∫ 768x2−64x3−448x4+56x5−256x6+e2x(144x2−84x4−48x6)+e3x(12x2−7x4−4x6)+ex(576x2−16x3−328x4+14x5−206x6+8x8)___________ 1024−3584x2+5184x4−3584x6+1024x8+e3x(16−56x2+81x4−56x6+16x8)+e2x(192−672x2+972x4−672x6+192x8)+ex(768−2688x2+3888x4−2688x6+768x8) dx

3.32.86 \(\int \frac {768 x^2-64 x^3-448 x^4+56 x^5-256 x^6+e^{2 x} (144 x^2-84 x^4-48 x^6)+e^{3 x} (12 x^2-7 x^4-4 x^6)+e^x (576 x^2-16 x^3-328 x^4+14 x^5-206 x^6+8 x^8)}{1024-3584 x^2+5184 x^4-3584 x^6+1024 x^8+e^{3 x} (16-56 x^2+81 x^4-56 x^6+16 x^8)+e^{2 x} (192-672 x^2+972 x^4-672 x^6+192 x^8)+e^x (768-2688 x^2+3888 x^4-2688 x^6+768 x^8)} \, dx\)

Optimal. Leaf size=30 \[ \frac {x-\frac {x^2}{\left (4+e^x\right )^2}}{1+\left (\frac {2}{x}-2 x\right )^2} \]

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Rubi [F] time = 8.35, 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 {768 x^2-64 x^3-448 x^4+56 x^5-256 x^6+e^{2 x} \left (144 x^2-84 x^4-48 x^6\right )+e^{3 x} \left (12 x^2-7 x^4-4 x^6\right )+e^x \left (576 x^2-16 x^3-328 x^4+14 x^5-206 x^6+8 x^8\right )}{1024-3584 x^2+5184 x^4-3584 x^6+1024 x^8+e^{3 x} \left (16-56 x^2+81 x^4-56 x^6+16 x^8\right )+e^{2 x} \left (192-672 x^2+972 x^4-672 x^6+192 x^8\right )+e^x \left (768-2688 x^2+3888 x^4-2688 x^6+768 x^8\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(768*x^2 - 64*x^3 - 448*x^4 + 56*x^5 - 256*x^6 + E^(2*x)*(144*x^2 - 84*x^4 - 48*x^6) + E^(3*x)*(12*x^2 - 7

*x^4 - 4*x^6) + E^x*(576*x^2 - 16*x^3 - 328*x^4 + 14*x^5 - 206*x^6 + 8*x^8))/(1024 - 3584*x^2 + 5184*x^4 - 358

4*x^6 + 1024*x^8 + E^(3*x)*(16 - 56*x^2 + 81*x^4 - 56*x^6 + 16*x^8) + E^(2*x)*(192 - 672*x^2 + 972*x^4 - 672*x

^6 + 192*x^8) + E^x*(768 - 2688*x^2 + 3888*x^4 - 2688*x^6 + 768*x^8)),x]

[Out]

-1/4*1/(4 + E^x)^2 + x^3/(4 - 7*x^2 + 4*x^4) - ((32*I)*Defer[Int][1/((4 + E^x)^3*(Sqrt[7 - I*Sqrt[15]] - 2*Sqr

t[2]*x)), x])/Sqrt[15*(7 - I*Sqrt[15])] + (7*(7*I + Sqrt[15])*Defer[Int][1/((4 + E^x)^3*(Sqrt[7 - I*Sqrt[15]]

- 2*Sqrt[2]*x)), x])/Sqrt[15*(7 - I*Sqrt[15])] - ((7*I)*Defer[Int][1/((4 + E^x)^2*(Sqrt[7 - I*Sqrt[15]] - 2*Sq

rt[2]*x)), x])/Sqrt[30] + ((8*I)*Defer[Int][1/((4 + E^x)^2*(Sqrt[7 - I*Sqrt[15]] - 2*Sqrt[2]*x)), x])/Sqrt[15*

(7 - I*Sqrt[15])] - (7*(7*I + Sqrt[15])*Defer[Int][1/((4 + E^x)^2*(Sqrt[7 - I*Sqrt[15]] - 2*Sqrt[2]*x)), x])/(

4*Sqrt[15*(7 - I*Sqrt[15])]) + ((32*I)*Defer[Int][1/((4 + E^x)^3*(Sqrt[7 + I*Sqrt[15]] - 2*Sqrt[2]*x)), x])/Sq

rt[15*(7 + I*Sqrt[15])] - (7*(7*I - Sqrt[15])*Defer[Int][1/((4 + E^x)^3*(Sqrt[7 + I*Sqrt[15]] - 2*Sqrt[2]*x)),

x])/Sqrt[15*(7 + I*Sqrt[15])] + ((7*I)*Defer[Int][1/((4 + E^x)^2*(Sqrt[7 + I*Sqrt[15]] - 2*Sqrt[2]*x)), x])/S

qrt[30] - ((8*I)*Defer[Int][1/((4 + E^x)^2*(Sqrt[7 + I*Sqrt[15]] - 2*Sqrt[2]*x)), x])/Sqrt[15*(7 + I*Sqrt[15])

] + (7*(7*I - Sqrt[15])*Defer[Int][1/((4 + E^x)^2*(Sqrt[7 + I*Sqrt[15]] - 2*Sqrt[2]*x)), x])/(4*Sqrt[15*(7 + I

*Sqrt[15])]) - ((32*I)*Defer[Int][1/((4 + E^x)^3*(Sqrt[7 - I*Sqrt[15]] + 2*Sqrt[2]*x)), x])/Sqrt[15*(7 - I*Sqr

t[15])] + (7*(7*I + Sqrt[15])*Defer[Int][1/((4 + E^x)^3*(Sqrt[7 - I*Sqrt[15]] + 2*Sqrt[2]*x)), x])/Sqrt[15*(7

- I*Sqrt[15])] + ((7*I)*Defer[Int][1/((4 + E^x)^2*(Sqrt[7 - I*Sqrt[15]] + 2*Sqrt[2]*x)), x])/Sqrt[30] + ((8*I)

*Defer[Int][1/((4 + E^x)^2*(Sqrt[7 - I*Sqrt[15]] + 2*Sqrt[2]*x)), x])/Sqrt[15*(7 - I*Sqrt[15])] - (7*(7*I + Sq

rt[15])*Defer[Int][1/((4 + E^x)^2*(Sqrt[7 - I*Sqrt[15]] + 2*Sqrt[2]*x)), x])/(4*Sqrt[15*(7 - I*Sqrt[15])]) + (

(32*I)*Defer[Int][1/((4 + E^x)^3*(Sqrt[7 + I*Sqrt[15]] + 2*Sqrt[2]*x)), x])/Sqrt[15*(7 + I*Sqrt[15])] - (7*(7*

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

((7*I)*Defer[Int][1/((4 + E^x)^2*(Sqrt[7 + I*Sqrt[15]] + 2*Sqrt[2]*x)), x])/Sqrt[30] - ((8*I)*Defer[Int][1/((

4 + E^x)^2*(Sqrt[7 + I*Sqrt[15]] + 2*Sqrt[2]*x)), x])/Sqrt[15*(7 + I*Sqrt[15])] + (7*(7*I - Sqrt[15])*Defer[In

t][1/((4 + E^x)^2*(Sqrt[7 + I*Sqrt[15]] + 2*Sqrt[2]*x)), x])/(4*Sqrt[15*(7 + I*Sqrt[15])]) - 14*Defer[Int][x/(

(4 + E^x)^2*(4 - 7*x^2 + 4*x^4)^2), x] + (17*Defer[Int][x^3/((4 + E^x)^2*(4 - 7*x^2 + 4*x^4)^2), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (-12 e^{2 x} \left (-12+7 x^2+4 x^4\right )-e^{3 x} \left (-12+7 x^2+4 x^4\right )-8 \left (-96+8 x+56 x^2-7 x^3+32 x^4\right )+2 e^x \left (288-8 x-164 x^2+7 x^3-103 x^4+4 x^6\right )\right )}{\left (4+e^x\right )^3 \left (4-7 x^2+4 x^4\right )^2} \, dx\\ &=\int \left (-\frac {8 x^4}{\left (4+e^x\right )^3 \left (4-7 x^2+4 x^4\right )}-\frac {x^2 \left (-12+7 x^2+4 x^4\right )}{\left (4-7 x^2+4 x^4\right )^2}+\frac {2 x^3 \left (-8+4 x+7 x^2-7 x^3+4 x^5\right )}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2}\right ) \, dx\\ &=2 \int \frac {x^3 \left (-8+4 x+7 x^2-7 x^3+4 x^5\right )}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2} \, dx-8 \int \frac {x^4}{\left (4+e^x\right )^3 \left (4-7 x^2+4 x^4\right )} \, dx-\int \frac {x^2 \left (-12+7 x^2+4 x^4\right )}{\left (4-7 x^2+4 x^4\right )^2} \, dx\\ &=\frac {x^3}{4-7 x^2+4 x^4}+2 \int \left (\frac {1}{4 \left (4+e^x\right )^2}+\frac {x \left (-28+17 x^2\right )}{4 \left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2}+\frac {-4+7 x+7 x^2}{4 \left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )}\right ) \, dx-8 \int \left (\frac {1}{4 \left (4+e^x\right )^3}+\frac {-4+7 x^2}{4 \left (4+e^x\right )^3 \left (4-7 x^2+4 x^4\right )}\right ) \, dx\\ &=\frac {x^3}{4-7 x^2+4 x^4}+\frac {1}{2} \int \frac {1}{\left (4+e^x\right )^2} \, dx+\frac {1}{2} \int \frac {x \left (-28+17 x^2\right )}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2} \, dx+\frac {1}{2} \int \frac {-4+7 x+7 x^2}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )} \, dx-2 \int \frac {1}{\left (4+e^x\right )^3} \, dx-2 \int \frac {-4+7 x^2}{\left (4+e^x\right )^3 \left (4-7 x^2+4 x^4\right )} \, dx\\ &=\frac {x^3}{4-7 x^2+4 x^4}+\frac {1}{2} \int \left (-\frac {28 x}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2}+\frac {17 x^3}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2}\right ) \, dx+\frac {1}{2} \int \left (-\frac {4}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )}+\frac {7 x}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )}+\frac {7 x^2}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )}\right ) \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (4+x)^2} \, dx,x,e^x\right )-2 \int \left (-\frac {4}{\left (4+e^x\right )^3 \left (4-7 x^2+4 x^4\right )}+\frac {7 x^2}{\left (4+e^x\right )^3 \left (4-7 x^2+4 x^4\right )}\right ) \, dx-2 \operatorname {Subst}\left (\int \frac {1}{x (4+x)^3} \, dx,x,e^x\right )\\ &=\frac {x^3}{4-7 x^2+4 x^4}+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{16 x}-\frac {1}{4 (4+x)^2}-\frac {1}{16 (4+x)}\right ) \, dx,x,e^x\right )-2 \int \frac {1}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )} \, dx-2 \operatorname {Subst}\left (\int \left (\frac {1}{64 x}-\frac {1}{4 (4+x)^3}-\frac {1}{16 (4+x)^2}-\frac {1}{64 (4+x)}\right ) \, dx,x,e^x\right )+\frac {7}{2} \int \frac {x}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )} \, dx+\frac {7}{2} \int \frac {x^2}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )} \, dx+8 \int \frac {1}{\left (4+e^x\right )^3 \left (4-7 x^2+4 x^4\right )} \, dx+\frac {17}{2} \int \frac {x^3}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2} \, dx-14 \int \frac {x}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2} \, dx-14 \int \frac {x^2}{\left (4+e^x\right )^3 \left (4-7 x^2+4 x^4\right )} \, dx\\ &=-\frac {1}{4 \left (4+e^x\right )^2}+\frac {x^3}{4-7 x^2+4 x^4}-2 \int \left (\frac {8 i}{\sqrt {15} \left (4+e^x\right )^2 \left (7+i \sqrt {15}-8 x^2\right )}+\frac {8 i}{\sqrt {15} \left (4+e^x\right )^2 \left (-7+i \sqrt {15}+8 x^2\right )}\right ) \, dx+\frac {7}{2} \int \left (\frac {1-\frac {7 i}{\sqrt {15}}}{\left (4+e^x\right )^2 \left (-7-i \sqrt {15}+8 x^2\right )}+\frac {1+\frac {7 i}{\sqrt {15}}}{\left (4+e^x\right )^2 \left (-7+i \sqrt {15}+8 x^2\right )}\right ) \, dx+\frac {7}{2} \int \left (\frac {8 i x}{\sqrt {15} \left (4+e^x\right )^2 \left (7+i \sqrt {15}-8 x^2\right )}+\frac {8 i x}{\sqrt {15} \left (4+e^x\right )^2 \left (-7+i \sqrt {15}+8 x^2\right )}\right ) \, dx+8 \int \left (\frac {8 i}{\sqrt {15} \left (4+e^x\right )^3 \left (7+i \sqrt {15}-8 x^2\right )}+\frac {8 i}{\sqrt {15} \left (4+e^x\right )^3 \left (-7+i \sqrt {15}+8 x^2\right )}\right ) \, dx+\frac {17}{2} \int \frac {x^3}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2} \, dx-14 \int \frac {x}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2} \, dx-14 \int \left (\frac {1-\frac {7 i}{\sqrt {15}}}{\left (4+e^x\right )^3 \left (-7-i \sqrt {15}+8 x^2\right )}+\frac {1+\frac {7 i}{\sqrt {15}}}{\left (4+e^x\right )^3 \left (-7+i \sqrt {15}+8 x^2\right )}\right ) \, dx\\ &=-\frac {1}{4 \left (4+e^x\right )^2}+\frac {x^3}{4-7 x^2+4 x^4}+\frac {17}{2} \int \frac {x^3}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2} \, dx-14 \int \frac {x}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )^2} \, dx-\frac {(16 i) \int \frac {1}{\left (4+e^x\right )^2 \left (7+i \sqrt {15}-8 x^2\right )} \, dx}{\sqrt {15}}-\frac {(16 i) \int \frac {1}{\left (4+e^x\right )^2 \left (-7+i \sqrt {15}+8 x^2\right )} \, dx}{\sqrt {15}}+\frac {(28 i) \int \frac {x}{\left (4+e^x\right )^2 \left (7+i \sqrt {15}-8 x^2\right )} \, dx}{\sqrt {15}}+\frac {(28 i) \int \frac {x}{\left (4+e^x\right )^2 \left (-7+i \sqrt {15}+8 x^2\right )} \, dx}{\sqrt {15}}+\frac {(64 i) \int \frac {1}{\left (4+e^x\right )^3 \left (7+i \sqrt {15}-8 x^2\right )} \, dx}{\sqrt {15}}+\frac {(64 i) \int \frac {1}{\left (4+e^x\right )^3 \left (-7+i \sqrt {15}+8 x^2\right )} \, dx}{\sqrt {15}}+\frac {1}{30} \left (7 \left (15-7 i \sqrt {15}\right )\right ) \int \frac {1}{\left (4+e^x\right )^2 \left (-7-i \sqrt {15}+8 x^2\right )} \, dx-\frac {1}{15} \left (14 \left (15-7 i \sqrt {15}\right )\right ) \int \frac {1}{\left (4+e^x\right )^3 \left (-7-i \sqrt {15}+8 x^2\right )} \, dx+\frac {1}{30} \left (7 \left (15+7 i \sqrt {15}\right )\right ) \int \frac {1}{\left (4+e^x\right )^2 \left (-7+i \sqrt {15}+8 x^2\right )} \, dx-\frac {1}{15} \left (14 \left (15+7 i \sqrt {15}\right )\right ) \int \frac {1}{\left (4+e^x\right )^3 \left (-7+i \sqrt {15}+8 x^2\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 0.06, size = 40, normalized size = 1.33 \begin {gather*} \frac {\left (16+8 e^x+e^{2 x}-x\right ) x^3}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(768*x^2 - 64*x^3 - 448*x^4 + 56*x^5 - 256*x^6 + E^(2*x)*(144*x^2 - 84*x^4 - 48*x^6) + E^(3*x)*(12*x

^2 - 7*x^4 - 4*x^6) + E^x*(576*x^2 - 16*x^3 - 328*x^4 + 14*x^5 - 206*x^6 + 8*x^8))/(1024 - 3584*x^2 + 5184*x^4

- 3584*x^6 + 1024*x^8 + E^(3*x)*(16 - 56*x^2 + 81*x^4 - 56*x^6 + 16*x^8) + E^(2*x)*(192 - 672*x^2 + 972*x^4 -

672*x^6 + 192*x^8) + E^x*(768 - 2688*x^2 + 3888*x^4 - 2688*x^6 + 768*x^8)),x]

[Out]

((16 + 8*E^x + E^(2*x) - x)*x^3)/((4 + E^x)^2*(4 - 7*x^2 + 4*x^4))

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fricas [B] time = 0.54, size = 74, normalized size = 2.47 \begin {gather*} -\frac {x^{4} - x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} - 16 \, x^{3}}{64 \, x^{4} - 112 \, x^{2} + {\left (4 \, x^{4} - 7 \, x^{2} + 4\right )} e^{\left (2 \, x\right )} + 8 \, {\left (4 \, x^{4} - 7 \, x^{2} + 4\right )} e^{x} + 64} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^6-7*x^4+12*x^2)*exp(x)^3+(-48*x^6-84*x^4+144*x^2)*exp(x)^2+(8*x^8-206*x^6+14*x^5-328*x^4-16*x

^3+576*x^2)*exp(x)-256*x^6+56*x^5-448*x^4-64*x^3+768*x^2)/((16*x^8-56*x^6+81*x^4-56*x^2+16)*exp(x)^3+(192*x^8-

672*x^6+972*x^4-672*x^2+192)*exp(x)^2+(768*x^8-2688*x^6+3888*x^4-2688*x^2+768)*exp(x)+1024*x^8-3584*x^6+5184*x

^4-3584*x^2+1024),x, algorithm="fricas")

[Out]

-(x^4 - x^3*e^(2*x) - 8*x^3*e^x - 16*x^3)/(64*x^4 - 112*x^2 + (4*x^4 - 7*x^2 + 4)*e^(2*x) + 8*(4*x^4 - 7*x^2 +

4)*e^x + 64)

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giac [B] time = 0.35, size = 83, normalized size = 2.77 \begin {gather*} -\frac {x^{4} - x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} - 16 \, x^{3}}{4 \, x^{4} e^{\left (2 \, x\right )} + 32 \, x^{4} e^{x} + 64 \, x^{4} - 7 \, x^{2} e^{\left (2 \, x\right )} - 56 \, x^{2} e^{x} - 112 \, x^{2} + 4 \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 64} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^6-7*x^4+12*x^2)*exp(x)^3+(-48*x^6-84*x^4+144*x^2)*exp(x)^2+(8*x^8-206*x^6+14*x^5-328*x^4-16*x

^3+576*x^2)*exp(x)-256*x^6+56*x^5-448*x^4-64*x^3+768*x^2)/((16*x^8-56*x^6+81*x^4-56*x^2+16)*exp(x)^3+(192*x^8-

672*x^6+972*x^4-672*x^2+192)*exp(x)^2+(768*x^8-2688*x^6+3888*x^4-2688*x^2+768)*exp(x)+1024*x^8-3584*x^6+5184*x

^4-3584*x^2+1024),x, algorithm="giac")

[Out]

-(x^4 - x^3*e^(2*x) - 8*x^3*e^x - 16*x^3)/(4*x^4*e^(2*x) + 32*x^4*e^x + 64*x^4 - 7*x^2*e^(2*x) - 56*x^2*e^x -

112*x^2 + 4*e^(2*x) + 32*e^x + 64)

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maple [A] time = 0.05, size = 44, normalized size = 1.47


method	result	size

risch	\(\frac {x^{3}}{4 x^{4}-7 x^{2}+4}-\frac {x^{4}}{\left (4 x^{4}-7 x^{2}+4\right ) \left ({\mathrm e}^{x}+4\right )^{2}}\)	\(44\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^6-7*x^4+12*x^2)*exp(x)^3+(-48*x^6-84*x^4+144*x^2)*exp(x)^2+(8*x^8-206*x^6+14*x^5-328*x^4-16*x^3+576

*x^2)*exp(x)-256*x^6+56*x^5-448*x^4-64*x^3+768*x^2)/((16*x^8-56*x^6+81*x^4-56*x^2+16)*exp(x)^3+(192*x^8-672*x^

6+972*x^4-672*x^2+192)*exp(x)^2+(768*x^8-2688*x^6+3888*x^4-2688*x^2+768)*exp(x)+1024*x^8-3584*x^6+5184*x^4-358

4*x^2+1024),x,method=_RETURNVERBOSE)

[Out]

1/4*x^3/(x^4-7/4*x^2+1)-x^4/(4*x^4-7*x^2+4)/(exp(x)+4)^2

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maxima [B] time = 1.00, size = 74, normalized size = 2.47 \begin {gather*} -\frac {x^{4} - x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} - 16 \, x^{3}}{64 \, x^{4} - 112 \, x^{2} + {\left (4 \, x^{4} - 7 \, x^{2} + 4\right )} e^{\left (2 \, x\right )} + 8 \, {\left (4 \, x^{4} - 7 \, x^{2} + 4\right )} e^{x} + 64} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^6-7*x^4+12*x^2)*exp(x)^3+(-48*x^6-84*x^4+144*x^2)*exp(x)^2+(8*x^8-206*x^6+14*x^5-328*x^4-16*x

^3+576*x^2)*exp(x)-256*x^6+56*x^5-448*x^4-64*x^3+768*x^2)/((16*x^8-56*x^6+81*x^4-56*x^2+16)*exp(x)^3+(192*x^8-

672*x^6+972*x^4-672*x^2+192)*exp(x)^2+(768*x^8-2688*x^6+3888*x^4-2688*x^2+768)*exp(x)+1024*x^8-3584*x^6+5184*x

^4-3584*x^2+1024),x, algorithm="maxima")

[Out]

-(x^4 - x^3*e^(2*x) - 8*x^3*e^x - 16*x^3)/(64*x^4 - 112*x^2 + (4*x^4 - 7*x^2 + 4)*e^(2*x) + 8*(4*x^4 - 7*x^2 +

4)*e^x + 64)

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mupad [B] time = 2.36, size = 37, normalized size = 1.23 \begin {gather*} \frac {x^3\,\left ({\mathrm {e}}^{2\,x}-x+8\,{\mathrm {e}}^x+16\right )}{{\left ({\mathrm {e}}^x+4\right )}^2\,\left (4\,x^4-7\,x^2+4\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3*x)*(7*x^4 - 12*x^2 + 4*x^6) + exp(2*x)*(84*x^4 - 144*x^2 + 48*x^6) - 768*x^2 + 64*x^3 + 448*x^4 -

56*x^5 + 256*x^6 - exp(x)*(576*x^2 - 16*x^3 - 328*x^4 + 14*x^5 - 206*x^6 + 8*x^8))/(exp(x)*(3888*x^4 - 2688*x^

2 - 2688*x^6 + 768*x^8 + 768) + exp(3*x)*(81*x^4 - 56*x^2 - 56*x^6 + 16*x^8 + 16) + exp(2*x)*(972*x^4 - 672*x^

2 - 672*x^6 + 192*x^8 + 192) - 3584*x^2 + 5184*x^4 - 3584*x^6 + 1024*x^8 + 1024),x)

[Out]

(x^3*(exp(2*x) - x + 8*exp(x) + 16))/((exp(x) + 4)^2*(4*x^4 - 7*x^2 + 4))

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sympy [B] time = 0.31, size = 61, normalized size = 2.03 \begin {gather*} - \frac {x^{4}}{64 x^{4} - 112 x^{2} + \left (4 x^{4} - 7 x^{2} + 4\right ) e^{2 x} + \left (32 x^{4} - 56 x^{2} + 32\right ) e^{x} + 64} + \frac {x^{3}}{4 x^{4} - 7 x^{2} + 4} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**6-7*x**4+12*x**2)*exp(x)**3+(-48*x**6-84*x**4+144*x**2)*exp(x)**2+(8*x**8-206*x**6+14*x**5-3

28*x**4-16*x**3+576*x**2)*exp(x)-256*x**6+56*x**5-448*x**4-64*x**3+768*x**2)/((16*x**8-56*x**6+81*x**4-56*x**2

+16)*exp(x)**3+(192*x**8-672*x**6+972*x**4-672*x**2+192)*exp(x)**2+(768*x**8-2688*x**6+3888*x**4-2688*x**2+768

)*exp(x)+1024*x**8-3584*x**6+5184*x**4-3584*x**2+1024),x)

[Out]

-x**4/(64*x**4 - 112*x**2 + (4*x**4 - 7*x**2 + 4)*exp(2*x) + (32*x**4 - 56*x**2 + 32)*exp(x) + 64) + x**3/(4*x

**4 - 7*x**2 + 4)

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