Optimal. Leaf size=25 \[ 3 x+\left (5+\frac {4+e^{x^2}}{-x+x^3}\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(258\) vs. \(2(25)=50\).
time = 1.56, antiderivative size = 258, normalized size of antiderivative = 10.32, number of steps
used = 55, number of rules used = 16, integrand size = 108, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used =
{6873, 6874, 205, 213, 272, 46, 296, 331, 294, 327, 2208, 2209, 2252, 2235, 2245, 2241}
\begin {gather*} -\frac {609 x}{8 \left (1-x^2\right )}-\frac {23 x}{2 \left (1-x^2\right )^2}+\frac {e^{x^2}}{1-x}+\frac {11 e^{x^2}}{x+1}+\frac {e^{2 x^2}}{1-x^2}+\frac {16}{1-x^2}+\frac {2 e^{x^2}}{(1-x)^2}+\frac {2 e^{x^2}}{(x+1)^2}+\frac {e^{2 x^2}}{\left (1-x^2\right )^2}+\frac {16}{\left (1-x^2\right )^2}-\frac {10 e^{x^2}}{x}+\frac {25}{\left (1-x^2\right ) x}+\frac {10}{\left (1-x^2\right )^2 x}+\frac {8 e^{x^2}}{x^2}+\frac {e^{2 x^2}}{x^2}+\frac {16}{x^2}-\frac {3 x^5}{4 \left (1-x^2\right )^2}+\frac {15 x^3}{8 \left (1-x^2\right )}+\frac {9 x^3}{4 \left (1-x^2\right )^2}+\frac {45 x}{8}-\frac {75}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 205
Rule 213
Rule 272
Rule 294
Rule 296
Rule 327
Rule 331
Rule 2208
Rule 2209
Rule 2235
Rule 2241
Rule 2245
Rule 2252
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-32+40 x+96 x^2-157 x^3+111 x^5+9 x^7-3 x^9-e^{2 x^2} \left (2-10 x^2+4 x^4\right )-e^{x^2} \left (16-10 x-64 x^2+60 x^3+16 x^4-70 x^5+20 x^7\right )}{x^3 \left (1-x^2\right )^3} \, dx\\ &=\int \left (\frac {157}{\left (-1+x^2\right )^3}+\frac {32}{x^3 \left (-1+x^2\right )^3}-\frac {40}{x^2 \left (-1+x^2\right )^3}-\frac {96}{x \left (-1+x^2\right )^3}-\frac {111 x^2}{\left (-1+x^2\right )^3}-\frac {9 x^4}{\left (-1+x^2\right )^3}+\frac {3 x^6}{\left (-1+x^2\right )^3}+\frac {2 e^{2 x^2} \left (1-5 x^2+2 x^4\right )}{x^3 \left (-1+x^2\right )^3}+\frac {2 e^{x^2} \left (8-5 x-32 x^2+30 x^3+8 x^4-35 x^5+10 x^7\right )}{x^3 \left (-1+x^2\right )^3}\right ) \, dx\\ &=2 \int \frac {e^{2 x^2} \left (1-5 x^2+2 x^4\right )}{x^3 \left (-1+x^2\right )^3} \, dx+2 \int \frac {e^{x^2} \left (8-5 x-32 x^2+30 x^3+8 x^4-35 x^5+10 x^7\right )}{x^3 \left (-1+x^2\right )^3} \, dx+3 \int \frac {x^6}{\left (-1+x^2\right )^3} \, dx-9 \int \frac {x^4}{\left (-1+x^2\right )^3} \, dx+32 \int \frac {1}{x^3 \left (-1+x^2\right )^3} \, dx-40 \int \frac {1}{x^2 \left (-1+x^2\right )^3} \, dx-96 \int \frac {1}{x \left (-1+x^2\right )^3} \, dx-111 \int \frac {x^2}{\left (-1+x^2\right )^3} \, dx+157 \int \frac {1}{\left (-1+x^2\right )^3} \, dx\\ &=\frac {10}{x \left (1-x^2\right )^2}-\frac {23 x}{2 \left (1-x^2\right )^2}+\frac {9 x^3}{4 \left (1-x^2\right )^2}-\frac {3 x^5}{4 \left (1-x^2\right )^2}+2 \int \left (-\frac {2 e^{x^2}}{(-1+x)^3}+\frac {5 e^{x^2}}{2 (-1+x)^2}+\frac {e^{x^2}}{-1+x}-\frac {8 e^{x^2}}{x^3}+\frac {5 e^{x^2}}{x^2}+\frac {8 e^{x^2}}{x}-\frac {2 e^{x^2}}{(1+x)^3}-\frac {15 e^{x^2}}{2 (1+x)^2}-\frac {9 e^{x^2}}{1+x}\right ) \, dx+\frac {15}{4} \int \frac {x^4}{\left (-1+x^2\right )^2} \, dx-\frac {27}{4} \int \frac {x^2}{\left (-1+x^2\right )^2} \, dx+16 \text {Subst}\left (\int \frac {1}{(-1+x)^3 x^2} \, dx,x,x^2\right )-\frac {111}{4} \int \frac {1}{\left (-1+x^2\right )^2} \, dx-48 \text {Subst}\left (\int \frac {1}{(-1+x)^3 x} \, dx,x,x^2\right )+50 \int \frac {1}{x^2 \left (-1+x^2\right )^2} \, dx-\frac {471}{4} \int \frac {1}{\left (-1+x^2\right )^2} \, dx+\text {Subst}\left (\int \frac {e^{2 x} \left (1-5 x+2 x^2\right )}{(-1+x)^3 x^2} \, dx,x,x^2\right )\\ &=\frac {10}{x \left (1-x^2\right )^2}-\frac {23 x}{2 \left (1-x^2\right )^2}+\frac {9 x^3}{4 \left (1-x^2\right )^2}-\frac {3 x^5}{4 \left (1-x^2\right )^2}+\frac {25}{x \left (1-x^2\right )}-\frac {609 x}{8 \left (1-x^2\right )}+\frac {15 x^3}{8 \left (1-x^2\right )}+2 \int \frac {e^{x^2}}{-1+x} \, dx-\frac {27}{8} \int \frac {1}{-1+x^2} \, dx-4 \int \frac {e^{x^2}}{(-1+x)^3} \, dx-4 \int \frac {e^{x^2}}{(1+x)^3} \, dx+5 \int \frac {e^{x^2}}{(-1+x)^2} \, dx+\frac {45}{8} \int \frac {x^2}{-1+x^2} \, dx+10 \int \frac {e^{x^2}}{x^2} \, dx+\frac {111}{8} \int \frac {1}{-1+x^2} \, dx-15 \int \frac {e^{x^2}}{(1+x)^2} \, dx-16 \int \frac {e^{x^2}}{x^3} \, dx+16 \int \frac {e^{x^2}}{x} \, dx+16 \text {Subst}\left (\int \left (\frac {1}{(-1+x)^3}-\frac {2}{(-1+x)^2}+\frac {3}{-1+x}-\frac {1}{x^2}-\frac {3}{x}\right ) \, dx,x,x^2\right )-18 \int \frac {e^{x^2}}{1+x} \, dx-48 \text {Subst}\left (\int \left (\frac {1}{(-1+x)^3}-\frac {1}{(-1+x)^2}+\frac {1}{-1+x}-\frac {1}{x}\right ) \, dx,x,x^2\right )+\frac {471}{8} \int \frac {1}{-1+x^2} \, dx-75 \int \frac {1}{x^2 \left (-1+x^2\right )} \, dx+\text {Subst}\left (\int \left (-\frac {2 e^{2 x}}{(-1+x)^3}+\frac {3 e^{2 x}}{(-1+x)^2}-\frac {2 e^{2 x}}{-1+x}-\frac {e^{2 x}}{x^2}+\frac {2 e^{2 x}}{x}\right ) \, dx,x,x^2\right )\\ &=\frac {2 e^{x^2}}{(1-x)^2}+\frac {5 e^{x^2}}{1-x}+\frac {16}{x^2}+\frac {8 e^{x^2}}{x^2}-\frac {75}{x}-\frac {10 e^{x^2}}{x}+\frac {45 x}{8}+\frac {2 e^{x^2}}{(1+x)^2}+\frac {15 e^{x^2}}{1+x}+\frac {16}{\left (1-x^2\right )^2}+\frac {10}{x \left (1-x^2\right )^2}-\frac {23 x}{2 \left (1-x^2\right )^2}+\frac {9 x^3}{4 \left (1-x^2\right )^2}-\frac {3 x^5}{4 \left (1-x^2\right )^2}+\frac {16}{1-x^2}+\frac {25}{x \left (1-x^2\right )}-\frac {609 x}{8 \left (1-x^2\right )}+\frac {15 x^3}{8 \left (1-x^2\right )}-\frac {555}{8} \tanh ^{-1}(x)+8 \text {Ei}\left (x^2\right )+2 \int \frac {e^{x^2}}{-1+x} \, dx-2 \text {Subst}\left (\int \frac {e^{2 x}}{(-1+x)^3} \, dx,x,x^2\right )-2 \text {Subst}\left (\int \frac {e^{2 x}}{-1+x} \, dx,x,x^2\right )+2 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,x^2\right )+3 \text {Subst}\left (\int \frac {e^{2 x}}{(-1+x)^2} \, dx,x,x^2\right )-4 \int \frac {e^{x^2}}{(-1+x)^2} \, dx-4 \int \frac {e^{x^2}}{-1+x} \, dx+4 \int \frac {e^{x^2}}{(1+x)^2} \, dx-4 \int \frac {e^{x^2}}{1+x} \, dx+\frac {45}{8} \int \frac {1}{-1+x^2} \, dx+10 \int e^{x^2} \, dx+10 \int \frac {e^{x^2}}{-1+x} \, dx-16 \int \frac {e^{x^2}}{x} \, dx-18 \int \frac {e^{x^2}}{1+x} \, dx+20 \int e^{x^2} \, dx-30 \int e^{x^2} \, dx+30 \int \frac {e^{x^2}}{1+x} \, dx-75 \int \frac {1}{-1+x^2} \, dx-\text {Subst}\left (\int \frac {e^{2 x}}{x^2} \, dx,x,x^2\right )\\ &=\frac {2 e^{x^2}}{(1-x)^2}+\frac {e^{x^2}}{1-x}+\frac {16}{x^2}+\frac {8 e^{x^2}}{x^2}+\frac {e^{2 x^2}}{x^2}-\frac {75}{x}-\frac {10 e^{x^2}}{x}+\frac {45 x}{8}+\frac {2 e^{x^2}}{(1+x)^2}+\frac {11 e^{x^2}}{1+x}+\frac {16}{\left (1-x^2\right )^2}+\frac {e^{2 x^2}}{\left (1-x^2\right )^2}+\frac {10}{x \left (1-x^2\right )^2}-\frac {23 x}{2 \left (1-x^2\right )^2}+\frac {9 x^3}{4 \left (1-x^2\right )^2}-\frac {3 x^5}{4 \left (1-x^2\right )^2}+\frac {16}{1-x^2}+\frac {3 e^{2 x^2}}{1-x^2}+\frac {25}{x \left (1-x^2\right )}-\frac {609 x}{8 \left (1-x^2\right )}+\frac {15 x^3}{8 \left (1-x^2\right )}+2 \text {Ei}\left (2 x^2\right )-2 e^2 \text {Ei}\left (-2 \left (1-x^2\right )\right )+2 \int \frac {e^{x^2}}{-1+x} \, dx-2 \text {Subst}\left (\int \frac {e^{2 x}}{(-1+x)^2} \, dx,x,x^2\right )-2 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,x^2\right )-4 \int \frac {e^{x^2}}{-1+x} \, dx-4 \int \frac {e^{x^2}}{1+x} \, dx+6 \text {Subst}\left (\int \frac {e^{2 x}}{-1+x} \, dx,x,x^2\right )-8 \int \frac {e^{x^2}}{-1+x} \, dx-8 \int \frac {e^{x^2}}{1+x} \, dx+10 \int \frac {e^{x^2}}{-1+x} \, dx-18 \int \frac {e^{x^2}}{1+x} \, dx+30 \int \frac {e^{x^2}}{1+x} \, dx\\ &=\frac {2 e^{x^2}}{(1-x)^2}+\frac {e^{x^2}}{1-x}+\frac {16}{x^2}+\frac {8 e^{x^2}}{x^2}+\frac {e^{2 x^2}}{x^2}-\frac {75}{x}-\frac {10 e^{x^2}}{x}+\frac {45 x}{8}+\frac {2 e^{x^2}}{(1+x)^2}+\frac {11 e^{x^2}}{1+x}+\frac {16}{\left (1-x^2\right )^2}+\frac {e^{2 x^2}}{\left (1-x^2\right )^2}+\frac {10}{x \left (1-x^2\right )^2}-\frac {23 x}{2 \left (1-x^2\right )^2}+\frac {9 x^3}{4 \left (1-x^2\right )^2}-\frac {3 x^5}{4 \left (1-x^2\right )^2}+\frac {16}{1-x^2}+\frac {e^{2 x^2}}{1-x^2}+\frac {25}{x \left (1-x^2\right )}-\frac {609 x}{8 \left (1-x^2\right )}+\frac {15 x^3}{8 \left (1-x^2\right )}+4 e^2 \text {Ei}\left (-2 \left (1-x^2\right )\right )+2 \int \frac {e^{x^2}}{-1+x} \, dx-4 \int \frac {e^{x^2}}{-1+x} \, dx-4 \int \frac {e^{x^2}}{1+x} \, dx-4 \text {Subst}\left (\int \frac {e^{2 x}}{-1+x} \, dx,x,x^2\right )-8 \int \frac {e^{x^2}}{-1+x} \, dx-8 \int \frac {e^{x^2}}{1+x} \, dx+10 \int \frac {e^{x^2}}{-1+x} \, dx-18 \int \frac {e^{x^2}}{1+x} \, dx+30 \int \frac {e^{x^2}}{1+x} \, dx\\ &=\frac {2 e^{x^2}}{(1-x)^2}+\frac {e^{x^2}}{1-x}+\frac {16}{x^2}+\frac {8 e^{x^2}}{x^2}+\frac {e^{2 x^2}}{x^2}-\frac {75}{x}-\frac {10 e^{x^2}}{x}+\frac {45 x}{8}+\frac {2 e^{x^2}}{(1+x)^2}+\frac {11 e^{x^2}}{1+x}+\frac {16}{\left (1-x^2\right )^2}+\frac {e^{2 x^2}}{\left (1-x^2\right )^2}+\frac {10}{x \left (1-x^2\right )^2}-\frac {23 x}{2 \left (1-x^2\right )^2}+\frac {9 x^3}{4 \left (1-x^2\right )^2}-\frac {3 x^5}{4 \left (1-x^2\right )^2}+\frac {16}{1-x^2}+\frac {e^{2 x^2}}{1-x^2}+\frac {25}{x \left (1-x^2\right )}-\frac {609 x}{8 \left (1-x^2\right )}+\frac {15 x^3}{8 \left (1-x^2\right )}+2 \int \frac {e^{x^2}}{-1+x} \, dx-4 \int \frac {e^{x^2}}{-1+x} \, dx-4 \int \frac {e^{x^2}}{1+x} \, dx-8 \int \frac {e^{x^2}}{-1+x} \, dx-8 \int \frac {e^{x^2}}{1+x} \, dx+10 \int \frac {e^{x^2}}{-1+x} \, dx-18 \int \frac {e^{x^2}}{1+x} \, dx+30 \int \frac {e^{x^2}}{1+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(25)=50\).
time = 8.33, size = 55, normalized size = 2.20 \begin {gather*} \frac {16+e^{2 x^2}-40 x+43 x^3-6 x^5+3 x^7+2 e^{x^2} \left (4-5 x+5 x^3\right )}{x^2 \left (-1+x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs.
\(2(24)=48\).
time = 0.35, size = 60, normalized size = 2.40
method | result | size |
norman | \(\frac {16+{\mathrm e}^{2 x^{2}}-40 x +43 x^{3}-6 x^{5}+3 x^{7}+10 x^{3} {\mathrm e}^{x^{2}}-10 \,{\mathrm e}^{x^{2}} x +8 \,{\mathrm e}^{x^{2}}}{x^{2} \left (x^{2}-1\right )^{2}}\) | \(60\) |
risch | \(3 x +\frac {40 x^{3}-40 x +16}{x^{2} \left (x^{4}-2 x^{2}+1\right )}+\frac {{\mathrm e}^{2 x^{2}}}{x^{2} \left (x^{2}-1\right )^{2}}+\frac {2 \left (5 x^{3}-5 x +4\right ) {\mathrm e}^{x^{2}}}{x^{2} \left (x^{2}-1\right )^{2}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 205 vs.
\(2 (24) = 48\).
time = 0.31, size = 205, normalized size = 8.20 \begin {gather*} 3 \, x - \frac {5 \, {\left (15 \, x^{4} - 25 \, x^{2} + 8\right )}}{x^{5} - 2 \, x^{3} + x} + \frac {8 \, {\left (6 \, x^{4} - 9 \, x^{2} + 2\right )}}{x^{6} - 2 \, x^{4} + x^{2}} - \frac {3 \, {\left (9 \, x^{3} - 7 \, x\right )}}{8 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} + \frac {9 \, {\left (5 \, x^{3} - 3 \, x\right )}}{8 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} + \frac {157 \, {\left (3 \, x^{3} - 5 \, x\right )}}{8 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} + \frac {111 \, {\left (x^{3} + x\right )}}{8 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} - \frac {24 \, {\left (2 \, x^{2} - 3\right )}}{x^{4} - 2 \, x^{2} + 1} + \frac {2 \, {\left (5 \, x^{3} - 5 \, x + 4\right )} e^{\left (x^{2}\right )} + e^{\left (2 \, x^{2}\right )}}{x^{6} - 2 \, x^{4} + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (24) = 48\).
time = 0.36, size = 57, normalized size = 2.28 \begin {gather*} \frac {3 \, x^{7} - 6 \, x^{5} + 43 \, x^{3} + 2 \, {\left (5 \, x^{3} - 5 \, x + 4\right )} e^{\left (x^{2}\right )} - 40 \, x + e^{\left (2 \, x^{2}\right )} + 16}{x^{6} - 2 \, x^{4} + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs.
\(2 (17) = 34\).
time = 0.11, size = 105, normalized size = 4.20 \begin {gather*} 3 x + \frac {\left (x^{6} - 2 x^{4} + x^{2}\right ) e^{2 x^{2}} + \left (10 x^{9} - 30 x^{7} + 8 x^{6} + 30 x^{5} - 16 x^{4} - 10 x^{3} + 8 x^{2}\right ) e^{x^{2}}}{x^{12} - 4 x^{10} + 6 x^{8} - 4 x^{6} + x^{4}} + \frac {40 x^{3} - 40 x + 16}{x^{6} - 2 x^{4} + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (24) = 48\).
time = 0.41, size = 63, normalized size = 2.52 \begin {gather*} \frac {3 \, x^{7} - 6 \, x^{5} + 10 \, x^{3} e^{\left (x^{2}\right )} + 43 \, x^{3} - 10 \, x e^{\left (x^{2}\right )} - 40 \, x + e^{\left (2 \, x^{2}\right )} + 8 \, e^{\left (x^{2}\right )} + 16}{x^{6} - 2 \, x^{4} + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 52, normalized size = 2.08 \begin {gather*} 3\,x+\frac {8\,{\mathrm {e}}^{x^2}+{\mathrm {e}}^{2\,x^2}+x^3\,\left (10\,{\mathrm {e}}^{x^2}+40\right )-x\,\left (10\,{\mathrm {e}}^{x^2}+40\right )+16}{x^2\,{\left (x^2-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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