Optimal. Leaf size=31 \[ -\frac {1-6 a x}{210 a^3 c^{19} (1-a x)^{21} (a x+1)^{15}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 81} \[ -\frac {1-6 a x}{210 a^3 c^{19} (1-a x)^{21} (a x+1)^{15}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 81
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{6 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{19}} \, dx &=\frac {\int \frac {x^2}{(1-a x)^{22} (1+a x)^{16}} \, dx}{c^{19}}\\ &=-\frac {1-6 a x}{210 a^3 c^{19} (1-a x)^{21} (1+a x)^{15}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.27, size = 30, normalized size = 0.97 \[ \frac {1-6 a x}{210 a^3 c^{19} (a x-1)^{21} (a x+1)^{15}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.48, size = 379, normalized size = 12.23 \[ -\frac {6 \, a x - 1}{210 \, {\left (a^{39} c^{19} x^{36} - 6 \, a^{38} c^{19} x^{35} + 70 \, a^{36} c^{19} x^{33} - 105 \, a^{35} c^{19} x^{32} - 336 \, a^{34} c^{19} x^{31} + 896 \, a^{33} c^{19} x^{30} + 720 \, a^{32} c^{19} x^{29} - 3900 \, a^{31} c^{19} x^{28} + 280 \, a^{30} c^{19} x^{27} + 10752 \, a^{29} c^{19} x^{26} - 6552 \, a^{28} c^{19} x^{25} - 20020 \, a^{27} c^{19} x^{24} + 21840 \, a^{26} c^{19} x^{23} + 24960 \, a^{25} c^{19} x^{22} - 43472 \, a^{24} c^{19} x^{21} - 18018 \, a^{23} c^{19} x^{20} + 60060 \, a^{22} c^{19} x^{19} - 60060 \, a^{20} c^{19} x^{17} + 18018 \, a^{19} c^{19} x^{16} + 43472 \, a^{18} c^{19} x^{15} - 24960 \, a^{17} c^{19} x^{14} - 21840 \, a^{16} c^{19} x^{13} + 20020 \, a^{15} c^{19} x^{12} + 6552 \, a^{14} c^{19} x^{11} - 10752 \, a^{13} c^{19} x^{10} - 280 \, a^{12} c^{19} x^{9} + 3900 \, a^{11} c^{19} x^{8} - 720 \, a^{10} c^{19} x^{7} - 896 \, a^{9} c^{19} x^{6} + 336 \, a^{8} c^{19} x^{5} + 105 \, a^{7} c^{19} x^{4} - 70 \, a^{6} c^{19} x^{3} + 6 \, a^{4} c^{19} x - a^{3} c^{19}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.25, size = 299, normalized size = 9.65 \[ -\frac {358229025 \, a^{14} x^{14} + 5340869100 \, a^{13} x^{13} + 37114698075 \, a^{12} x^{12} + 159416118225 \, a^{11} x^{11} + 473088806190 \, a^{10} x^{10} + 1026819468675 \, a^{9} x^{9} + 1682288472150 \, a^{8} x^{8} + 2115551402250 \, a^{7} x^{7} + 2054435046125 \, a^{6} x^{6} + 1535397250002 \, a^{5} x^{5} + 870854759775 \, a^{4} x^{4} + 364307533205 \, a^{3} x^{3} + 106553746740 \, a^{2} x^{2} + 19571887695 \, a x + 1710785408}{901943132160 \, {\left (a x + 1\right )}^{15} a^{3} c^{19}} + \frac {358229025 \, a^{20} x^{20} - 7555375800 \, a^{19} x^{19} + 75901131600 \, a^{18} x^{18} - 483051354975 \, a^{17} x^{17} + 2184946607340 \, a^{16} x^{16} - 7469205450840 \, a^{15} x^{15} + 20031221295000 \, a^{14} x^{14} - 43177004037300 \, a^{13} x^{13} + 76013078916950 \, a^{12} x^{12} - 110448380006328 \, a^{11} x^{11} + 133277726128008 \, a^{10} x^{10} - 133908931763530 \, a^{9} x^{9} + 111933156213900 \, a^{8} x^{8} - 77492989590120 \, a^{7} x^{7} + 44041557267624 \, a^{6} x^{6} - 20244576347604 \, a^{5} x^{5} + 7349182966545 \, a^{4} x^{4} - 2026362494800 \, a^{3} x^{3} + 396520754280 \, a^{2} x^{2} - 48177926223 \, a x + 2584181888}{901943132160 \, {\left (a x - 1\right )}^{21} a^{3} c^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 426, normalized size = 13.74 \[ \frac {-\frac {3991995}{8589934592 a^{3} \left (a x -1\right )^{4}}+\frac {1964315}{4294967296 a^{3} \left (a x -1\right )^{3}}-\frac {930465}{2147483648 a^{3} \left (a x -1\right )^{2}}-\frac {1}{65536 a^{3} \left (a x -1\right )^{19}}-\frac {858429}{4294967296 a^{3} \left (a x +1\right )^{5}}-\frac {2211105}{8589934592 a^{3} \left (a x +1\right )^{4}}-\frac {1344005}{4294967296 a^{3} \left (a x +1\right )^{3}}-\frac {1550775}{4294967296 a^{3} \left (a x +1\right )^{2}}-\frac {13}{16777216 a^{3} \left (a x +1\right )^{13}}-\frac {1}{62914560 a^{3} \left (a x +1\right )^{15}}-\frac {9}{58720256 a^{3} \left (a x +1\right )^{14}}-\frac {275}{100663296 a^{3} \left (a x +1\right )^{12}}-\frac {253}{33554432 a^{3} \left (a x +1\right )^{11}}-\frac {5819}{335544320 a^{3} \left (a x +1\right )^{10}}-\frac {13915}{402653184 a^{3} \left (a x +1\right )^{9}}-\frac {16445}{268435456 a^{3} \left (a x +1\right )^{8}}-\frac {740025}{7516192768 a^{3} \left (a x +1\right )^{7}}-\frac {312455}{2147483648 a^{3} \left (a x +1\right )^{6}}-\frac {1}{1376256 a^{3} \left (a x -1\right )^{21}}+\frac {3}{655360 a^{3} \left (a x -1\right )^{20}}+\frac {7}{196608 a^{3} \left (a x -1\right )^{18}}-\frac {17}{262144 a^{3} \left (a x -1\right )^{17}}+\frac {51}{524288 a^{3} \left (a x -1\right )^{16}}-\frac {323}{2621440 a^{3} \left (a x -1\right )^{15}}+\frac {969}{7340032 a^{3} \left (a x -1\right )^{14}}-\frac {969}{8388608 a^{3} \left (a x -1\right )^{13}}+\frac {3553}{50331648 a^{3} \left (a x -1\right )^{12}}-\frac {7429}{83886080 a^{3} \left (a x -1\right )^{10}}+\frac {37145}{201326592 a^{3} \left (a x -1\right )^{9}}-\frac {37145}{134217728 a^{3} \left (a x -1\right )^{8}}+\frac {334305}{939524096 a^{3} \left (a x -1\right )^{7}}-\frac {111435}{268435456 a^{3} \left (a x -1\right )^{6}}+\frac {1938969}{4294967296 a^{3} \left (a x -1\right )^{5}}+\frac {3411705}{8589934592 a^{3} \left (a x -1\right )}-\frac {3411705}{8589934592 a^{3} \left (a x +1\right )}}{c^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.57, size = 379, normalized size = 12.23 \[ -\frac {6 \, a x - 1}{210 \, {\left (a^{39} c^{19} x^{36} - 6 \, a^{38} c^{19} x^{35} + 70 \, a^{36} c^{19} x^{33} - 105 \, a^{35} c^{19} x^{32} - 336 \, a^{34} c^{19} x^{31} + 896 \, a^{33} c^{19} x^{30} + 720 \, a^{32} c^{19} x^{29} - 3900 \, a^{31} c^{19} x^{28} + 280 \, a^{30} c^{19} x^{27} + 10752 \, a^{29} c^{19} x^{26} - 6552 \, a^{28} c^{19} x^{25} - 20020 \, a^{27} c^{19} x^{24} + 21840 \, a^{26} c^{19} x^{23} + 24960 \, a^{25} c^{19} x^{22} - 43472 \, a^{24} c^{19} x^{21} - 18018 \, a^{23} c^{19} x^{20} + 60060 \, a^{22} c^{19} x^{19} - 60060 \, a^{20} c^{19} x^{17} + 18018 \, a^{19} c^{19} x^{16} + 43472 \, a^{18} c^{19} x^{15} - 24960 \, a^{17} c^{19} x^{14} - 21840 \, a^{16} c^{19} x^{13} + 20020 \, a^{15} c^{19} x^{12} + 6552 \, a^{14} c^{19} x^{11} - 10752 \, a^{13} c^{19} x^{10} - 280 \, a^{12} c^{19} x^{9} + 3900 \, a^{11} c^{19} x^{8} - 720 \, a^{10} c^{19} x^{7} - 896 \, a^{9} c^{19} x^{6} + 336 \, a^{8} c^{19} x^{5} + 105 \, a^{7} c^{19} x^{4} - 70 \, a^{6} c^{19} x^{3} + 6 \, a^{4} c^{19} x - a^{3} c^{19}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.03 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 3.15, size = 405, normalized size = 13.06 \[ \frac {- 6 a x + 1}{210 a^{39} c^{19} x^{36} - 1260 a^{38} c^{19} x^{35} + 14700 a^{36} c^{19} x^{33} - 22050 a^{35} c^{19} x^{32} - 70560 a^{34} c^{19} x^{31} + 188160 a^{33} c^{19} x^{30} + 151200 a^{32} c^{19} x^{29} - 819000 a^{31} c^{19} x^{28} + 58800 a^{30} c^{19} x^{27} + 2257920 a^{29} c^{19} x^{26} - 1375920 a^{28} c^{19} x^{25} - 4204200 a^{27} c^{19} x^{24} + 4586400 a^{26} c^{19} x^{23} + 5241600 a^{25} c^{19} x^{22} - 9129120 a^{24} c^{19} x^{21} - 3783780 a^{23} c^{19} x^{20} + 12612600 a^{22} c^{19} x^{19} - 12612600 a^{20} c^{19} x^{17} + 3783780 a^{19} c^{19} x^{16} + 9129120 a^{18} c^{19} x^{15} - 5241600 a^{17} c^{19} x^{14} - 4586400 a^{16} c^{19} x^{13} + 4204200 a^{15} c^{19} x^{12} + 1375920 a^{14} c^{19} x^{11} - 2257920 a^{13} c^{19} x^{10} - 58800 a^{12} c^{19} x^{9} + 819000 a^{11} c^{19} x^{8} - 151200 a^{10} c^{19} x^{7} - 188160 a^{9} c^{19} x^{6} + 70560 a^{8} c^{19} x^{5} + 22050 a^{7} c^{19} x^{4} - 14700 a^{6} c^{19} x^{3} + 1260 a^{4} c^{19} x - 210 a^{3} c^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________