Optimal. Leaf size=27 \[ \frac {\left (3+e^{\frac {x}{\frac {21}{25}-e^4+x}}\right )^2}{9 x^2} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(421\) vs. \(2(27)=54\).
time = 7.82, antiderivative size = 421, normalized size of antiderivative = 15.59, number of steps
used = 107, number of rules used = 13, integrand size = 150, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6, 6873,
12, 6874, 46, 78, 2262, 2240, 2241, 2264, 2263, 2265, 2209} \begin {gather*} \frac {2 e^{\frac {25 x}{25 x-25 e^4+21}}}{3 x^2}+\frac {e^{\frac {50 x}{25 x-25 e^4+21}}}{9 x^2}+\frac {441+625 e^8}{\left (21-25 e^4\right )^2 x^2}-\frac {1050 e^4}{\left (21-25 e^4\right )^2 x^2}-\frac {100 \left (441+625 e^8\right )}{\left (21-25 e^4\right )^3 x}+\frac {2500 e^4 \left (21+25 e^4\right )}{\left (21-25 e^4\right )^3 x}+\frac {2100}{\left (21-25 e^4\right )^2 x}-\frac {1250 \left (441+625 e^8\right )}{\left (21-25 e^4\right )^3 \left (25 x-25 e^4+21\right )}-\frac {1250}{\left (21-25 e^4\right ) \left (25 x-25 e^4+21\right )}+\frac {52500}{\left (21-25 e^4\right )^2 \left (25 x-25 e^4+21\right )}+\frac {1562500 e^8}{\left (21-25 e^4\right )^3 \left (25 x-25 e^4+21\right )}-\frac {3750 \left (441+625 e^8\right ) \log (x)}{\left (21-25 e^4\right )^4}+\frac {62500 e^4 \left (21+50 e^4\right ) \log (x)}{\left (21-25 e^4\right )^4}-\frac {1250 \log (x)}{\left (21-25 e^4\right )^2}+\frac {105000 \log (x)}{\left (21-25 e^4\right )^3}+\frac {3750 \left (441+625 e^8\right ) \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^4}-\frac {62500 e^4 \left (21+50 e^4\right ) \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^4}+\frac {1250 \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^2}-\frac {105000 \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 46
Rule 78
Rule 2209
Rule 2240
Rule 2241
Rule 2262
Rule 2263
Rule 2264
Rule 2265
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{\left (3969+5625 e^8\right ) x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx\\ &=\int \frac {-7938 \left (1+\frac {625 e^8}{441}\right )-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{9 x^3 \left (21-25 e^4+25 x\right )^2} \, dx\\ &=\frac {1}{9} \int \frac {-7938 \left (1+\frac {625 e^8}{441}\right )-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{x^3 \left (21-25 e^4+25 x\right )^2} \, dx\\ &=\frac {1}{9} \int \left (-\frac {18 \left (441+625 e^8\right )}{\left (-21+25 e^4-25 x\right )^2 x^3}+\frac {900 e^4 (21+25 x)}{\left (-21+25 e^4-25 x\right )^2 x^3}-\frac {18900}{x^2 \left (21-25 e^4+25 x\right )^2}-\frac {11250}{x \left (21-25 e^4+25 x\right )^2}+\frac {6 e^{\frac {25 x}{21-25 e^4+25 x}} \left (-2 \left (21-25 e^4\right )^2-75 \left (21-25 e^4\right ) x-1250 x^2\right )}{x^3 \left (21-25 e^4+25 x\right )^2}+\frac {2 e^{\frac {50 x}{21-25 e^4+25 x}} \left (-\left (21-25 e^4\right )^2-25 \left (21-25 e^4\right ) x-625 x^2\right )}{x^3 \left (21-25 e^4+25 x\right )^2}\right ) \, dx\\ &=\frac {2}{9} \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}} \left (-\left (21-25 e^4\right )^2-25 \left (21-25 e^4\right ) x-625 x^2\right )}{x^3 \left (21-25 e^4+25 x\right )^2} \, dx+\frac {2}{3} \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}} \left (-2 \left (21-25 e^4\right )^2-75 \left (21-25 e^4\right ) x-1250 x^2\right )}{x^3 \left (21-25 e^4+25 x\right )^2} \, dx-1250 \int \frac {1}{x \left (21-25 e^4+25 x\right )^2} \, dx-2100 \int \frac {1}{x^2 \left (21-25 e^4+25 x\right )^2} \, dx+\left (100 e^4\right ) \int \frac {21+25 x}{\left (-21+25 e^4-25 x\right )^2 x^3} \, dx-\left (2 \left (441+625 e^8\right )\right ) \int \frac {1}{\left (-21+25 e^4-25 x\right )^2 x^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 28, normalized size = 1.04 \begin {gather*} \frac {\left (3+e^{\frac {25 x}{21-25 e^4+25 x}}\right )^2}{9 x^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. \(264\) vs.
\(2(21)=42\).
time = 2.02, size = 265, normalized size = 9.81
method | result | size |
risch | \(\frac {1}{x^{2}}+\frac {{\mathrm e}^{-\frac {50 x}{25 \,{\mathrm e}^{4}-25 x -21}}}{9 x^{2}}+\frac {2 \,{\mathrm e}^{-\frac {25 x}{25 \,{\mathrm e}^{4}-25 x -21}}}{3 x^{2}}\) | \(45\) |
norman | \(\frac {\left (\frac {25 \,{\mathrm e}^{4}}{9}-\frac {7}{3}\right ) {\mathrm e}^{-\frac {50 x}{25 \,{\mathrm e}^{4}-25 x -21}}+\left (\frac {50 \,{\mathrm e}^{4}}{3}-14\right ) {\mathrm e}^{-\frac {25 x}{25 \,{\mathrm e}^{4}-25 x -21}}-25 x -\frac {50 \,{\mathrm e}^{-\frac {25 x}{25 \,{\mathrm e}^{4}-25 x -21}} x}{3}-\frac {25 \,{\mathrm e}^{-\frac {50 x}{25 \,{\mathrm e}^{4}-25 x -21}} x}{9}-21+25 \,{\mathrm e}^{4}}{x^{2} \left (25 \,{\mathrm e}^{4}-25 x -21\right )}\) | \(109\) |
derivativedivides | \(-\frac {\left (25 \,{\mathrm e}^{4}-21\right ) \left (-\frac {13781250 \left (\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}+68906250 \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}+24310125 \,{\mathrm e}^{4}-4084101}+\frac {32812500 \,{\mathrm e}^{4} \left (\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}+68906250 \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}+24310125 \,{\mathrm e}^{4}-4084101}-\frac {19531250 \,{\mathrm e}^{8} \left (\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}+68906250 \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}+24310125 \,{\mathrm e}^{4}-4084101}\right )}{25}\) | \(265\) |
default | \(-\frac {\left (25 \,{\mathrm e}^{4}-21\right ) \left (-\frac {13781250 \left (\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}+68906250 \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}+24310125 \,{\mathrm e}^{4}-4084101}+\frac {32812500 \,{\mathrm e}^{4} \left (\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}+68906250 \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}+24310125 \,{\mathrm e}^{4}-4084101}-\frac {19531250 \,{\mathrm e}^{8} \left (\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}+68906250 \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}+24310125 \,{\mathrm e}^{4}-4084101}\right )}{25}\) | \(265\) |
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. 716 vs.
\(2 (24) = 48\).
time = 0.63, size = 716, normalized size = 26.52 \begin {gather*} 625 \, {\left (\frac {3750 \, x^{2} - 75 \, x {\left (25 \, e^{4} - 21\right )} - 625 \, e^{8} + 1050 \, e^{4} - 441}{25 \, x^{3} {\left (15625 \, e^{12} - 39375 \, e^{8} + 33075 \, e^{4} - 9261\right )} - x^{2} {\left (390625 \, e^{16} - 1312500 \, e^{12} + 1653750 \, e^{8} - 926100 \, e^{4} + 194481\right )}} + \frac {3750 \, \log \left (25 \, x - 25 \, e^{4} + 21\right )}{390625 \, e^{16} - 1312500 \, e^{12} + 1653750 \, e^{8} - 926100 \, e^{4} + 194481} - \frac {3750 \, \log \left (x\right )}{390625 \, e^{16} - 1312500 \, e^{12} + 1653750 \, e^{8} - 926100 \, e^{4} + 194481}\right )} e^{8} - 1050 \, {\left (\frac {3750 \, x^{2} - 75 \, x {\left (25 \, e^{4} - 21\right )} - 625 \, e^{8} + 1050 \, e^{4} - 441}{25 \, x^{3} {\left (15625 \, e^{12} - 39375 \, e^{8} + 33075 \, e^{4} - 9261\right )} - x^{2} {\left (390625 \, e^{16} - 1312500 \, e^{12} + 1653750 \, e^{8} - 926100 \, e^{4} + 194481\right )}} + \frac {3750 \, \log \left (25 \, x - 25 \, e^{4} + 21\right )}{390625 \, e^{16} - 1312500 \, e^{12} + 1653750 \, e^{8} - 926100 \, e^{4} + 194481} - \frac {3750 \, \log \left (x\right )}{390625 \, e^{16} - 1312500 \, e^{12} + 1653750 \, e^{8} - 926100 \, e^{4} + 194481}\right )} e^{4} - 2500 \, {\left (\frac {50 \, x - 25 \, e^{4} + 21}{25 \, x^{2} {\left (625 \, e^{8} - 1050 \, e^{4} + 441\right )} - x {\left (15625 \, e^{12} - 39375 \, e^{8} + 33075 \, e^{4} - 9261\right )}} + \frac {50 \, \log \left (25 \, x - 25 \, e^{4} + 21\right )}{15625 \, e^{12} - 39375 \, e^{8} + 33075 \, e^{4} - 9261} - \frac {50 \, \log \left (x\right )}{15625 \, e^{12} - 39375 \, e^{8} + 33075 \, e^{4} - 9261}\right )} e^{4} + \frac {441 \, {\left (3750 \, x^{2} - 75 \, x {\left (25 \, e^{4} - 21\right )} - 625 \, e^{8} + 1050 \, e^{4} - 441\right )}}{25 \, x^{3} {\left (15625 \, e^{12} - 39375 \, e^{8} + 33075 \, e^{4} - 9261\right )} - x^{2} {\left (390625 \, e^{16} - 1312500 \, e^{12} + 1653750 \, e^{8} - 926100 \, e^{4} + 194481\right )}} + \frac {2100 \, {\left (50 \, x - 25 \, e^{4} + 21\right )}}{25 \, x^{2} {\left (625 \, e^{8} - 1050 \, e^{4} + 441\right )} - x {\left (15625 \, e^{12} - 39375 \, e^{8} + 33075 \, e^{4} - 9261\right )}} + \frac {{\left (e^{\left (\frac {50 \, e^{4}}{25 \, x - 25 \, e^{4} + 21} + 2\right )} + 6 \, e^{\left (\frac {25 \, e^{4}}{25 \, x - 25 \, e^{4} + 21} + \frac {21}{25 \, x - 25 \, e^{4} + 21} + 1\right )}\right )} e^{\left (-\frac {42}{25 \, x - 25 \, e^{4} + 21}\right )}}{9 \, x^{2}} + \frac {1653750 \, \log \left (25 \, x - 25 \, e^{4} + 21\right )}{390625 \, e^{16} - 1312500 \, e^{12} + 1653750 \, e^{8} - 926100 \, e^{4} + 194481} + \frac {105000 \, \log \left (25 \, x - 25 \, e^{4} + 21\right )}{15625 \, e^{12} - 39375 \, e^{8} + 33075 \, e^{4} - 9261} + \frac {1250 \, \log \left (25 \, x - 25 \, e^{4} + 21\right )}{625 \, e^{8} - 1050 \, e^{4} + 441} - \frac {1653750 \, \log \left (x\right )}{390625 \, e^{16} - 1312500 \, e^{12} + 1653750 \, e^{8} - 926100 \, e^{4} + 194481} - \frac {105000 \, \log \left (x\right )}{15625 \, e^{12} - 39375 \, e^{8} + 33075 \, e^{4} - 9261} - \frac {1250 \, \log \left (x\right )}{625 \, e^{8} - 1050 \, e^{4} + 441} + \frac {1250}{25 \, x {\left (25 \, e^{4} - 21\right )} - 625 \, e^{8} + 1050 \, e^{4} - 441} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 39, normalized size = 1.44 \begin {gather*} \frac {e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} + 6 \, e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} + 9}{9 \, 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. 49 vs.
\(2 (19) = 38\).
time = 0.21, size = 49, normalized size = 1.81 \begin {gather*} \frac {1}{x^{2}} + \frac {18 x^{2} e^{- \frac {25 x}{- 25 x - 21 + 25 e^{4}}} + 3 x^{2} e^{- \frac {50 x}{- 25 x - 21 + 25 e^{4}}}}{27 x^{4}} \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. 625 vs.
\(2 (24) = 48\).
time = 3.10, size = 625, normalized size = 23.15 \begin {gather*} -\frac {\frac {281250 \, x e^{8}}{25 \, x - 25 \, e^{4} + 21} - \frac {472500 \, x e^{4}}{25 \, x - 25 \, e^{4} + 21} + \frac {22050 \, x e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{25 \, x - 25 \, e^{4} + 21} - \frac {275625 \, x^{2} e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {132300 \, x e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{25 \, x - 25 \, e^{4} + 21} - \frac {1653750 \, x^{2} e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {31250 \, x e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )}}{25 \, x - 25 \, e^{4} + 21} - \frac {390625 \, x^{2} e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} - \frac {52500 \, x e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )}}{25 \, x - 25 \, e^{4} + 21} + \frac {656250 \, x^{2} e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {187500 \, x e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )}}{25 \, x - 25 \, e^{4} + 21} - \frac {2343750 \, x^{2} e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} - \frac {315000 \, x e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )}}{25 \, x - 25 \, e^{4} + 21} + \frac {3937500 \, x^{2} e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {198450 \, x}{25 \, x - 25 \, e^{4} + 21} - 5625 \, e^{8} + 9450 \, e^{4} - 441 \, e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} - 2646 \, e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} - 625 \, e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )} + 1050 \, e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )} - 3750 \, e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )} + 6300 \, e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )} - 3969}{9 \, {\left (\frac {15625 \, x^{2} e^{12}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} - \frac {39375 \, x^{2} e^{8}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {33075 \, x^{2} e^{4}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} - \frac {9261 \, x^{2}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}}\right )} {\left (25 \, e^{4} - 21\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.98, size = 24, normalized size = 0.89 \begin {gather*} \frac {{\left ({\mathrm {e}}^{\frac {25\,x}{25\,x-25\,{\mathrm {e}}^4+21}}+3\right )}^2}{9\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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