Optimal. Leaf size=317 \[ -\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac {40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac {\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{5 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1642, 648, 632,
210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {5 x+1}{\sqrt {14}}\right ) \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right )}{5 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (5 x^2+2 x+3\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac {40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1642
Rubi steps
\begin {align*} \int \frac {2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left (3+2 x+5 x^2\right )} \, dx &=\int \left (\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e^2 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^3}+\frac {-40 d^5-d^4 e-28 d^3 e^2-44 d^2 e^3+2 d e^4-e^5}{e^2 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}+\frac {100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6}{e^2 \left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}+\frac {7 d^3+816 d^2 e-339 d e^2-118 e^3+\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) x}{\left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac {40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\int \frac {7 d^3+816 d^2 e-339 d e^2-118 e^3+\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) x}{3+2 x+5 x^2} \, dx}{\left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac {40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \int \frac {2+10 x}{3+2 x+5 x^2} \, dx}{10 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac {\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \int \frac {1}{3+2 x+5 x^2} \, dx}{5 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac {40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (2 \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right )\right ) \text {Subst}\left (\int \frac {1}{-56-x^2} \, dx,x,2+10 x\right )}{5 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac {4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac {40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac {\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{5 \sqrt {14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac {\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 278, normalized size = 0.88 \begin {gather*} -\frac {\frac {35 \left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e^3 (d+e x)^2}-\frac {70 \left (5 d^2-2 d e+3 e^2\right ) \left (40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5\right )}{e^3 (d+e x)}+\sqrt {14} \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )+\frac {70 \left (-100 d^6+120 d^5 e-228 d^4 e^2+242 d^3 e^3-141 d^2 e^4-120 d e^5+e^6\right ) \log (d+e x)}{e^3}-7 \left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{70 \left (5 d^2-2 d e+3 e^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 298, normalized size = 0.94
method | result | size |
default | \(-\frac {4 d^{4}+5 d^{3} e +3 d^{2} e^{2}-d \,e^{3}+2 e^{4}}{2 e^{3} \left (5 d^{2}-2 d e +3 e^{2}\right ) \left (e x +d \right )^{2}}-\frac {-40 d^{5}-d^{4} e -28 d^{3} e^{2}-44 d^{2} e^{3}+2 d \,e^{4}-e^{5}}{e^{3} \left (5 d^{2}-2 d e +3 e^{2}\right )^{2} \left (e x +d \right )}+\frac {\left (100 d^{6}-120 d^{5} e +228 d^{4} e^{2}-242 d^{3} e^{3}+141 d^{2} e^{4}+120 d \,e^{5}-e^{6}\right ) \ln \left (e x +d \right )}{e^{3} \left (5 d^{2}-2 d e +3 e^{2}\right )^{3}}+\frac {\frac {\left (458 d^{3}-21 d^{2} e -816 d \,e^{2}+113 e^{3}\right ) \ln \left (5 x^{2}+2 x +3\right )}{10}+\frac {\left (-\frac {423}{5} d^{3}+\frac {4101}{5} d^{2} e -\frac {879}{5} d \,e^{2}-\frac {703}{5} e^{3}\right ) \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{14}}{\left (5 d^{2}-2 d e +3 e^{2}\right )^{3}}\) | \(298\) |
risch | \(\text {Expression too large to display}\) | \(24519\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 455, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {14} {\left (423 \, d^{3} - 4101 \, d^{2} e + 879 \, d e^{2} + 703 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{70 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac {{\left (458 \, d^{3} - 21 \, d^{2} e - 816 \, d e^{2} + 113 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{10 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac {{\left (100 \, d^{6} - 120 \, d^{5} e + 228 \, d^{4} e^{2} - 242 \, d^{3} e^{3} + 141 \, d^{2} e^{4} + 120 \, d e^{5} - e^{6}\right )} \log \left (x e + d\right )}{125 \, d^{6} e^{3} - 150 \, d^{5} e^{4} + 285 \, d^{4} e^{5} - 188 \, d^{3} e^{6} + 171 \, d^{2} e^{7} - 54 \, d e^{8} + 27 \, e^{9}} + \frac {60 \, d^{6} - 15 \, d^{5} e + 39 \, d^{4} e^{2} + 84 \, d^{3} e^{3} - 25 \, d^{2} e^{4} + 2 \, {\left (40 \, d^{5} e + d^{4} e^{2} + 28 \, d^{3} e^{3} + 44 \, d^{2} e^{4} - 2 \, d e^{5} + e^{6}\right )} x + 9 \, d e^{5} - 6 \, e^{6}}{2 \, {\left (25 \, d^{6} e^{3} - 20 \, d^{5} e^{4} + 34 \, d^{4} e^{5} - 12 \, d^{3} e^{6} + {\left (25 \, d^{4} e^{5} - 20 \, d^{3} e^{6} + 34 \, d^{2} e^{7} - 12 \, d e^{8} + 9 \, e^{9}\right )} x^{2} + 9 \, d^{2} e^{7} + 2 \, {\left (25 \, d^{5} e^{4} - 20 \, d^{4} e^{5} + 34 \, d^{3} e^{6} - 12 \, d^{2} e^{7} + 9 \, d e^{8}\right )} x\right )}} \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. 675 vs.
\(2 (297) = 594\).
time = 0.55, size = 675, normalized size = 2.13 \begin {gather*} \frac {10500 \, d^{8} - \sqrt {14} {\left (423 \, d^{5} e^{3} + 703 \, x^{2} e^{8} + {\left (879 \, d x^{2} + 1406 \, d x\right )} e^{7} - {\left (4101 \, d^{2} x^{2} - 1758 \, d^{2} x - 703 \, d^{2}\right )} e^{6} + 3 \, {\left (141 \, d^{3} x^{2} - 2734 \, d^{3} x + 293 \, d^{3}\right )} e^{5} + 3 \, {\left (282 \, d^{4} x - 1367 \, d^{4}\right )} e^{4}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + 210 \, {\left (x - 3\right )} e^{8} - 35 \, {\left (16 \, d x - 39 \, d\right )} e^{7} + 105 \, {\left (94 \, d^{2} x - 41 \, d^{2}\right )} e^{6} - 35 \, {\left (28 \, d^{3} x - 347 \, d^{3}\right )} e^{5} + 70 \, {\left (167 \, d^{4} x - 88 \, d^{4}\right )} e^{4} + 105 \, {\left (172 \, d^{5} x + 99 \, d^{5}\right )} e^{3} - 525 \, {\left (10 \, d^{6} x - 27 \, d^{6}\right )} e^{2} + 175 \, {\left (80 \, d^{7} x - 39 \, d^{7}\right )} e + 7 \, {\left (458 \, d^{5} e^{3} + 113 \, x^{2} e^{8} - 2 \, {\left (408 \, d x^{2} - 113 \, d x\right )} e^{7} - {\left (21 \, d^{2} x^{2} + 1632 \, d^{2} x - 113 \, d^{2}\right )} e^{6} + 2 \, {\left (229 \, d^{3} x^{2} - 21 \, d^{3} x - 408 \, d^{3}\right )} e^{5} + {\left (916 \, d^{4} x - 21 \, d^{4}\right )} e^{4}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + 70 \, {\left (100 \, d^{8} - x^{2} e^{8} + 2 \, {\left (60 \, d x^{2} - d x\right )} e^{7} + {\left (141 \, d^{2} x^{2} + 240 \, d^{2} x - d^{2}\right )} e^{6} - 2 \, {\left (121 \, d^{3} x^{2} - 141 \, d^{3} x - 60 \, d^{3}\right )} e^{5} + {\left (228 \, d^{4} x^{2} - 484 \, d^{4} x + 141 \, d^{4}\right )} e^{4} - 2 \, {\left (60 \, d^{5} x^{2} - 228 \, d^{5} x + 121 \, d^{5}\right )} e^{3} + 4 \, {\left (25 \, d^{6} x^{2} - 60 \, d^{6} x + 57 \, d^{6}\right )} e^{2} + 40 \, {\left (5 \, d^{7} x - 3 \, d^{7}\right )} e\right )} \log \left (x e + d\right )}{70 \, {\left (125 \, d^{8} e^{3} + 27 \, x^{2} e^{11} - 54 \, {\left (d x^{2} - d x\right )} e^{10} + 9 \, {\left (19 \, d^{2} x^{2} - 12 \, d^{2} x + 3 \, d^{2}\right )} e^{9} - 2 \, {\left (94 \, d^{3} x^{2} - 171 \, d^{3} x + 27 \, d^{3}\right )} e^{8} + {\left (285 \, d^{4} x^{2} - 376 \, d^{4} x + 171 \, d^{4}\right )} e^{7} - 2 \, {\left (75 \, d^{5} x^{2} - 285 \, d^{5} x + 94 \, d^{5}\right )} e^{6} + 5 \, {\left (25 \, d^{6} x^{2} - 60 \, d^{6} x + 57 \, d^{6}\right )} e^{5} + 50 \, {\left (5 \, d^{7} x - 3 \, d^{7}\right )} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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.
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Giac [A]
time = 3.71, size = 406, normalized size = 1.28 \begin {gather*} -\frac {\sqrt {14} {\left (423 \, d^{3} - 4101 \, d^{2} e + 879 \, d e^{2} + 703 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right )}{70 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac {{\left (458 \, d^{3} - 21 \, d^{2} e - 816 \, d e^{2} + 113 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{10 \, {\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac {{\left (100 \, d^{6} - 120 \, d^{5} e + 228 \, d^{4} e^{2} - 242 \, d^{3} e^{3} + 141 \, d^{2} e^{4} + 120 \, d e^{5} - e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{125 \, d^{6} e^{3} - 150 \, d^{5} e^{4} + 285 \, d^{4} e^{5} - 188 \, d^{3} e^{6} + 171 \, d^{2} e^{7} - 54 \, d e^{8} + 27 \, e^{9}} + \frac {{\left (2 \, {\left (200 \, d^{7} - 75 \, d^{6} e + 258 \, d^{5} e^{2} + 167 \, d^{4} e^{3} - 14 \, d^{3} e^{4} + 141 \, d^{2} e^{5} - 8 \, d e^{6} + 3 \, e^{7}\right )} x + {\left (300 \, d^{8} - 195 \, d^{7} e + 405 \, d^{6} e^{2} + 297 \, d^{5} e^{3} - 176 \, d^{4} e^{4} + 347 \, d^{3} e^{5} - 123 \, d^{2} e^{6} + 39 \, d e^{7} - 18 \, e^{8}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \, {\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}^{3} {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.76, size = 493, normalized size = 1.56 \begin {gather*} \frac {\frac {60\,d^6-15\,d^5\,e+39\,d^4\,e^2+84\,d^3\,e^3-25\,d^2\,e^4+9\,d\,e^5-6\,e^6}{2\,e^3\,\left (25\,d^4-20\,d^3\,e+34\,d^2\,e^2-12\,d\,e^3+9\,e^4\right )}+\frac {x\,\left (40\,d^5+d^4\,e+28\,d^3\,e^2+44\,d^2\,e^3-2\,d\,e^4+e^5\right )}{e^2\,\left (25\,d^4-20\,d^3\,e+34\,d^2\,e^2-12\,d\,e^3+9\,e^4\right )}}{d^2+2\,d\,e\,x+e^2\,x^2}-\frac {\ln \left (x+\frac {1}{5}-\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {423\,\sqrt {14}}{140}-\frac {229}{5}{}\mathrm {i}\right )\,d^3+\left (-\frac {4101\,\sqrt {14}}{140}+\frac {21}{10}{}\mathrm {i}\right )\,d^2\,e+\left (\frac {879\,\sqrt {14}}{140}+\frac {408}{5}{}\mathrm {i}\right )\,d\,e^2+\left (\frac {703\,\sqrt {14}}{140}-\frac {113}{10}{}\mathrm {i}\right )\,e^3\right )}{d^6\,125{}\mathrm {i}-d^5\,e\,150{}\mathrm {i}+d^4\,e^2\,285{}\mathrm {i}-d^3\,e^3\,188{}\mathrm {i}+d^2\,e^4\,171{}\mathrm {i}-d\,e^5\,54{}\mathrm {i}+e^6\,27{}\mathrm {i}}+\frac {\ln \left (x+\frac {1}{5}+\frac {\sqrt {14}\,1{}\mathrm {i}}{5}\right )\,\left (\left (\frac {423\,\sqrt {14}}{140}+\frac {229}{5}{}\mathrm {i}\right )\,d^3+\left (-\frac {4101\,\sqrt {14}}{140}-\frac {21}{10}{}\mathrm {i}\right )\,d^2\,e+\left (\frac {879\,\sqrt {14}}{140}-\frac {408}{5}{}\mathrm {i}\right )\,d\,e^2+\left (\frac {703\,\sqrt {14}}{140}+\frac {113}{10}{}\mathrm {i}\right )\,e^3\right )}{d^6\,125{}\mathrm {i}-d^5\,e\,150{}\mathrm {i}+d^4\,e^2\,285{}\mathrm {i}-d^3\,e^3\,188{}\mathrm {i}+d^2\,e^4\,171{}\mathrm {i}-d\,e^5\,54{}\mathrm {i}+e^6\,27{}\mathrm {i}}+\frac {\ln \left (d+e\,x\right )\,\left (100\,d^6-120\,d^5\,e+228\,d^4\,e^2-242\,d^3\,e^3+141\,d^2\,e^4+120\,d\,e^5-e^6\right )}{e^3\,{\left (5\,d^2-2\,d\,e+3\,e^2\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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