Optimal. Leaf size=38 \[ x \left (-1-\left (4-\frac {-5+x}{\left (5+e^{x^2}\right ) x}\right )^2+\frac {x}{5 \log (1+x)}\right ) \]
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Rubi [F] time = 2.13, 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 {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {125-1930 x^2-17 e^{3 x^2} x^2-e^{2 x^2} x^2 \left (247-80 x+16 x^2\right )+e^{x^2} \left (25-1096 x^2+360 x^3-76 x^4\right )}{\left (5+e^{x^2}\right )^3 x^2}-\frac {x^2}{5 (1+x) \log ^2(1+x)}+\frac {2 x}{5 \log (1+x)}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {x^2}{(1+x) \log ^2(1+x)} \, dx\right )+\frac {2}{5} \int \frac {x}{\log (1+x)} \, dx+\int \frac {125-1930 x^2-17 e^{3 x^2} x^2-e^{2 x^2} x^2 \left (247-80 x+16 x^2\right )+e^{x^2} \left (25-1096 x^2+360 x^3-76 x^4\right )}{\left (5+e^{x^2}\right )^3 x^2} \, dx\\ &=-\left (\frac {1}{5} \operatorname {Subst}\left (\int \frac {(-1+x)^2}{x \log ^2(x)} \, dx,x,1+x\right )\right )+\frac {2}{5} \int \left (-\frac {1}{\log (1+x)}+\frac {1+x}{\log (1+x)}\right ) \, dx+\int \left (-17-\frac {20 (-5+x)^2}{\left (5+e^{x^2}\right )^3}-\frac {8 \left (-1-10 x+2 x^2\right )}{5+e^{x^2}}+\frac {25+99 x^2-440 x^3+84 x^4}{\left (5+e^{x^2}\right )^2 x^2}\right ) \, dx\\ &=-17 x-\frac {1}{5} \operatorname {Subst}\left (\int \left (-\frac {2}{\log ^2(x)}+\frac {1}{x \log ^2(x)}+\frac {x}{\log ^2(x)}\right ) \, dx,x,1+x\right )-\frac {2}{5} \int \frac {1}{\log (1+x)} \, dx+\frac {2}{5} \int \frac {1+x}{\log (1+x)} \, dx-8 \int \frac {-1-10 x+2 x^2}{5+e^{x^2}} \, dx-20 \int \frac {(-5+x)^2}{\left (5+e^{x^2}\right )^3} \, dx+\int \frac {25+99 x^2-440 x^3+84 x^4}{\left (5+e^{x^2}\right )^2 x^2} \, dx\\ &=-17 x-\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,1+x\right )-\frac {1}{5} \operatorname {Subst}\left (\int \frac {x}{\log ^2(x)} \, dx,x,1+x\right )+\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,1+x\right )-\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,1+x\right )+\frac {2}{5} \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,1+x\right )-8 \int \left (-\frac {1}{5+e^{x^2}}-\frac {10 x}{5+e^{x^2}}+\frac {2 x^2}{5+e^{x^2}}\right ) \, dx-20 \int \left (\frac {25}{\left (5+e^{x^2}\right )^3}-\frac {10 x}{\left (5+e^{x^2}\right )^3}+\frac {x^2}{\left (5+e^{x^2}\right )^3}\right ) \, dx+\int \left (\frac {99}{\left (5+e^{x^2}\right )^2}+\frac {25}{\left (5+e^{x^2}\right )^2 x^2}-\frac {440 x}{\left (5+e^{x^2}\right )^2}+\frac {84 x^2}{\left (5+e^{x^2}\right )^2}\right ) \, dx\\ &=-17 x-\frac {2 (1+x)}{5 \log (1+x)}+\frac {(1+x)^2}{5 \log (1+x)}-\frac {2 \text {li}(1+x)}{5}-\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (1+x)\right )+\frac {2}{5} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (1+x)\right )+\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,1+x\right )-\frac {2}{5} \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,1+x\right )+8 \int \frac {1}{5+e^{x^2}} \, dx-16 \int \frac {x^2}{5+e^{x^2}} \, dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3} \, dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2} \, dx+80 \int \frac {x}{5+e^{x^2}} \, dx+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2} \, dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2} \, dx+200 \int \frac {x}{\left (5+e^{x^2}\right )^3} \, dx-440 \int \frac {x}{\left (5+e^{x^2}\right )^2} \, dx-500 \int \frac {1}{\left (5+e^{x^2}\right )^3} \, dx\\ &=-17 x+\frac {2}{5} \text {Ei}(2 \log (1+x))+\frac {1}{5 \log (1+x)}-\frac {2 (1+x)}{5 \log (1+x)}+\frac {(1+x)^2}{5 \log (1+x)}-\frac {2}{5} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (1+x)\right )+8 \int \frac {1}{5+e^{x^2}} \, dx-16 \int \frac {x^2}{5+e^{x^2}} \, dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3} \, dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2} \, dx+40 \operatorname {Subst}\left (\int \frac {1}{5+e^x} \, dx,x,x^2\right )+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2} \, dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2} \, dx+100 \operatorname {Subst}\left (\int \frac {1}{\left (5+e^x\right )^3} \, dx,x,x^2\right )-220 \operatorname {Subst}\left (\int \frac {1}{\left (5+e^x\right )^2} \, dx,x,x^2\right )-500 \int \frac {1}{\left (5+e^{x^2}\right )^3} \, dx\\ &=-17 x+\frac {1}{5 \log (1+x)}-\frac {2 (1+x)}{5 \log (1+x)}+\frac {(1+x)^2}{5 \log (1+x)}+8 \int \frac {1}{5+e^{x^2}} \, dx-16 \int \frac {x^2}{5+e^{x^2}} \, dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3} \, dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2} \, dx+40 \operatorname {Subst}\left (\int \frac {1}{x (5+x)} \, dx,x,e^{x^2}\right )+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2} \, dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2} \, dx+100 \operatorname {Subst}\left (\int \frac {1}{x (5+x)^3} \, dx,x,e^{x^2}\right )-220 \operatorname {Subst}\left (\int \frac {1}{x (5+x)^2} \, dx,x,e^{x^2}\right )-500 \int \frac {1}{\left (5+e^{x^2}\right )^3} \, dx\\ &=-17 x+\frac {1}{5 \log (1+x)}-\frac {2 (1+x)}{5 \log (1+x)}+\frac {(1+x)^2}{5 \log (1+x)}+8 \int \frac {1}{5+e^{x^2}} \, dx+8 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x^2}\right )-8 \operatorname {Subst}\left (\int \frac {1}{5+x} \, dx,x,e^{x^2}\right )-16 \int \frac {x^2}{5+e^{x^2}} \, dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3} \, dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2} \, dx+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2} \, dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2} \, dx+100 \operatorname {Subst}\left (\int \left (\frac {1}{125 x}-\frac {1}{5 (5+x)^3}-\frac {1}{25 (5+x)^2}-\frac {1}{125 (5+x)}\right ) \, dx,x,e^{x^2}\right )-220 \operatorname {Subst}\left (\int \left (\frac {1}{25 x}-\frac {1}{5 (5+x)^2}-\frac {1}{25 (5+x)}\right ) \, dx,x,e^{x^2}\right )-500 \int \frac {1}{\left (5+e^{x^2}\right )^3} \, dx\\ &=\frac {10}{\left (5+e^{x^2}\right )^2}-\frac {40}{5+e^{x^2}}-17 x+\frac {1}{5 \log (1+x)}-\frac {2 (1+x)}{5 \log (1+x)}+\frac {(1+x)^2}{5 \log (1+x)}+8 \int \frac {1}{5+e^{x^2}} \, dx-16 \int \frac {x^2}{5+e^{x^2}} \, dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3} \, dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2} \, dx+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2} \, dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2} \, dx-500 \int \frac {1}{\left (5+e^{x^2}\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [C] time = 0.54, size = 67, normalized size = 1.76 \begin {gather*} \frac {8 (-5+x)}{5+e^{x^2}}-\frac {(-5+x)^2}{\left (5+e^{x^2}\right )^2 x}-17 x-\frac {2}{5} \text {Ei}(\log (1+x))+\frac {x^2}{5 \log (1+x)}+\frac {2 \text {li}(1+x)}{5} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 96, normalized size = 2.53 \begin {gather*} \frac {x^{3} e^{\left (2 \, x^{2}\right )} + 10 \, x^{3} e^{\left (x^{2}\right )} + 25 \, x^{3} - 5 \, {\left (17 \, x^{2} e^{\left (2 \, x^{2}\right )} + 386 \, x^{2} + 2 \, {\left (81 \, x^{2} + 20 \, x\right )} e^{\left (x^{2}\right )} + 190 \, x + 25\right )} \log \left (x + 1\right )}{5 \, {\left (x e^{\left (2 \, x^{2}\right )} + 10 \, x e^{\left (x^{2}\right )} + 25 \, x\right )} \log \left (x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.03, size = 121, normalized size = 3.18 \begin {gather*} \frac {x^{3} e^{\left (2 \, x^{2}\right )} + 10 \, x^{3} e^{\left (x^{2}\right )} - 85 \, x^{2} e^{\left (2 \, x^{2}\right )} \log \left (x + 1\right ) - 810 \, x^{2} e^{\left (x^{2}\right )} \log \left (x + 1\right ) + 25 \, x^{3} - 1930 \, x^{2} \log \left (x + 1\right ) - 200 \, x e^{\left (x^{2}\right )} \log \left (x + 1\right ) - 950 \, x \log \left (x + 1\right ) - 125 \, \log \left (x + 1\right )}{5 \, {\left (x e^{\left (2 \, x^{2}\right )} \log \left (x + 1\right ) + 10 \, x e^{\left (x^{2}\right )} \log \left (x + 1\right ) + 25 \, x \log \left (x + 1\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 63, normalized size = 1.66
method | result | size |
risch | \(-\frac {17 \,{\mathrm e}^{2 x^{2}} x^{2}+162 x^{2} {\mathrm e}^{x^{2}}+386 x^{2}+40 \,{\mathrm e}^{x^{2}} x +190 x +25}{x \left (5+{\mathrm e}^{x^{2}}\right )^{2}}+\frac {x^{2}}{5 \ln \left (x +1\right )}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 102, normalized size = 2.68 \begin {gather*} \frac {25 \, x^{3} + {\left (x^{3} - 85 \, x^{2} \log \left (x + 1\right )\right )} e^{\left (2 \, x^{2}\right )} + 10 \, {\left (x^{3} - {\left (81 \, x^{2} + 20 \, x\right )} \log \left (x + 1\right )\right )} e^{\left (x^{2}\right )} - 5 \, {\left (386 \, x^{2} + 190 \, x + 25\right )} \log \left (x + 1\right )}{5 \, {\left (x e^{\left (2 \, x^{2}\right )} \log \left (x + 1\right ) + 10 \, x e^{\left (x^{2}\right )} \log \left (x + 1\right ) + 25 \, x \log \left (x + 1\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 89, normalized size = 2.34 \begin {gather*} \frac {\frac {x^2}{5}-\frac {2\,x\,\ln \left (x+1\right )\,\left (x+1\right )}{5}}{\ln \left (x+1\right )}-\frac {83\,x}{5}+\frac {2\,x^2}{5}-\frac {8\,\left (5\,x-x^2\right )}{x\,\left ({\mathrm {e}}^{x^2}+5\right )}-\frac {x^4-10\,x^3+25\,x^2}{x^3\,\left (10\,{\mathrm {e}}^{x^2}+{\mathrm {e}}^{2\,x^2}+25\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.46, size = 56, normalized size = 1.47 \begin {gather*} \frac {x^{2}}{5 \log {\left (x + 1 \right )}} - 17 x + \frac {39 x^{2} - 190 x + \left (8 x^{2} - 40 x\right ) e^{x^{2}} - 25}{x e^{2 x^{2}} + 10 x e^{x^{2}} + 25 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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