3.93.7 \(\int \frac {50+20 x^2+2 x^4+(-100-20 x^2) \log (5)+50 \log ^2(5)+e^{\frac {9 x+x^3-5 x \log (5)}{-5-x^2+5 \log (5)}} (-45-6 x^2-x^4+(70+10 x^2) \log (5)-25 \log ^2(5))}{25+10 x^2+x^4+(-50-10 x^2) \log (5)+25 \log ^2(5)} \, dx\)

Optimal. Leaf size=30 \[ e^{-x+\frac {4}{-\frac {5}{x}-x+\frac {5 \log (5)}{x}}}+2 x \]

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Rubi [F]  time = 7.54, 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 {50+20 x^2+2 x^4+\left (-100-20 x^2\right ) \log (5)+50 \log ^2(5)+e^{\frac {9 x+x^3-5 x \log (5)}{-5-x^2+5 \log (5)}} \left (-45-6 x^2-x^4+\left (70+10 x^2\right ) \log (5)-25 \log ^2(5)\right )}{25+10 x^2+x^4+\left (-50-10 x^2\right ) \log (5)+25 \log ^2(5)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(50 + 20*x^2 + 2*x^4 + (-100 - 20*x^2)*Log[5] + 50*Log[5]^2 + E^((9*x + x^3 - 5*x*Log[5])/(-5 - x^2 + 5*Lo
g[5]))*(-45 - 6*x^2 - x^4 + (70 + 10*x^2)*Log[5] - 25*Log[5]^2))/(25 + 10*x^2 + x^4 + (-50 - 10*x^2)*Log[5] +
25*Log[5]^2),x]

[Out]

3*x - (10*x)/(x^2 + 5*(1 - Log[5])) - x^3/(x^2 + 5*(1 - Log[5])) - 2*ArcTanh[x/Sqrt[5*(-1 + Log[5])]]*Sqrt[5/(
-1 + Log[5])] - 3*ArcTanh[x/Sqrt[5*(-1 + Log[5])]]*Sqrt[5*(-1 + Log[5])] - (2*Sqrt[5]*ArcTanh[x/Sqrt[5*(-1 + L
og[5])]]*(2 - Log[5])*Log[5])/(-1 + Log[5])^(3/2) - (10*x*Log[5]^2)/((x^2 + 5*(1 - Log[5]))*(1 - Log[5])) + (5
*x*(1 + Log[5]^2))/((x^2 + 5*(1 - Log[5]))*(1 - Log[5])) + (Sqrt[5]*ArcTanh[x/Sqrt[5*(-1 + Log[5])]]*(1 + Log[
5]^2))/(-1 + Log[5])^(3/2) - Defer[Int][5^((5*x)/(5 + x^2 - 5*Log[5]))/E^((x*(9 + x^2))/(5 + x^2 - 5*Log[5])),
x] + 2*Defer[Int][5^((5*x)/(5 + x^2 - 5*Log[5]))/(E^((x*(9 + x^2))/(5 + x^2 - 5*Log[5]))*(-x + Sqrt[5*(-1 + L
og[5])])^2), x] + (2*Defer[Int][5^(-1/2 + (5*x)/(5 + x^2 - 5*Log[5]))/(E^((x*(9 + x^2))/(5 + x^2 - 5*Log[5]))*
(-x + Sqrt[5*(-1 + Log[5])])), x])/Sqrt[-1 + Log[5]] - (2*Defer[Int][5^(1/2 + (5*x)/(5 + x^2 - 5*Log[5]))/(E^(
(x*(9 + x^2))/(5 + x^2 - 5*Log[5]))*(-x + Sqrt[5*(-1 + Log[5])])), x])/(5*Sqrt[-1 + Log[5]]) + 2*Defer[Int][5^
((5*x)/(5 + x^2 - 5*Log[5]))/(E^((x*(9 + x^2))/(5 + x^2 - 5*Log[5]))*(x + Sqrt[5*(-1 + Log[5])])^2), x] + (2*D
efer[Int][5^(-1/2 + (5*x)/(5 + x^2 - 5*Log[5]))/(E^((x*(9 + x^2))/(5 + x^2 - 5*Log[5]))*(x + Sqrt[5*(-1 + Log[
5])])), x])/Sqrt[-1 + Log[5]] - (2*Defer[Int][5^(1/2 + (5*x)/(5 + x^2 - 5*Log[5]))/(E^((x*(9 + x^2))/(5 + x^2
- 5*Log[5]))*(x + Sqrt[5*(-1 + Log[5])])), x])/(5*Sqrt[-1 + Log[5]])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20 x^2+2 x^4+\left (-100-20 x^2\right ) \log (5)+e^{\frac {9 x+x^3-5 x \log (5)}{-5-x^2+5 \log (5)}} \left (-45-6 x^2-x^4+\left (70+10 x^2\right ) \log (5)-25 \log ^2(5)\right )+50 \left (1+\log ^2(5)\right )}{x^4+10 x^2 (1-\log (5))+25 (1-\log (5))^2} \, dx\\ &=\int \frac {20 x^2+2 x^4+\left (-100-20 x^2\right ) \log (5)+e^{\frac {9 x+x^3-5 x \log (5)}{-5-x^2+5 \log (5)}} \left (-45-6 x^2-x^4+\left (70+10 x^2\right ) \log (5)-25 \log ^2(5)\right )+50 \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right )^2} \, dx\\ &=\int \left (\frac {20 x^2}{\left (5+x^2-5 \log (5)\right )^2}+\frac {2 x^4}{\left (5+x^2-5 \log (5)\right )^2}+\frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \left (-x^4-2 x^2 (3-5 \log (5))-5 (9-5 \log (5)) (1-\log (5))\right )}{\left (5+x^2-5 \log (5)\right )^2}-\frac {20 \left (5+x^2\right ) \log (5)}{\left (5+x^2-5 \log (5)\right )^2}+\frac {50 \left (1+\log ^2(5)\right )}{\left (5+x^2-5 \log (5)\right )^2}\right ) \, dx\\ &=2 \int \frac {x^4}{\left (5+x^2-5 \log (5)\right )^2} \, dx+20 \int \frac {x^2}{\left (5+x^2-5 \log (5)\right )^2} \, dx-(20 \log (5)) \int \frac {5+x^2}{\left (5+x^2-5 \log (5)\right )^2} \, dx+\left (50 \left (1+\log ^2(5)\right )\right ) \int \frac {1}{\left (5+x^2-5 \log (5)\right )^2} \, dx+\int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \left (-x^4-2 x^2 (3-5 \log (5))-5 (9-5 \log (5)) (1-\log (5))\right )}{\left (5+x^2-5 \log (5)\right )^2} \, dx\\ &=-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+3 \int \frac {x^2}{5+x^2-5 \log (5)} \, dx+10 \int \frac {1}{5+x^2-5 \log (5)} \, dx-\frac {(10 (2-\log (5)) \log (5)) \int \frac {1}{5+x^2-5 \log (5)} \, dx}{1-\log (5)}+\frac {\left (5 \left (1+\log ^2(5)\right )\right ) \int \frac {1}{5+x^2-5 \log (5)} \, dx}{1-\log (5)}+\int \left (-5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}+\frac {4\ 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{5+x^2-5 \log (5)}+\frac {8\ 5^{1+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} (-1+\log (5))}{\left (5+x^2-5 \log (5)\right )^2}\right ) \, dx\\ &=3 x-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {\frac {5}{-1+\log (5)}}-\frac {2 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) (2-\log (5)) \log (5)}{(-1+\log (5))^{3/2}}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {\sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \left (1+\log ^2(5)\right )}{(-1+\log (5))^{3/2}}+4 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{5+x^2-5 \log (5)} \, dx-(8 (1-\log (5))) \int \frac {5^{1+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (5+x^2-5 \log (5)\right )^2} \, dx-(15 (1-\log (5))) \int \frac {1}{5+x^2-5 \log (5)} \, dx-\int 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \, dx\\ &=3 x-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {\frac {5}{-1+\log (5)}}-3 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {5 (-1+\log (5))}-\frac {2 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) (2-\log (5)) \log (5)}{(-1+\log (5))^{3/2}}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {\sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \left (1+\log ^2(5)\right )}{(-1+\log (5))^{3/2}}+4 \int \left (\frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \sqrt {-1+\log (5)}}{2 \left (-x+\sqrt {5 (-1+\log (5))}\right ) (5-5 \log (5))}+\frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \sqrt {-1+\log (5)}}{2 \left (x+\sqrt {5 (-1+\log (5))}\right ) (5-5 \log (5))}\right ) \, dx-(8 (1-\log (5))) \int \left (\frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{2 \left (-x^2+5 (-1+\log (5))\right ) (-1+\log (5))}+\frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{4 \left (-x+\sqrt {5 (-1+\log (5))}\right )^2 (-1+\log (5))}+\frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{4 \left (x+\sqrt {5 (-1+\log (5))}\right )^2 (-1+\log (5))}\right ) \, dx-\int 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \, dx\\ &=3 x-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {\frac {5}{-1+\log (5)}}-3 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {5 (-1+\log (5))}-\frac {2 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) (2-\log (5)) \log (5)}{(-1+\log (5))^{3/2}}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {\sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \left (1+\log ^2(5)\right )}{(-1+\log (5))^{3/2}}+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (-x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx+4 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{-x^2+5 (-1+\log (5))} \, dx-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{-x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}-\int 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \, dx\\ &=3 x-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {\frac {5}{-1+\log (5)}}-3 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {5 (-1+\log (5))}-\frac {2 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) (2-\log (5)) \log (5)}{(-1+\log (5))^{3/2}}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {\sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \left (1+\log ^2(5)\right )}{(-1+\log (5))^{3/2}}+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (-x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx+4 \int \left (\frac {5^{-\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{2 \left (-x+\sqrt {5 (-1+\log (5))}\right ) \sqrt {-1+\log (5)}}+\frac {5^{-\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{2 \left (x+\sqrt {5 (-1+\log (5))}\right ) \sqrt {-1+\log (5)}}\right ) \, dx-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{-x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}-\int 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \, dx\\ &=3 x-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {\frac {5}{-1+\log (5)}}-3 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {5 (-1+\log (5))}-\frac {2 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) (2-\log (5)) \log (5)}{(-1+\log (5))^{3/2}}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {\sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \left (1+\log ^2(5)\right )}{(-1+\log (5))^{3/2}}+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (-x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{-x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}+\frac {2 \int \frac {5^{-\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{-x+\sqrt {5 (-1+\log (5))}} \, dx}{\sqrt {-1+\log (5)}}+\frac {2 \int \frac {5^{-\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{x+\sqrt {5 (-1+\log (5))}} \, dx}{\sqrt {-1+\log (5)}}-\int 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 24, normalized size = 0.80 \begin {gather*} e^{-x-\frac {4 x}{5+x^2-5 \log (5)}}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(50 + 20*x^2 + 2*x^4 + (-100 - 20*x^2)*Log[5] + 50*Log[5]^2 + E^((9*x + x^3 - 5*x*Log[5])/(-5 - x^2
+ 5*Log[5]))*(-45 - 6*x^2 - x^4 + (70 + 10*x^2)*Log[5] - 25*Log[5]^2))/(25 + 10*x^2 + x^4 + (-50 - 10*x^2)*Log
[5] + 25*Log[5]^2),x]

[Out]

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

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fricas [A]  time = 1.06, size = 30, normalized size = 1.00 \begin {gather*} 2 \, x + e^{\left (-\frac {x^{3} - 5 \, x \log \relax (5) + 9 \, x}{x^{2} - 5 \, \log \relax (5) + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*log(5)^2+(10*x^2+70)*log(5)-x^4-6*x^2-45)*exp((-5*x*log(5)+x^3+9*x)/(5*log(5)-x^2-5))+50*log(5
)^2+(-20*x^2-100)*log(5)+2*x^4+20*x^2+50)/(25*log(5)^2+(-10*x^2-50)*log(5)+x^4+10*x^2+25),x, algorithm="fricas
")

[Out]

2*x + e^(-(x^3 - 5*x*log(5) + 9*x)/(x^2 - 5*log(5) + 5))

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giac [A]  time = 0.37, size = 30, normalized size = 1.00 \begin {gather*} 2 \, x + e^{\left (-\frac {x^{3} - 5 \, x \log \relax (5) + 9 \, x}{x^{2} - 5 \, \log \relax (5) + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*log(5)^2+(10*x^2+70)*log(5)-x^4-6*x^2-45)*exp((-5*x*log(5)+x^3+9*x)/(5*log(5)-x^2-5))+50*log(5
)^2+(-20*x^2-100)*log(5)+2*x^4+20*x^2+50)/(25*log(5)^2+(-10*x^2-50)*log(5)+x^4+10*x^2+25),x, algorithm="giac")

[Out]

2*x + e^(-(x^3 - 5*x*log(5) + 9*x)/(x^2 - 5*log(5) + 5))

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maple [A]  time = 0.31, size = 33, normalized size = 1.10

method	result	size
risch	\(2 x +{\mathrm e}^{-\frac {x \left (-x^{2}+5 \ln \relax (5)-9\right )}{5 \ln \relax (5)-x^{2}-5}}\)	\(33\)
norman	\(\frac {\left (5 \ln \relax (5)-5\right ) {\mathrm e}^{\frac {-5 x \ln \relax (5)+x^{3}+9 x}{5 \ln \relax (5)-x^{2}-5}}+\left (10 \ln \relax (5)-10\right ) x -2 x^{3}-x^{2} {\mathrm e}^{\frac {-5 x \ln \relax (5)+x^{3}+9 x}{5 \ln \relax (5)-x^{2}-5}}}{5 \ln \relax (5)-x^{2}-5}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-25*ln(5)^2+(10*x^2+70)*ln(5)-x^4-6*x^2-45)*exp((-5*x*ln(5)+x^3+9*x)/(5*ln(5)-x^2-5))+50*ln(5)^2+(-20*x^
2-100)*ln(5)+2*x^4+20*x^2+50)/(25*ln(5)^2+(-10*x^2-50)*ln(5)+x^4+10*x^2+25),x,method=_RETURNVERBOSE)

[Out]

2*x+exp(-x*(-x^2+5*ln(5)-9)/(5*ln(5)-x^2-5))

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maxima [B]  time = 0.57, size = 396, normalized size = 13.20 \begin {gather*} -\frac {5}{2} \, {\left (\frac {2 \, x}{x^{2} {\left (\log \relax (5) - 1\right )} - 5 \, \log \relax (5)^{2} + 10 \, \log \relax (5) - 5} + \frac {\log \left (\frac {x - \sqrt {5 \, \log \relax (5) - 5}}{x + \sqrt {5 \, \log \relax (5) - 5}}\right )}{\sqrt {5 \, \log \relax (5) - 5} {\left (\log \relax (5) - 1\right )}}\right )} \log \relax (5)^{2} - 5 \, {\left (\frac {\log \left (\frac {x - \sqrt {5 \, \log \relax (5) - 5}}{x + \sqrt {5 \, \log \relax (5) - 5}}\right )}{\sqrt {5 \, \log \relax (5) - 5}} - \frac {2 \, x}{x^{2} - 5 \, \log \relax (5) + 5}\right )} \log \relax (5) + 5 \, {\left (\frac {2 \, x}{x^{2} {\left (\log \relax (5) - 1\right )} - 5 \, \log \relax (5)^{2} + 10 \, \log \relax (5) - 5} + \frac {\log \left (\frac {x - \sqrt {5 \, \log \relax (5) - 5}}{x + \sqrt {5 \, \log \relax (5) - 5}}\right )}{\sqrt {5 \, \log \relax (5) - 5} {\left (\log \relax (5) - 1\right )}}\right )} \log \relax (5) + \frac {15 \, {\left (\log \relax (5) - 1\right )} \log \left (\frac {x - \sqrt {5 \, \log \relax (5) - 5}}{x + \sqrt {5 \, \log \relax (5) - 5}}\right )}{2 \, \sqrt {5 \, \log \relax (5) - 5}} + 2 \, x - \frac {5 \, x {\left (\log \relax (5) - 1\right )}}{x^{2} - 5 \, \log \relax (5) + 5} + \frac {5 \, \log \left (\frac {x - \sqrt {5 \, \log \relax (5) - 5}}{x + \sqrt {5 \, \log \relax (5) - 5}}\right )}{\sqrt {5 \, \log \relax (5) - 5}} - \frac {5 \, x}{x^{2} {\left (\log \relax (5) - 1\right )} - 5 \, \log \relax (5)^{2} + 10 \, \log \relax (5) - 5} - \frac {10 \, x}{x^{2} - 5 \, \log \relax (5) + 5} - \frac {5 \, \log \left (\frac {x - \sqrt {5 \, \log \relax (5) - 5}}{x + \sqrt {5 \, \log \relax (5) - 5}}\right )}{2 \, \sqrt {5 \, \log \relax (5) - 5} {\left (\log \relax (5) - 1\right )}} + e^{\left (-x - \frac {4 \, x}{x^{2} - 5 \, \log \relax (5) + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*log(5)^2+(10*x^2+70)*log(5)-x^4-6*x^2-45)*exp((-5*x*log(5)+x^3+9*x)/(5*log(5)-x^2-5))+50*log(5
)^2+(-20*x^2-100)*log(5)+2*x^4+20*x^2+50)/(25*log(5)^2+(-10*x^2-50)*log(5)+x^4+10*x^2+25),x, algorithm="maxima
")

[Out]

-5/2*(2*x/(x^2*(log(5) - 1) - 5*log(5)^2 + 10*log(5) - 5) + log((x - sqrt(5*log(5) - 5))/(x + sqrt(5*log(5) -
5)))/(sqrt(5*log(5) - 5)*(log(5) - 1)))*log(5)^2 - 5*(log((x - sqrt(5*log(5) - 5))/(x + sqrt(5*log(5) - 5)))/s
qrt(5*log(5) - 5) - 2*x/(x^2 - 5*log(5) + 5))*log(5) + 5*(2*x/(x^2*(log(5) - 1) - 5*log(5)^2 + 10*log(5) - 5)
+ log((x - sqrt(5*log(5) - 5))/(x + sqrt(5*log(5) - 5)))/(sqrt(5*log(5) - 5)*(log(5) - 1)))*log(5) + 15/2*(log
(5) - 1)*log((x - sqrt(5*log(5) - 5))/(x + sqrt(5*log(5) - 5)))/sqrt(5*log(5) - 5) + 2*x - 5*x*(log(5) - 1)/(x
^2 - 5*log(5) + 5) + 5*log((x - sqrt(5*log(5) - 5))/(x + sqrt(5*log(5) - 5)))/sqrt(5*log(5) - 5) - 5*x/(x^2*(l
og(5) - 1) - 5*log(5)^2 + 10*log(5) - 5) - 10*x/(x^2 - 5*log(5) + 5) - 5/2*log((x - sqrt(5*log(5) - 5))/(x + s
qrt(5*log(5) - 5)))/(sqrt(5*log(5) - 5)*(log(5) - 1)) + e^(-x - 4*x/(x^2 - 5*log(5) + 5))

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mupad [B]  time = 1.12, size = 53, normalized size = 1.77 \begin {gather*} 2\,x+5^{\frac {5\,x}{x^2-5\,\ln \relax (5)+5}}\,{\mathrm {e}}^{-\frac {9\,x}{x^2-5\,\ln \relax (5)+5}}\,{\mathrm {e}}^{-\frac {x^3}{x^2-5\,\ln \relax (5)+5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*log(5)^2 - log(5)*(20*x^2 + 100) - exp(-(9*x - 5*x*log(5) + x^3)/(x^2 - 5*log(5) + 5))*(25*log(5)^2 -
log(5)*(10*x^2 + 70) + 6*x^2 + x^4 + 45) + 20*x^2 + 2*x^4 + 50)/(25*log(5)^2 - log(5)*(10*x^2 + 50) + 10*x^2 +
x^4 + 25),x)

[Out]

2*x + 5^((5*x)/(x^2 - 5*log(5) + 5))*exp(-(9*x)/(x^2 - 5*log(5) + 5))*exp(-x^3/(x^2 - 5*log(5) + 5))

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sympy [A]  time = 0.55, size = 27, normalized size = 0.90 \begin {gather*} 2 x + e^{\frac {x^{3} - 5 x \log {\relax (5 )} + 9 x}{- x^{2} - 5 + 5 \log {\relax (5 )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*ln(5)**2+(10*x**2+70)*ln(5)-x**4-6*x**2-45)*exp((-5*x*ln(5)+x**3+9*x)/(5*ln(5)-x**2-5))+50*ln(
5)**2+(-20*x**2-100)*ln(5)+2*x**4+20*x**2+50)/(25*ln(5)**2+(-10*x**2-50)*ln(5)+x**4+10*x**2+25),x)

[Out]

2*x + exp((x**3 - 5*x*log(5) + 9*x)/(-x**2 - 5 + 5*log(5)))

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