3.80.18 \(\int \frac {-48-160 x-144 x^2-80 x^3-264 x^4-320 x^5-220 x^6-120 x^7-36 x^8+e^{2 x} (-32 x^2-96 x^3-88 x^4-48 x^5-36 x^6)+e^x (32+128 x+80 x^2+208 x^3+368 x^4+272 x^5+168 x^6+72 x^7)}{36 x^2+120 x^3+148 x^4+20 x^5-120 x^6-100 x^7+x^8+30 x^9+9 x^{10}+e^{2 x} (16 x^2+48 x^3+20 x^4-48 x^5-32 x^6+12 x^7+9 x^8)+e^x (-48 x^2-152 x^3-128 x^4+68 x^5+160 x^6+40 x^7-42 x^8-18 x^9)} \, dx\)

Optimal. Leaf size=36 \[ \frac {4}{x-\frac {2 x}{x^2-\frac {1}{\left (1-e^x+x\right ) \left (1+\frac {3 x}{2}\right )}}} \]

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Rubi [F]  time = 20.52, 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 {-48-160 x-144 x^2-80 x^3-264 x^4-320 x^5-220 x^6-120 x^7-36 x^8+e^{2 x} \left (-32 x^2-96 x^3-88 x^4-48 x^5-36 x^6\right )+e^x \left (32+128 x+80 x^2+208 x^3+368 x^4+272 x^5+168 x^6+72 x^7\right )}{36 x^2+120 x^3+148 x^4+20 x^5-120 x^6-100 x^7+x^8+30 x^9+9 x^{10}+e^{2 x} \left (16 x^2+48 x^3+20 x^4-48 x^5-32 x^6+12 x^7+9 x^8\right )+e^x \left (-48 x^2-152 x^3-128 x^4+68 x^5+160 x^6+40 x^7-42 x^8-18 x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-48 - 160*x - 144*x^2 - 80*x^3 - 264*x^4 - 320*x^5 - 220*x^6 - 120*x^7 - 36*x^8 + E^(2*x)*(-32*x^2 - 96*x
^3 - 88*x^4 - 48*x^5 - 36*x^6) + E^x*(32 + 128*x + 80*x^2 + 208*x^3 + 368*x^4 + 272*x^5 + 168*x^6 + 72*x^7))/(
36*x^2 + 120*x^3 + 148*x^4 + 20*x^5 - 120*x^6 - 100*x^7 + x^8 + 30*x^9 + 9*x^10 + E^(2*x)*(16*x^2 + 48*x^3 + 2
0*x^4 - 48*x^5 - 32*x^6 + 12*x^7 + 9*x^8) + E^x*(-48*x^2 - 152*x^3 - 128*x^4 + 68*x^5 + 160*x^6 + 40*x^7 - 42*
x^8 - 18*x^9)),x]

[Out]

(-4*x)/(2 - x^2) + 32*Defer[Int][(6 - 4*E^x + 10*x - 6*E^x*x + 4*x^2 + 2*E^x*x^2 - 5*x^3 + 3*E^x*x^3 - 3*x^4)^
(-2), x] + (32*Defer[Int][1/((Sqrt[2] - x)*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^
3 + 3*x^4)^2), x])/7 + (36*Sqrt[2]*Defer[Int][1/((Sqrt[2] - x)*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^
2 + 5*x^3 - 3*E^x*x^3 + 3*x^4)^2), x])/7 + 40*Defer[Int][1/(x*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2
+ 5*x^3 - 3*E^x*x^3 + 3*x^4)^2), x] + 48*Defer[Int][x/(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^
3 - 3*E^x*x^3 + 3*x^4)^2, x] - (32*Defer[Int][1/((Sqrt[2] + x)*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^
2 + 5*x^3 - 3*E^x*x^3 + 3*x^4)^2), x])/7 + (36*Sqrt[2]*Defer[Int][1/((Sqrt[2] + x)*(-6 + 4*E^x - 10*x + 6*E^x*
x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^3 + 3*x^4)^2), x])/7 - (648*Defer[Int][1/((2 + 3*x)*(-6 + 4*E^x - 10*x
+ 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^3 + 3*x^4)^2), x])/7 - 64*Defer[Int][1/((-2 + x^2)^2*(-6 + 4*
E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^3 + 3*x^4)^2), x] + (16*Defer[Int][1/((Sqrt[2] - x)
*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^3 + 3*x^4)), x])/7 + (32*Sqrt[2]*Defer[Int
][1/((Sqrt[2] - x)*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^3 + 3*x^4)), x])/7 + 8*D
efer[Int][1/(x^2*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^3 + 3*x^4)), x] + 20*Defer
[Int][1/(x*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^3 + 3*x^4)), x] - (16*Defer[Int]
[1/((Sqrt[2] + x)*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^3 + 3*x^4)), x])/7 + (32*
Sqrt[2]*Defer[Int][1/((Sqrt[2] + x)*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^3 + 3*x
^4)), x])/7 - (324*Defer[Int][1/((2 + 3*x)*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 - 3*E^x*x^
3 + 3*x^4)), x])/7 - 64*Defer[Int][1/((-2 + x^2)^2*(-6 + 4*E^x - 10*x + 6*E^x*x - 4*x^2 - 2*E^x*x^2 + 5*x^3 -
3*E^x*x^3 + 3*x^4)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-12-40 x-36 x^2-20 x^3-66 x^4-80 x^5-55 x^6-30 x^7-9 x^8-e^{2 x} x^2 (2+3 x)^2 \left (2+x^2\right )+2 e^x \left (4+16 x+10 x^2+26 x^3+46 x^4+34 x^5+21 x^6+9 x^7\right )\right )}{x^2 \left (6+10 x+4 x^2-5 x^3-3 x^4+e^x \left (-4-6 x+2 x^2+3 x^3\right )\right )^2} \, dx\\ &=4 \int \frac {-12-40 x-36 x^2-20 x^3-66 x^4-80 x^5-55 x^6-30 x^7-9 x^8-e^{2 x} x^2 (2+3 x)^2 \left (2+x^2\right )+2 e^x \left (4+16 x+10 x^2+26 x^3+46 x^4+34 x^5+21 x^6+9 x^7\right )}{x^2 \left (6+10 x+4 x^2-5 x^3-3 x^4+e^x \left (-4-6 x+2 x^2+3 x^3\right )\right )^2} \, dx\\ &=4 \int \left (\frac {-2-x^2}{\left (-2+x^2\right )^2}-\frac {4 \left (-4-16 x+4 x^2+20 x^3+3 x^4\right )}{x^2 (2+3 x) \left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )}+\frac {4 \left (20+20 x+26 x^2+14 x^3-48 x^4-32 x^5+12 x^6+9 x^7\right )}{x (2+3 x) \left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2}\right ) \, dx\\ &=4 \int \frac {-2-x^2}{\left (-2+x^2\right )^2} \, dx-16 \int \frac {-4-16 x+4 x^2+20 x^3+3 x^4}{x^2 (2+3 x) \left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+16 \int \frac {20+20 x+26 x^2+14 x^3-48 x^4-32 x^5+12 x^6+9 x^7}{x (2+3 x) \left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx\\ &=-\frac {4 x}{2-x^2}+16 \int \left (\frac {2}{\left (6-4 e^x+10 x-6 e^x x+4 x^2+2 e^x x^2-5 x^3+3 e^x x^3-3 x^4\right )^2}+\frac {5}{2 x \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2}+\frac {3 x}{\left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2}-\frac {81}{14 (2+3 x) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2}-\frac {4}{\left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2}-\frac {9+4 x}{7 \left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2}\right ) \, dx-16 \int \left (-\frac {1}{2 x^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )}-\frac {5}{4 x \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )}+\frac {81}{28 (2+3 x) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )}+\frac {4}{\left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )}+\frac {2 (4+x)}{7 \left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )}\right ) \, dx\\ &=-\frac {4 x}{2-x^2}-\frac {16}{7} \int \frac {9+4 x}{\left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-\frac {32}{7} \int \frac {4+x}{\left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+8 \int \frac {1}{x^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+20 \int \frac {1}{x \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+32 \int \frac {1}{\left (6-4 e^x+10 x-6 e^x x+4 x^2+2 e^x x^2-5 x^3+3 e^x x^3-3 x^4\right )^2} \, dx+40 \int \frac {1}{x \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-\frac {324}{7} \int \frac {1}{(2+3 x) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+48 \int \frac {x}{\left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-64 \int \frac {1}{\left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-64 \int \frac {1}{\left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx-\frac {648}{7} \int \frac {1}{(2+3 x) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx\\ &=-\frac {4 x}{2-x^2}-\frac {16}{7} \int \left (\frac {9}{\left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2}+\frac {4 x}{\left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2}\right ) \, dx-\frac {32}{7} \int \left (\frac {4}{\left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )}+\frac {x}{\left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )}\right ) \, dx+8 \int \frac {1}{x^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+20 \int \frac {1}{x \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+32 \int \frac {1}{\left (6-4 e^x+10 x-6 e^x x+4 x^2+2 e^x x^2-5 x^3+3 e^x x^3-3 x^4\right )^2} \, dx+40 \int \frac {1}{x \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-\frac {324}{7} \int \frac {1}{(2+3 x) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+48 \int \frac {x}{\left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-64 \int \frac {1}{\left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-64 \int \frac {1}{\left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx-\frac {648}{7} \int \frac {1}{(2+3 x) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx\\ &=-\frac {4 x}{2-x^2}-\frac {32}{7} \int \frac {x}{\left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+8 \int \frac {1}{x^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx-\frac {64}{7} \int \frac {x}{\left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-\frac {128}{7} \int \frac {1}{\left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+20 \int \frac {1}{x \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx-\frac {144}{7} \int \frac {1}{\left (-2+x^2\right ) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx+32 \int \frac {1}{\left (6-4 e^x+10 x-6 e^x x+4 x^2+2 e^x x^2-5 x^3+3 e^x x^3-3 x^4\right )^2} \, dx+40 \int \frac {1}{x \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-\frac {324}{7} \int \frac {1}{(2+3 x) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx+48 \int \frac {x}{\left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-64 \int \frac {1}{\left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx-64 \int \frac {1}{\left (-2+x^2\right )^2 \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )} \, dx-\frac {648}{7} \int \frac {1}{(2+3 x) \left (-6+4 e^x-10 x+6 e^x x-4 x^2-2 e^x x^2+5 x^3-3 e^x x^3+3 x^4\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 61, normalized size = 1.69 \begin {gather*} -\frac {4 \left (-x^2+\frac {4}{6+10 x+4 x^2-5 x^3-3 x^4+e^x \left (-4-6 x+2 x^2+3 x^3\right )}\right )}{x \left (-2+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-48 - 160*x - 144*x^2 - 80*x^3 - 264*x^4 - 320*x^5 - 220*x^6 - 120*x^7 - 36*x^8 + E^(2*x)*(-32*x^2
- 96*x^3 - 88*x^4 - 48*x^5 - 36*x^6) + E^x*(32 + 128*x + 80*x^2 + 208*x^3 + 368*x^4 + 272*x^5 + 168*x^6 + 72*x
^7))/(36*x^2 + 120*x^3 + 148*x^4 + 20*x^5 - 120*x^6 - 100*x^7 + x^8 + 30*x^9 + 9*x^10 + E^(2*x)*(16*x^2 + 48*x
^3 + 20*x^4 - 48*x^5 - 32*x^6 + 12*x^7 + 9*x^8) + E^x*(-48*x^2 - 152*x^3 - 128*x^4 + 68*x^5 + 160*x^6 + 40*x^7
- 42*x^8 - 18*x^9)),x]

[Out]

(-4*(-x^2 + 4/(6 + 10*x + 4*x^2 - 5*x^3 - 3*x^4 + E^x*(-4 - 6*x + 2*x^2 + 3*x^3))))/(x*(-2 + x^2))

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fricas [B]  time = 0.74, size = 83, normalized size = 2.31 \begin {gather*} \frac {4 \, {\left (3 \, x^{4} + 5 \, x^{3} + 2 \, x^{2} - {\left (3 \, x^{3} + 2 \, x^{2}\right )} e^{x} - 2\right )}}{3 \, x^{5} + 5 \, x^{4} - 4 \, x^{3} - 10 \, x^{2} - {\left (3 \, x^{4} + 2 \, x^{3} - 6 \, x^{2} - 4 \, x\right )} e^{x} - 6 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x^6-48*x^5-88*x^4-96*x^3-32*x^2)*exp(x)^2+(72*x^7+168*x^6+272*x^5+368*x^4+208*x^3+80*x^2+128*x
+32)*exp(x)-36*x^8-120*x^7-220*x^6-320*x^5-264*x^4-80*x^3-144*x^2-160*x-48)/((9*x^8+12*x^7-32*x^6-48*x^5+20*x^
4+48*x^3+16*x^2)*exp(x)^2+(-18*x^9-42*x^8+40*x^7+160*x^6+68*x^5-128*x^4-152*x^3-48*x^2)*exp(x)+9*x^10+30*x^9+x
^8-100*x^7-120*x^6+20*x^5+148*x^4+120*x^3+36*x^2),x, algorithm="fricas")

[Out]

4*(3*x^4 + 5*x^3 + 2*x^2 - (3*x^3 + 2*x^2)*e^x - 2)/(3*x^5 + 5*x^4 - 4*x^3 - 10*x^2 - (3*x^4 + 2*x^3 - 6*x^2 -
4*x)*e^x - 6*x)

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giac [B]  time = 0.20, size = 85, normalized size = 2.36 \begin {gather*} \frac {4 \, {\left (3 \, x^{4} - 3 \, x^{3} e^{x} + 5 \, x^{3} - 2 \, x^{2} e^{x} + 2 \, x^{2} - 2\right )}}{3 \, x^{5} - 3 \, x^{4} e^{x} + 5 \, x^{4} - 2 \, x^{3} e^{x} - 4 \, x^{3} + 6 \, x^{2} e^{x} - 10 \, x^{2} + 4 \, x e^{x} - 6 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x^6-48*x^5-88*x^4-96*x^3-32*x^2)*exp(x)^2+(72*x^7+168*x^6+272*x^5+368*x^4+208*x^3+80*x^2+128*x
+32)*exp(x)-36*x^8-120*x^7-220*x^6-320*x^5-264*x^4-80*x^3-144*x^2-160*x-48)/((9*x^8+12*x^7-32*x^6-48*x^5+20*x^
4+48*x^3+16*x^2)*exp(x)^2+(-18*x^9-42*x^8+40*x^7+160*x^6+68*x^5-128*x^4-152*x^3-48*x^2)*exp(x)+9*x^10+30*x^9+x
^8-100*x^7-120*x^6+20*x^5+148*x^4+120*x^3+36*x^2),x, algorithm="giac")

[Out]

4*(3*x^4 - 3*x^3*e^x + 5*x^3 - 2*x^2*e^x + 2*x^2 - 2)/(3*x^5 - 3*x^4*e^x + 5*x^4 - 2*x^3*e^x - 4*x^3 + 6*x^2*e
^x - 10*x^2 + 4*x*e^x - 6*x)

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maple [B]  time = 0.17, size = 69, normalized size = 1.92

method	result	size
risch	\(\frac {4 x}{x^{2}-2}+\frac {16}{\left (x^{2}-2\right ) x \left (3 x^{4}-3 \,{\mathrm e}^{x} x^{3}+5 x^{3}-2 \,{\mathrm e}^{x} x^{2}-4 x^{2}+6 \,{\mathrm e}^{x} x -10 x +4 \,{\mathrm e}^{x}-6\right )}\)	\(69\)
norman	\(\frac {-8+8 x^{2}+12 x^{4}+20 x^{3}-12 \,{\mathrm e}^{x} x^{3}-8 \,{\mathrm e}^{x} x^{2}}{x \left (3 x^{4}-3 \,{\mathrm e}^{x} x^{3}+5 x^{3}-2 \,{\mathrm e}^{x} x^{2}-4 x^{2}+6 \,{\mathrm e}^{x} x -10 x +4 \,{\mathrm e}^{x}-6\right )}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-36*x^6-48*x^5-88*x^4-96*x^3-32*x^2)*exp(x)^2+(72*x^7+168*x^6+272*x^5+368*x^4+208*x^3+80*x^2+128*x+32)*e
xp(x)-36*x^8-120*x^7-220*x^6-320*x^5-264*x^4-80*x^3-144*x^2-160*x-48)/((9*x^8+12*x^7-32*x^6-48*x^5+20*x^4+48*x
^3+16*x^2)*exp(x)^2+(-18*x^9-42*x^8+40*x^7+160*x^6+68*x^5-128*x^4-152*x^3-48*x^2)*exp(x)+9*x^10+30*x^9+x^8-100
*x^7-120*x^6+20*x^5+148*x^4+120*x^3+36*x^2),x,method=_RETURNVERBOSE)

[Out]

4*x/(x^2-2)+16/(x^2-2)/x/(3*x^4-3*exp(x)*x^3+5*x^3-2*exp(x)*x^2-4*x^2+6*exp(x)*x-10*x+4*exp(x)-6)

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maxima [B]  time = 0.50, size = 83, normalized size = 2.31 \begin {gather*} \frac {4 \, {\left (3 \, x^{4} + 5 \, x^{3} + 2 \, x^{2} - {\left (3 \, x^{3} + 2 \, x^{2}\right )} e^{x} - 2\right )}}{3 \, x^{5} + 5 \, x^{4} - 4 \, x^{3} - 10 \, x^{2} - {\left (3 \, x^{4} + 2 \, x^{3} - 6 \, x^{2} - 4 \, x\right )} e^{x} - 6 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x^6-48*x^5-88*x^4-96*x^3-32*x^2)*exp(x)^2+(72*x^7+168*x^6+272*x^5+368*x^4+208*x^3+80*x^2+128*x
+32)*exp(x)-36*x^8-120*x^7-220*x^6-320*x^5-264*x^4-80*x^3-144*x^2-160*x-48)/((9*x^8+12*x^7-32*x^6-48*x^5+20*x^
4+48*x^3+16*x^2)*exp(x)^2+(-18*x^9-42*x^8+40*x^7+160*x^6+68*x^5-128*x^4-152*x^3-48*x^2)*exp(x)+9*x^10+30*x^9+x
^8-100*x^7-120*x^6+20*x^5+148*x^4+120*x^3+36*x^2),x, algorithm="maxima")

[Out]

4*(3*x^4 + 5*x^3 + 2*x^2 - (3*x^3 + 2*x^2)*e^x - 2)/(3*x^5 + 5*x^4 - 4*x^3 - 10*x^2 - (3*x^4 + 2*x^3 - 6*x^2 -
4*x)*e^x - 6*x)

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mupad [B]  time = 5.41, size = 76, normalized size = 2.11 \begin {gather*} \frac {x^2\,\left (8\,{\mathrm {e}}^x-8\right )+x^3\,\left (12\,{\mathrm {e}}^x-20\right )-12\,x^4+8}{x\,\left (10\,x-4\,{\mathrm {e}}^x+2\,x^2\,{\mathrm {e}}^x+3\,x^3\,{\mathrm {e}}^x-6\,x\,{\mathrm {e}}^x+4\,x^2-5\,x^3-3\,x^4+6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(160*x + exp(2*x)*(32*x^2 + 96*x^3 + 88*x^4 + 48*x^5 + 36*x^6) - exp(x)*(128*x + 80*x^2 + 208*x^3 + 368*x
^4 + 272*x^5 + 168*x^6 + 72*x^7 + 32) + 144*x^2 + 80*x^3 + 264*x^4 + 320*x^5 + 220*x^6 + 120*x^7 + 36*x^8 + 48
)/(exp(2*x)*(16*x^2 + 48*x^3 + 20*x^4 - 48*x^5 - 32*x^6 + 12*x^7 + 9*x^8) - exp(x)*(48*x^2 + 152*x^3 + 128*x^4
- 68*x^5 - 160*x^6 - 40*x^7 + 42*x^8 + 18*x^9) + 36*x^2 + 120*x^3 + 148*x^4 + 20*x^5 - 120*x^6 - 100*x^7 + x^
8 + 30*x^9 + 9*x^10),x)

[Out]

(x^2*(8*exp(x) - 8) + x^3*(12*exp(x) - 20) - 12*x^4 + 8)/(x*(10*x - 4*exp(x) + 2*x^2*exp(x) + 3*x^3*exp(x) - 6
*x*exp(x) + 4*x^2 - 5*x^3 - 3*x^4 + 6))

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sympy [B]  time = 0.61, size = 75, normalized size = 2.08 \begin {gather*} \frac {4 x}{x^{2} - 2} - \frac {16}{- 3 x^{7} - 5 x^{6} + 10 x^{5} + 20 x^{4} - 2 x^{3} - 20 x^{2} - 12 x + \left (3 x^{6} + 2 x^{5} - 12 x^{4} - 8 x^{3} + 12 x^{2} + 8 x\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x**6-48*x**5-88*x**4-96*x**3-32*x**2)*exp(x)**2+(72*x**7+168*x**6+272*x**5+368*x**4+208*x**3+8
0*x**2+128*x+32)*exp(x)-36*x**8-120*x**7-220*x**6-320*x**5-264*x**4-80*x**3-144*x**2-160*x-48)/((9*x**8+12*x**
7-32*x**6-48*x**5+20*x**4+48*x**3+16*x**2)*exp(x)**2+(-18*x**9-42*x**8+40*x**7+160*x**6+68*x**5-128*x**4-152*x
**3-48*x**2)*exp(x)+9*x**10+30*x**9+x**8-100*x**7-120*x**6+20*x**5+148*x**4+120*x**3+36*x**2),x)

[Out]

4*x/(x**2 - 2) - 16/(-3*x**7 - 5*x**6 + 10*x**5 + 20*x**4 - 2*x**3 - 20*x**2 - 12*x + (3*x**6 + 2*x**5 - 12*x*
*4 - 8*x**3 + 12*x**2 + 8*x)*exp(x))

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