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3.42.91 \(\int \frac {200 x+300 x^2+80 x^3+120 x^4+8 x^5+12 x^6+e^{\frac {10}{5+x^2}} (-50 x-20 x^3-2 x^5)+e^{\frac {5}{5+x^2}} (150 x-75 x^2+10 x^3-40 x^4+6 x^5-3 x^6)+e^x (-200 x-100 x^2-80 x^3-40 x^4-8 x^5-4 x^6+e^{\frac {5}{5+x^2}} (50 x+25 x^2+30 x^3+10 x^4+2 x^5+x^6))}{800+420 x^2+100 x^3+72 x^4+40 x^5+4 x^6+4 x^7+e^{\frac {10}{5+x^2}} (50-5 x^2-8 x^4-x^6)+e^{\frac {5}{5+x^2}} (-400-85 x^2-25 x^3+14 x^4-10 x^5+3 x^6-x^7)+e^x (-100 x^2-40 x^4-4 x^6+e^{\frac {5}{5+x^2}} (25 x^2+10 x^4+x^6))} \, dx\)

Optimal. Leaf size=36 \[ \log \left (-2+x^2-\frac {x^2 \left (5-e^x+x\right )}{4-e^{\frac {5}{5+x^2}}}\right ) \]

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Rubi [F]  time = 128.61, 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 {200 x+300 x^2+80 x^3+120 x^4+8 x^5+12 x^6+e^{\frac {10}{5+x^2}} \left (-50 x-20 x^3-2 x^5\right )+e^{\frac {5}{5+x^2}} \left (150 x-75 x^2+10 x^3-40 x^4+6 x^5-3 x^6\right )+e^x \left (-200 x-100 x^2-80 x^3-40 x^4-8 x^5-4 x^6+e^{\frac {5}{5+x^2}} \left (50 x+25 x^2+30 x^3+10 x^4+2 x^5+x^6\right )\right )}{800+420 x^2+100 x^3+72 x^4+40 x^5+4 x^6+4 x^7+e^{\frac {10}{5+x^2}} \left (50-5 x^2-8 x^4-x^6\right )+e^{\frac {5}{5+x^2}} \left (-400-85 x^2-25 x^3+14 x^4-10 x^5+3 x^6-x^7\right )+e^x \left (-100 x^2-40 x^4-4 x^6+e^{\frac {5}{5+x^2}} \left (25 x^2+10 x^4+x^6\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(200*x + 300*x^2 + 80*x^3 + 120*x^4 + 8*x^5 + 12*x^6 + E^(10/(5 + x^2))*(-50*x - 20*x^3 - 2*x^5) + E^(5/(5
 + x^2))*(150*x - 75*x^2 + 10*x^3 - 40*x^4 + 6*x^5 - 3*x^6) + E^x*(-200*x - 100*x^2 - 80*x^3 - 40*x^4 - 8*x^5
- 4*x^6 + E^(5/(5 + x^2))*(50*x + 25*x^2 + 30*x^3 + 10*x^4 + 2*x^5 + x^6)))/(800 + 420*x^2 + 100*x^3 + 72*x^4
+ 40*x^5 + 4*x^6 + 4*x^7 + E^(10/(5 + x^2))*(50 - 5*x^2 - 8*x^4 - x^6) + E^(5/(5 + x^2))*(-400 - 85*x^2 - 25*x
^3 + 14*x^4 - 10*x^5 + 3*x^6 - x^7) + E^x*(-100*x^2 - 40*x^4 - 4*x^6 + E^(5/(5 + x^2))*(25*x^2 + 10*x^4 + x^6)
)),x]

[Out]

5/(5 + x^2) - Log[4 - E^(5/(5 + x^2))] + Log[2 - x^2] + (50*Defer[Int][E^x/((Sqrt[2] - x)*(-8 + 2*E^(5/(5 + x^
2)) - x^2 + E^x*x^2 - E^(5/(5 + x^2))*x^2 - x^3)), x])/49 - (50*Defer[Int][E^x/((Sqrt[2] + x)*(-8 + 2*E^(5/(5
+ x^2)) - x^2 + E^x*x^2 - E^(5/(5 + x^2))*x^2 - x^3)), x])/49 + (50*Defer[Int][E^x/((I*Sqrt[5] + x)*(-8 + 2*E^
(5/(5 + x^2)) - x^2 + E^x*x^2 - E^(5/(5 + x^2))*x^2 - x^3)), x])/49 + (100*Defer[Int][(E^x*x)/((5 + x^2)^2*(-8
 + 2*E^(5/(5 + x^2)) - x^2 + E^x*x^2 - E^(5/(5 + x^2))*x^2 - x^3)), x])/7 + 6*Defer[Int][(8 - 2*E^(5/(5 + x^2)
) + x^2 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)^(-1), x] + 10*Defer[Int][1/((Sqrt[2] - x)*(8 - 2*E^(5/(5 + x^2)
) + x^2 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x] + 2*Sqrt[2]*Defer[Int][1/((Sqrt[2] - x)*(8 - 2*E^(5/(5 + x
^2)) + x^2 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x] - (48*Defer[Int][E^x/((Sqrt[2] - x)*(8 - 2*E^(5/(5 + x^
2)) + x^2 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x])/49 - 5*Defer[Int][1/((I*Sqrt[5] - x)*(8 - 2*E^(5/(5 + x
^2)) + x^2 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x] - (10*I)*Sqrt[5]*Defer[Int][1/((I*Sqrt[5] - x)*(8 - 2*E
^(5/(5 + x^2)) + x^2 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x] + (195*Defer[Int][E^x/((I*Sqrt[5] - x)*(8 - 2
*E^(5/(5 + x^2)) + x^2 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x])/49 + Defer[Int][x^2/(8 - 2*E^(5/(5 + x^2))
 + x^2 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3), x] - Defer[Int][(E^x*x^2)/(8 - 2*E^(5/(5 + x^2)) + x^2 - E^x*x^
2 + E^(5/(5 + x^2))*x^2 + x^3), x] - 10*Defer[Int][1/((Sqrt[2] + x)*(8 - 2*E^(5/(5 + x^2)) + x^2 - E^x*x^2 + E
^(5/(5 + x^2))*x^2 + x^3)), x] + 2*Sqrt[2]*Defer[Int][1/((Sqrt[2] + x)*(8 - 2*E^(5/(5 + x^2)) + x^2 - E^x*x^2
+ E^(5/(5 + x^2))*x^2 + x^3)), x] + (48*Defer[Int][E^x/((Sqrt[2] + x)*(8 - 2*E^(5/(5 + x^2)) + x^2 - E^x*x^2 +
 E^(5/(5 + x^2))*x^2 + x^3)), x])/49 + 5*Defer[Int][1/((I*Sqrt[5] + x)*(8 - 2*E^(5/(5 + x^2)) + x^2 - E^x*x^2
+ E^(5/(5 + x^2))*x^2 + x^3)), x] - (10*I)*Sqrt[5]*Defer[Int][1/((I*Sqrt[5] + x)*(8 - 2*E^(5/(5 + x^2)) + x^2
- E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x] - (195*Defer[Int][E^x/((I*Sqrt[5] + x)*(8 - 2*E^(5/(5 + x^2)) + x^
2 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x])/49 + 250*Defer[Int][1/((5 + x^2)^2*(8 - 2*E^(5/(5 + x^2)) + x^2
 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x] + 30*Defer[Int][x/((5 + x^2)^2*(8 - 2*E^(5/(5 + x^2)) + x^2 - E^x
*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x] + (450*Defer[Int][(E^x*x)/((5 + x^2)^2*(8 - 2*E^(5/(5 + x^2)) + x^2 - E
^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3)), x])/7 - (50*Defer[Int][E^x/(((-I)*Sqrt[5] + x)*(8 + x^2 - E^x*x^2 + x^3
+ E^(5/(5 + x^2))*(-2 + x^2))), x])/49

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (-2 e^{\frac {10}{5+x^2}} \left (5+x^2\right )^2-4 e^x (2+x) \left (5+x^2\right )^2+4 (2+3 x) \left (5+x^2\right )^2+e^{\frac {5}{5+x^2}} \left (150-75 x+10 x^2-40 x^3+6 x^4-3 x^5\right )+e^{x+\frac {5}{5+x^2}} \left (50+25 x+30 x^2+10 x^3+2 x^4+x^5\right )\right )}{\left (4-e^{\frac {5}{5+x^2}}\right ) \left (5+x^2\right )^2 \left (8+x^2-e^x x^2+x^3+e^{\frac {5}{5+x^2}} \left (-2+x^2\right )\right )} \, dx\\ &=\int \left (\frac {2 x}{-2+x^2}+\frac {40 x}{\left (-4+e^{\frac {5}{5+x^2}}\right ) \left (5+x^2\right )^2}-\frac {x \left (660-100 e^x+150 x-50 e^x x+140 x^2-60 e^x x^2+55 x^3+5 e^x x^3+10 x^4+6 e^x x^4-14 x^5+8 e^x x^5-x^7+e^x x^7\right )}{\left (-2+x^2\right ) \left (5+x^2\right )^2 \left (8-2 e^{\frac {5}{5+x^2}}+x^2-e^x x^2+e^{\frac {5}{5+x^2}} x^2+x^3\right )}\right ) \, dx\\ &=2 \int \frac {x}{-2+x^2} \, dx+40 \int \frac {x}{\left (-4+e^{\frac {5}{5+x^2}}\right ) \left (5+x^2\right )^2} \, dx-\int \frac {x \left (660-100 e^x+150 x-50 e^x x+140 x^2-60 e^x x^2+55 x^3+5 e^x x^3+10 x^4+6 e^x x^4-14 x^5+8 e^x x^5-x^7+e^x x^7\right )}{\left (-2+x^2\right ) \left (5+x^2\right )^2 \left (8-2 e^{\frac {5}{5+x^2}}+x^2-e^x x^2+e^{\frac {5}{5+x^2}} x^2+x^3\right )} \, dx\\ &=\log \left (2-x^2\right )+20 \operatorname {Subst}\left (\int \frac {1}{\left (-4+e^{\frac {5}{5+x}}\right ) (5+x)^2} \, dx,x,x^2\right )-\int \left (\frac {x \left (660-100 e^x+150 x-50 e^x x+140 x^2-60 e^x x^2+55 x^3+5 e^x x^3+10 x^4+6 e^x x^4-14 x^5+8 e^x x^5-x^7+e^x x^7\right )}{49 \left (-2+x^2\right ) \left (8-2 e^{\frac {5}{5+x^2}}+x^2-e^x x^2+e^{\frac {5}{5+x^2}} x^2+x^3\right )}-\frac {x \left (660-100 e^x+150 x-50 e^x x+140 x^2-60 e^x x^2+55 x^3+5 e^x x^3+10 x^4+6 e^x x^4-14 x^5+8 e^x x^5-x^7+e^x x^7\right )}{7 \left (5+x^2\right )^2 \left (8-2 e^{\frac {5}{5+x^2}}+x^2-e^x x^2+e^{\frac {5}{5+x^2}} x^2+x^3\right )}-\frac {x \left (660-100 e^x+150 x-50 e^x x+140 x^2-60 e^x x^2+55 x^3+5 e^x x^3+10 x^4+6 e^x x^4-14 x^5+8 e^x x^5-x^7+e^x x^7\right )}{49 \left (5+x^2\right ) \left (8-2 e^{\frac {5}{5+x^2}}+x^2-e^x x^2+e^{\frac {5}{5+x^2}} x^2+x^3\right )}\right ) \, dx\\ &=\log \left (2-x^2\right )-\frac {1}{49} \int \frac {x \left (660-100 e^x+150 x-50 e^x x+140 x^2-60 e^x x^2+55 x^3+5 e^x x^3+10 x^4+6 e^x x^4-14 x^5+8 e^x x^5-x^7+e^x x^7\right )}{\left (-2+x^2\right ) \left (8-2 e^{\frac {5}{5+x^2}}+x^2-e^x x^2+e^{\frac {5}{5+x^2}} x^2+x^3\right )} \, dx+\frac {1}{49} \int \frac {x \left (660-100 e^x+150 x-50 e^x x+140 x^2-60 e^x x^2+55 x^3+5 e^x x^3+10 x^4+6 e^x x^4-14 x^5+8 e^x x^5-x^7+e^x x^7\right )}{\left (5+x^2\right ) \left (8-2 e^{\frac {5}{5+x^2}}+x^2-e^x x^2+e^{\frac {5}{5+x^2}} x^2+x^3\right )} \, dx+\frac {1}{7} \int \frac {x \left (660-100 e^x+150 x-50 e^x x+140 x^2-60 e^x x^2+55 x^3+5 e^x x^3+10 x^4+6 e^x x^4-14 x^5+8 e^x x^5-x^7+e^x x^7\right )}{\left (5+x^2\right )^2 \left (8-2 e^{\frac {5}{5+x^2}}+x^2-e^x x^2+e^{\frac {5}{5+x^2}} x^2+x^3\right )} \, dx+20 \operatorname {Subst}\left (\int \frac {1}{\left (-4+e^{5/x}\right ) x^2} \, dx,x,5+x^2\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 64, normalized size = 1.78 \begin {gather*} -\log \left (4-e^{\frac {5}{5+x^2}}\right )+\log \left (8-2 e^{\frac {5}{5+x^2}}+x^2-e^x x^2+e^{\frac {5}{5+x^2}} x^2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(200*x + 300*x^2 + 80*x^3 + 120*x^4 + 8*x^5 + 12*x^6 + E^(10/(5 + x^2))*(-50*x - 20*x^3 - 2*x^5) + E
^(5/(5 + x^2))*(150*x - 75*x^2 + 10*x^3 - 40*x^4 + 6*x^5 - 3*x^6) + E^x*(-200*x - 100*x^2 - 80*x^3 - 40*x^4 -
8*x^5 - 4*x^6 + E^(5/(5 + x^2))*(50*x + 25*x^2 + 30*x^3 + 10*x^4 + 2*x^5 + x^6)))/(800 + 420*x^2 + 100*x^3 + 7
2*x^4 + 40*x^5 + 4*x^6 + 4*x^7 + E^(10/(5 + x^2))*(50 - 5*x^2 - 8*x^4 - x^6) + E^(5/(5 + x^2))*(-400 - 85*x^2
- 25*x^3 + 14*x^4 - 10*x^5 + 3*x^6 - x^7) + E^x*(-100*x^2 - 40*x^4 - 4*x^6 + E^(5/(5 + x^2))*(25*x^2 + 10*x^4
+ x^6))),x]

[Out]

-Log[4 - E^(5/(5 + x^2))] + Log[8 - 2*E^(5/(5 + x^2)) + x^2 - E^x*x^2 + E^(5/(5 + x^2))*x^2 + x^3]

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fricas [A]  time = 2.77, size = 57, normalized size = 1.58 \begin {gather*} 2 \, \log \relax (x) + \log \left (-\frac {x^{3} - x^{2} e^{x} + x^{2} + {\left (x^{2} - 2\right )} e^{\left (\frac {5}{x^{2} + 5}\right )} + 8}{x^{2}}\right ) - \log \left (e^{\left (\frac {5}{x^{2} + 5}\right )} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^6+2*x^5+10*x^4+30*x^3+25*x^2+50*x)*exp(5/(x^2+5))-4*x^6-8*x^5-40*x^4-80*x^3-100*x^2-200*x)*exp(
x)+(-2*x^5-20*x^3-50*x)*exp(5/(x^2+5))^2+(-3*x^6+6*x^5-40*x^4+10*x^3-75*x^2+150*x)*exp(5/(x^2+5))+12*x^6+8*x^5
+120*x^4+80*x^3+300*x^2+200*x)/(((x^6+10*x^4+25*x^2)*exp(5/(x^2+5))-4*x^6-40*x^4-100*x^2)*exp(x)+(-x^6-8*x^4-5
*x^2+50)*exp(5/(x^2+5))^2+(-x^7+3*x^6-10*x^5+14*x^4-25*x^3-85*x^2-400)*exp(5/(x^2+5))+4*x^7+4*x^6+40*x^5+72*x^
4+100*x^3+420*x^2+800),x, algorithm="fricas")

[Out]

2*log(x) + log(-(x^3 - x^2*e^x + x^2 + (x^2 - 2)*e^(5/(x^2 + 5)) + 8)/x^2) - log(e^(5/(x^2 + 5)) - 4)

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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((((x^6+2*x^5+10*x^4+30*x^3+25*x^2+50*x)*exp(5/(x^2+5))-4*x^6-8*x^5-40*x^4-80*x^3-100*x^2-200*x)*exp(
x)+(-2*x^5-20*x^3-50*x)*exp(5/(x^2+5))^2+(-3*x^6+6*x^5-40*x^4+10*x^3-75*x^2+150*x)*exp(5/(x^2+5))+12*x^6+8*x^5
+120*x^4+80*x^3+300*x^2+200*x)/(((x^6+10*x^4+25*x^2)*exp(5/(x^2+5))-4*x^6-40*x^4-100*x^2)*exp(x)+(-x^6-8*x^4-5
*x^2+50)*exp(5/(x^2+5))^2+(-x^7+3*x^6-10*x^5+14*x^4-25*x^3-85*x^2-400)*exp(5/(x^2+5))+4*x^7+4*x^6+40*x^5+72*x^
4+100*x^3+420*x^2+800),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.10, size = 58, normalized size = 1.61

method result size
risch \(\ln \left (x^{2}-2\right )+\ln \left ({\mathrm e}^{\frac {5}{x^{2}+5}}+\frac {x^{3}-{\mathrm e}^{x} x^{2}+x^{2}+8}{x^{2}-2}\right )-\ln \left ({\mathrm e}^{\frac {5}{x^{2}+5}}-4\right )\) \(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^6+2*x^5+10*x^4+30*x^3+25*x^2+50*x)*exp(5/(x^2+5))-4*x^6-8*x^5-40*x^4-80*x^3-100*x^2-200*x)*exp(x)+(-2
*x^5-20*x^3-50*x)*exp(5/(x^2+5))^2+(-3*x^6+6*x^5-40*x^4+10*x^3-75*x^2+150*x)*exp(5/(x^2+5))+12*x^6+8*x^5+120*x
^4+80*x^3+300*x^2+200*x)/(((x^6+10*x^4+25*x^2)*exp(5/(x^2+5))-4*x^6-40*x^4-100*x^2)*exp(x)+(-x^6-8*x^4-5*x^2+5
0)*exp(5/(x^2+5))^2+(-x^7+3*x^6-10*x^5+14*x^4-25*x^3-85*x^2-400)*exp(5/(x^2+5))+4*x^7+4*x^6+40*x^5+72*x^4+100*
x^3+420*x^2+800),x,method=_RETURNVERBOSE)

[Out]

ln(x^2-2)+ln(exp(5/(x^2+5))+(x^3-exp(x)*x^2+x^2+8)/(x^2-2))-ln(exp(5/(x^2+5))-4)

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maxima [A]  time = 0.56, size = 62, normalized size = 1.72 \begin {gather*} \log \left (x^{2} - 2\right ) + \log \left (\frac {x^{3} - x^{2} e^{x} + x^{2} + {\left (x^{2} - 2\right )} e^{\left (\frac {5}{x^{2} + 5}\right )} + 8}{x^{2} - 2}\right ) - \log \left (e^{\left (\frac {5}{x^{2} + 5}\right )} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^6+2*x^5+10*x^4+30*x^3+25*x^2+50*x)*exp(5/(x^2+5))-4*x^6-8*x^5-40*x^4-80*x^3-100*x^2-200*x)*exp(
x)+(-2*x^5-20*x^3-50*x)*exp(5/(x^2+5))^2+(-3*x^6+6*x^5-40*x^4+10*x^3-75*x^2+150*x)*exp(5/(x^2+5))+12*x^6+8*x^5
+120*x^4+80*x^3+300*x^2+200*x)/(((x^6+10*x^4+25*x^2)*exp(5/(x^2+5))-4*x^6-40*x^4-100*x^2)*exp(x)+(-x^6-8*x^4-5
*x^2+50)*exp(5/(x^2+5))^2+(-x^7+3*x^6-10*x^5+14*x^4-25*x^3-85*x^2-400)*exp(5/(x^2+5))+4*x^7+4*x^6+40*x^5+72*x^
4+100*x^3+420*x^2+800),x, algorithm="maxima")

[Out]

log(x^2 - 2) + log((x^3 - x^2*e^x + x^2 + (x^2 - 2)*e^(5/(x^2 + 5)) + 8)/(x^2 - 2)) - log(e^(5/(x^2 + 5)) - 4)

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mupad [B]  time = 4.82, size = 528, normalized size = 14.67 \begin {gather*} \ln \left (x^7+8\,x^5+6\,x^4+5\,x^3-60\,x^2-50\,x-100\right )-\ln \left (\frac {5280\,x-400\,x^2\,{\mathrm {e}}^x-480\,x^3\,{\mathrm {e}}^x+40\,x^4\,{\mathrm {e}}^x+48\,x^5\,{\mathrm {e}}^x+64\,x^6\,{\mathrm {e}}^x+8\,x^8\,{\mathrm {e}}^x-1320\,x\,{\mathrm {e}}^{\frac {5}{x^2+5}}-300\,x^2\,{\mathrm {e}}^{\frac {5}{x^2+5}}-280\,x^3\,{\mathrm {e}}^{\frac {5}{x^2+5}}-110\,x^4\,{\mathrm {e}}^{\frac {5}{x^2+5}}-20\,x^5\,{\mathrm {e}}^{\frac {5}{x^2+5}}+28\,x^6\,{\mathrm {e}}^{\frac {5}{x^2+5}}+2\,x^8\,{\mathrm {e}}^{\frac {5}{x^2+5}}-800\,x\,{\mathrm {e}}^x+1200\,x^2+1120\,x^3+440\,x^4+80\,x^5-112\,x^6-8\,x^8+200\,x\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x+100\,x^2\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x+120\,x^3\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x-10\,x^4\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x-12\,x^5\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x-16\,x^6\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x-2\,x^8\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x}{x^8+6\,x^6-11\,x^4-60\,x^2+100}\right )-\ln \left (x^2-2\right )+\ln \left (\frac {150\,x-100\,{\mathrm {e}}^x-60\,x^2\,{\mathrm {e}}^x+5\,x^3\,{\mathrm {e}}^x+6\,x^4\,{\mathrm {e}}^x+8\,x^5\,{\mathrm {e}}^x+x^7\,{\mathrm {e}}^x-50\,x\,{\mathrm {e}}^x+140\,x^2+55\,x^3+10\,x^4-14\,x^5-x^7+660}{x^7+8\,x^5+6\,x^4+5\,x^3-60\,x^2-50\,x-100}\right )+\ln \left (\frac {x\,\left (x^2\,{\mathrm {e}}^{\frac {5}{x^2+5}}-x^2\,{\mathrm {e}}^x-2\,{\mathrm {e}}^{\frac {5}{x^2+5}}+x^2+x^3+8\right )}{\left (x^2-2\right )\,{\left (x^2+5\right )}^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((200*x + exp(5/(x^2 + 5))*(150*x - 75*x^2 + 10*x^3 - 40*x^4 + 6*x^5 - 3*x^6) - exp(x)*(200*x - exp(5/(x^2
+ 5))*(50*x + 25*x^2 + 30*x^3 + 10*x^4 + 2*x^5 + x^6) + 100*x^2 + 80*x^3 + 40*x^4 + 8*x^5 + 4*x^6) - exp(10/(x
^2 + 5))*(50*x + 20*x^3 + 2*x^5) + 300*x^2 + 80*x^3 + 120*x^4 + 8*x^5 + 12*x^6)/(420*x^2 - exp(10/(x^2 + 5))*(
5*x^2 + 8*x^4 + x^6 - 50) - exp(x)*(100*x^2 - exp(5/(x^2 + 5))*(25*x^2 + 10*x^4 + x^6) + 40*x^4 + 4*x^6) - exp
(5/(x^2 + 5))*(85*x^2 + 25*x^3 - 14*x^4 + 10*x^5 - 3*x^6 + x^7 + 400) + 100*x^3 + 72*x^4 + 40*x^5 + 4*x^6 + 4*
x^7 + 800),x)

[Out]

log(5*x^3 - 60*x^2 - 50*x + 6*x^4 + 8*x^5 + x^7 - 100) - log((5280*x - 400*x^2*exp(x) - 480*x^3*exp(x) + 40*x^
4*exp(x) + 48*x^5*exp(x) + 64*x^6*exp(x) + 8*x^8*exp(x) - 1320*x*exp(5/(x^2 + 5)) - 300*x^2*exp(5/(x^2 + 5)) -
 280*x^3*exp(5/(x^2 + 5)) - 110*x^4*exp(5/(x^2 + 5)) - 20*x^5*exp(5/(x^2 + 5)) + 28*x^6*exp(5/(x^2 + 5)) + 2*x
^8*exp(5/(x^2 + 5)) - 800*x*exp(x) + 1200*x^2 + 1120*x^3 + 440*x^4 + 80*x^5 - 112*x^6 - 8*x^8 + 200*x*exp(5/(x
^2 + 5))*exp(x) + 100*x^2*exp(5/(x^2 + 5))*exp(x) + 120*x^3*exp(5/(x^2 + 5))*exp(x) - 10*x^4*exp(5/(x^2 + 5))*
exp(x) - 12*x^5*exp(5/(x^2 + 5))*exp(x) - 16*x^6*exp(5/(x^2 + 5))*exp(x) - 2*x^8*exp(5/(x^2 + 5))*exp(x))/(6*x
^6 - 11*x^4 - 60*x^2 + x^8 + 100)) - log(x^2 - 2) + log((150*x - 100*exp(x) - 60*x^2*exp(x) + 5*x^3*exp(x) + 6
*x^4*exp(x) + 8*x^5*exp(x) + x^7*exp(x) - 50*x*exp(x) + 140*x^2 + 55*x^3 + 10*x^4 - 14*x^5 - x^7 + 660)/(5*x^3
 - 60*x^2 - 50*x + 6*x^4 + 8*x^5 + x^7 - 100)) + log((x*(x^2*exp(5/(x^2 + 5)) - x^2*exp(x) - 2*exp(5/(x^2 + 5)
) + x^2 + x^3 + 8))/((x^2 - 2)*(x^2 + 5)^2))

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**6+2*x**5+10*x**4+30*x**3+25*x**2+50*x)*exp(5/(x**2+5))-4*x**6-8*x**5-40*x**4-80*x**3-100*x**2-
200*x)*exp(x)+(-2*x**5-20*x**3-50*x)*exp(5/(x**2+5))**2+(-3*x**6+6*x**5-40*x**4+10*x**3-75*x**2+150*x)*exp(5/(
x**2+5))+12*x**6+8*x**5+120*x**4+80*x**3+300*x**2+200*x)/(((x**6+10*x**4+25*x**2)*exp(5/(x**2+5))-4*x**6-40*x*
*4-100*x**2)*exp(x)+(-x**6-8*x**4-5*x**2+50)*exp(5/(x**2+5))**2+(-x**7+3*x**6-10*x**5+14*x**4-25*x**3-85*x**2-
400)*exp(5/(x**2+5))+4*x**7+4*x**6+40*x**5+72*x**4+100*x**3+420*x**2+800),x)

[Out]

Exception raised: PolynomialError

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