^2\right )^2} \, dx-(7 \log (4)) \int \frac {x}{\left (-4-e^x x+x^2+2 e^x x^2\right )^3} \, dx-\frac {1}{2} (15 \log (4)) \int \frac {1}{(-1+2 x) \left (-4-e^x x+x^2+2 e^x x^2\right )^3} \, dx-\frac {1}{2} (31 \log (4)) \int \frac {1}{\left (-4-e^x x+x^2+2 e^x x^2\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 30, normalized size = 0.88 \begin {gather*} \frac {5}{x}+x \left (3+\frac {\log (4)}{\left (-4+x^2+e^x x (-1+2 x)\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(320 - 432*x^2 + 204*x^4 - 41*x^6 + 3*x^8 + E^(3*x)*(5*x^3 - 30*x^4 + 57*x^5 - 22*x^6 - 36*x^7 + 24*
x^8) + E^(2*x)*(60*x^2 - 240*x^3 + 189*x^4 + 204*x^5 - 195*x^6 - 36*x^7 + 36*x^8) + (-4*x^2 - 3*x^4)*Log[4] +
E^x*(240*x - 480*x^2 - 264*x^3 + 528*x^4 + 87*x^5 - 174*x^6 - 9*x^7 + 18*x^8 + (x^3 - 4*x^4 - 4*x^5)*Log[4]))/
(-64*x^2 + 48*x^4 - 12*x^6 + x^8 + E^x*(-48*x^3 + 96*x^4 + 24*x^5 - 48*x^6 - 3*x^7 + 6*x^8) + E^(3*x)*(-x^5 +
6*x^6 - 12*x^7 + 8*x^8) + E^(2*x)*(-12*x^4 + 48*x^5 - 45*x^6 - 12*x^7 + 12*x^8)),x]

[Out]

5/x + x*(3 + Log[4]/(-4 + x^2 + E^x*x*(-1 + 2*x))^2)

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fricas [B]  time = 0.98, size = 147, normalized size = 4.32 \begin {gather*} \frac {3 \, x^{6} - 19 \, x^{4} + 2 \, x^{2} \log \relax (2) + 8 \, x^{2} + {\left (12 \, x^{6} - 12 \, x^{5} + 23 \, x^{4} - 20 \, x^{3} + 5 \, x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (6 \, x^{6} - 3 \, x^{5} - 14 \, x^{4} + 7 \, x^{3} - 40 \, x^{2} + 20 \, x\right )} e^{x} + 80}{x^{5} - 8 \, x^{3} + {\left (4 \, x^{5} - 4 \, x^{4} + x^{3}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (2 \, x^{5} - x^{4} - 8 \, x^{3} + 4 \, x^{2}\right )} e^{x} + 16 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^8-36*x^7-22*x^6+57*x^5-30*x^4+5*x^3)*exp(x)^3+(36*x^8-36*x^7-195*x^6+204*x^5+189*x^4-240*x^3+
60*x^2)*exp(x)^2+(2*(-4*x^5-4*x^4+x^3)*log(2)+18*x^8-9*x^7-174*x^6+87*x^5+528*x^4-264*x^3-480*x^2+240*x)*exp(x
)+2*(-3*x^4-4*x^2)*log(2)+3*x^8-41*x^6+204*x^4-432*x^2+320)/((8*x^8-12*x^7+6*x^6-x^5)*exp(x)^3+(12*x^8-12*x^7-
45*x^6+48*x^5-12*x^4)*exp(x)^2+(6*x^8-3*x^7-48*x^6+24*x^5+96*x^4-48*x^3)*exp(x)+x^8-12*x^6+48*x^4-64*x^2),x, a
lgorithm="fricas")

[Out]

(3*x^6 - 19*x^4 + 2*x^2*log(2) + 8*x^2 + (12*x^6 - 12*x^5 + 23*x^4 - 20*x^3 + 5*x^2)*e^(2*x) + 2*(6*x^6 - 3*x^
5 - 14*x^4 + 7*x^3 - 40*x^2 + 20*x)*e^x + 80)/(x^5 - 8*x^3 + (4*x^5 - 4*x^4 + x^3)*e^(2*x) + 2*(2*x^5 - x^4 -
8*x^3 + 4*x^2)*e^x + 16*x)

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giac [B]  time = 0.40, size = 178, normalized size = 5.24 \begin {gather*} \frac {12 \, x^{6} e^{\left (2 \, x\right )} + 12 \, x^{6} e^{x} + 3 \, x^{6} - 12 \, x^{5} e^{\left (2 \, x\right )} - 6 \, x^{5} e^{x} + 23 \, x^{4} e^{\left (2 \, x\right )} - 28 \, x^{4} e^{x} - 19 \, x^{4} - 20 \, x^{3} e^{\left (2 \, x\right )} + 14 \, x^{3} e^{x} + 5 \, x^{2} e^{\left (2 \, x\right )} - 80 \, x^{2} e^{x} + 2 \, x^{2} \log \relax (2) + 8 \, x^{2} + 40 \, x e^{x} + 80}{4 \, x^{5} e^{\left (2 \, x\right )} + 4 \, x^{5} e^{x} + x^{5} - 4 \, x^{4} e^{\left (2 \, x\right )} - 2 \, x^{4} e^{x} + x^{3} e^{\left (2 \, x\right )} - 16 \, x^{3} e^{x} - 8 \, x^{3} + 8 \, x^{2} e^{x} + 16 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^8-36*x^7-22*x^6+57*x^5-30*x^4+5*x^3)*exp(x)^3+(36*x^8-36*x^7-195*x^6+204*x^5+189*x^4-240*x^3+
60*x^2)*exp(x)^2+(2*(-4*x^5-4*x^4+x^3)*log(2)+18*x^8-9*x^7-174*x^6+87*x^5+528*x^4-264*x^3-480*x^2+240*x)*exp(x
)+2*(-3*x^4-4*x^2)*log(2)+3*x^8-41*x^6+204*x^4-432*x^2+320)/((8*x^8-12*x^7+6*x^6-x^5)*exp(x)^3+(12*x^8-12*x^7-
45*x^6+48*x^5-12*x^4)*exp(x)^2+(6*x^8-3*x^7-48*x^6+24*x^5+96*x^4-48*x^3)*exp(x)+x^8-12*x^6+48*x^4-64*x^2),x, a
lgorithm="giac")

[Out]

(12*x^6*e^(2*x) + 12*x^6*e^x + 3*x^6 - 12*x^5*e^(2*x) - 6*x^5*e^x + 23*x^4*e^(2*x) - 28*x^4*e^x - 19*x^4 - 20*
x^3*e^(2*x) + 14*x^3*e^x + 5*x^2*e^(2*x) - 80*x^2*e^x + 2*x^2*log(2) + 8*x^2 + 40*x*e^x + 80)/(4*x^5*e^(2*x) +
4*x^5*e^x + x^5 - 4*x^4*e^(2*x) - 2*x^4*e^x + x^3*e^(2*x) - 16*x^3*e^x - 8*x^3 + 8*x^2*e^x + 16*x)

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maple [A]  time = 0.07, size = 34, normalized size = 1.00

method	result	size
risch	\(3 x +\frac {5}{x}+\frac {2 x \ln \relax (2)}{\left (2 \,{\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +x^{2}-4\right )^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x^8-36*x^7-22*x^6+57*x^5-30*x^4+5*x^3)*exp(x)^3+(36*x^8-36*x^7-195*x^6+204*x^5+189*x^4-240*x^3+60*x^2
)*exp(x)^2+(2*(-4*x^5-4*x^4+x^3)*ln(2)+18*x^8-9*x^7-174*x^6+87*x^5+528*x^4-264*x^3-480*x^2+240*x)*exp(x)+2*(-3
*x^4-4*x^2)*ln(2)+3*x^8-41*x^6+204*x^4-432*x^2+320)/((8*x^8-12*x^7+6*x^6-x^5)*exp(x)^3+(12*x^8-12*x^7-45*x^6+4
8*x^5-12*x^4)*exp(x)^2+(6*x^8-3*x^7-48*x^6+24*x^5+96*x^4-48*x^3)*exp(x)+x^8-12*x^6+48*x^4-64*x^2),x,method=_RE
TURNVERBOSE)

[Out]

3*x+5/x+2*x*ln(2)/(2*exp(x)*x^2-exp(x)*x+x^2-4)^2

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maxima [B]  time = 1.08, size = 144, normalized size = 4.24 \begin {gather*} \frac {3 \, x^{6} - 19 \, x^{4} + 2 \, x^{2} {\left (\log \relax (2) + 4\right )} + {\left (12 \, x^{6} - 12 \, x^{5} + 23 \, x^{4} - 20 \, x^{3} + 5 \, x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (6 \, x^{6} - 3 \, x^{5} - 14 \, x^{4} + 7 \, x^{3} - 40 \, x^{2} + 20 \, x\right )} e^{x} + 80}{x^{5} - 8 \, x^{3} + {\left (4 \, x^{5} - 4 \, x^{4} + x^{3}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (2 \, x^{5} - x^{4} - 8 \, x^{3} + 4 \, x^{2}\right )} e^{x} + 16 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^8-36*x^7-22*x^6+57*x^5-30*x^4+5*x^3)*exp(x)^3+(36*x^8-36*x^7-195*x^6+204*x^5+189*x^4-240*x^3+
60*x^2)*exp(x)^2+(2*(-4*x^5-4*x^4+x^3)*log(2)+18*x^8-9*x^7-174*x^6+87*x^5+528*x^4-264*x^3-480*x^2+240*x)*exp(x
)+2*(-3*x^4-4*x^2)*log(2)+3*x^8-41*x^6+204*x^4-432*x^2+320)/((8*x^8-12*x^7+6*x^6-x^5)*exp(x)^3+(12*x^8-12*x^7-
45*x^6+48*x^5-12*x^4)*exp(x)^2+(6*x^8-3*x^7-48*x^6+24*x^5+96*x^4-48*x^3)*exp(x)+x^8-12*x^6+48*x^4-64*x^2),x, a
lgorithm="maxima")

[Out]

(3*x^6 - 19*x^4 + 2*x^2*(log(2) + 4) + (12*x^6 - 12*x^5 + 23*x^4 - 20*x^3 + 5*x^2)*e^(2*x) + 2*(6*x^6 - 3*x^5
- 14*x^4 + 7*x^3 - 40*x^2 + 20*x)*e^x + 80)/(x^5 - 8*x^3 + (4*x^5 - 4*x^4 + x^3)*e^(2*x) + 2*(2*x^5 - x^4 - 8*
x^3 + 4*x^2)*e^x + 16*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {{\mathrm {e}}^{3\,x}\,\left (24\,x^8-36\,x^7-22\,x^6+57\,x^5-30\,x^4+5\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (36\,x^8-36\,x^7-195\,x^6+204\,x^5+189\,x^4-240\,x^3+60\,x^2\right )-2\,\ln \relax (2)\,\left (3\,x^4+4\,x^2\right )-{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (4\,x^5+4\,x^4-x^3\right )-240\,x+480\,x^2+264\,x^3-528\,x^4-87\,x^5+174\,x^6+9\,x^7-18\,x^8\right )-432\,x^2+204\,x^4-41\,x^6+3\,x^8+320}{{\mathrm {e}}^{2\,x}\,\left (-12\,x^8+12\,x^7+45\,x^6-48\,x^5+12\,x^4\right )+{\mathrm {e}}^{3\,x}\,\left (-8\,x^8+12\,x^7-6\,x^6+x^5\right )+64\,x^2-48\,x^4+12\,x^6-x^8+{\mathrm {e}}^x\,\left (-6\,x^8+3\,x^7+48\,x^6-24\,x^5-96\,x^4+48\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3*x)*(5*x^3 - 30*x^4 + 57*x^5 - 22*x^6 - 36*x^7 + 24*x^8) + exp(2*x)*(60*x^2 - 240*x^3 + 189*x^4 + 2
04*x^5 - 195*x^6 - 36*x^7 + 36*x^8) - 2*log(2)*(4*x^2 + 3*x^4) - exp(x)*(2*log(2)*(4*x^4 - x^3 + 4*x^5) - 240*
x + 480*x^2 + 264*x^3 - 528*x^4 - 87*x^5 + 174*x^6 + 9*x^7 - 18*x^8) - 432*x^2 + 204*x^4 - 41*x^6 + 3*x^8 + 32
0)/(exp(2*x)*(12*x^4 - 48*x^5 + 45*x^6 + 12*x^7 - 12*x^8) + exp(3*x)*(x^5 - 6*x^6 + 12*x^7 - 8*x^8) + 64*x^2 -
48*x^4 + 12*x^6 - x^8 + exp(x)*(48*x^3 - 96*x^4 - 24*x^5 + 48*x^6 + 3*x^7 - 6*x^8)),x)

[Out]

-int((exp(3*x)*(5*x^3 - 30*x^4 + 57*x^5 - 22*x^6 - 36*x^7 + 24*x^8) + exp(2*x)*(60*x^2 - 240*x^3 + 189*x^4 + 2
04*x^5 - 195*x^6 - 36*x^7 + 36*x^8) - 2*log(2)*(4*x^2 + 3*x^4) - exp(x)*(2*log(2)*(4*x^4 - x^3 + 4*x^5) - 240*
x + 480*x^2 + 264*x^3 - 528*x^4 - 87*x^5 + 174*x^6 + 9*x^7 - 18*x^8) - 432*x^2 + 204*x^4 - 41*x^6 + 3*x^8 + 32
0)/(exp(2*x)*(12*x^4 - 48*x^5 + 45*x^6 + 12*x^7 - 12*x^8) + exp(3*x)*(x^5 - 6*x^6 + 12*x^7 - 8*x^8) + 64*x^2 -
48*x^4 + 12*x^6 - x^8 + exp(x)*(48*x^3 - 96*x^4 - 24*x^5 + 48*x^6 + 3*x^7 - 6*x^8)), x)

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sympy [B]  time = 0.59, size = 63, normalized size = 1.85 \begin {gather*} 3 x + \frac {2 x \log {\relax (2 )}}{x^{4} - 8 x^{2} + \left (4 x^{4} - 4 x^{3} + x^{2}\right ) e^{2 x} + \left (4 x^{4} - 2 x^{3} - 16 x^{2} + 8 x\right ) e^{x} + 16} + \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x**8-36*x**7-22*x**6+57*x**5-30*x**4+5*x**3)*exp(x)**3+(36*x**8-36*x**7-195*x**6+204*x**5+189*x
**4-240*x**3+60*x**2)*exp(x)**2+(2*(-4*x**5-4*x**4+x**3)*ln(2)+18*x**8-9*x**7-174*x**6+87*x**5+528*x**4-264*x*
*3-480*x**2+240*x)*exp(x)+2*(-3*x**4-4*x**2)*ln(2)+3*x**8-41*x**6+204*x**4-432*x**2+320)/((8*x**8-12*x**7+6*x*
*6-x**5)*exp(x)**3+(12*x**8-12*x**7-45*x**6+48*x**5-12*x**4)*exp(x)**2+(6*x**8-3*x**7-48*x**6+24*x**5+96*x**4-
48*x**3)*exp(x)+x**8-12*x**6+48*x**4-64*x**2),x)

[Out]

3*x + 2*x*log(2)/(x**4 - 8*x**2 + (4*x**4 - 4*x**3 + x**2)*exp(2*x) + (4*x**4 - 2*x**3 - 16*x**2 + 8*x)*exp(x)
+ 16) + 5/x

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