Optimal. Leaf size=327 \[ -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^4} \]
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Rubi [A]
time = 0.15, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6329, 105,
157, 12, 94, 214} \begin {gather*} \frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {\frac {1}{a x}+1}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (\frac {1}{a x}+1\right )^{7/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 94
Rule 105
Rule 157
Rule 214
Rule 6329
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{c^4}\\ &=\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {\text {Subst}\left (\int \frac {\frac {3}{a}-\frac {7 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {a \text {Subst}\left (\int \frac {-\frac {9}{a^2}+\frac {24 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{3 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^2 \text {Subst}\left (\int \frac {\frac {9}{a^3}-\frac {75 x}{a^4}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{3 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^3 \text {Subst}\left (\int \frac {\frac {81}{a^4}-\frac {336 x}{a^5}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{27 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^4 \text {Subst}\left (\int \frac {\frac {567}{a^5}-\frac {1251 x}{a^6}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{189 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^5 \text {Subst}\left (\int \frac {\frac {2835}{a^6}-\frac {3636 x}{a^7}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{945 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^6 \text {Subst}\left (\int \frac {\frac {8505}{a^7}-\frac {6471 x}{a^8}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2835 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^7 \text {Subst}\left (\int \frac {8505}{a^8 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2835 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {3 \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2 c^4}\\ &=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^4}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 117, normalized size = 0.36 \begin {gather*} \frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (1664+4047 a x-339 a^2 x^2-7399 a^3 x^3-4029 a^4 x^4+2967 a^5 x^5+2669 a^6 x^6+315 a^7 x^7\right )}{315 (-1+a x)^2 (1+a x)^5}-3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(765\) vs.
\(2(279)=558\).
time = 0.19, size = 766, normalized size = 2.34
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{4}}+\frac {\left (-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{8} \sqrt {a^{2}}}-\frac {691 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{315 a^{11} \left (x +\frac {1}{a}\right )^{2}}+\frac {113591 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{20160 a^{10} \left (x +\frac {1}{a}\right )}-\frac {31 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{192 a^{10} \left (x -\frac {1}{a}\right )}-\frac {59 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{252 a^{13} \left (x +\frac {1}{a}\right )^{4}}+\frac {1507 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{1680 a^{12} \left (x +\frac {1}{a}\right )^{3}}-\frac {\sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{96 a^{11} \left (x -\frac {1}{a}\right )^{2}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{36 a^{14} \left (x +\frac {1}{a}\right )^{5}}\right ) a^{8} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c^{4} \left (a x -1\right )}\) | \(355\) |
default | \(-\frac {\left (-138915 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{9} x^{9}+120960 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{10} x^{9}+98595 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{7} x^{7}-416745 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{8} x^{8}+362880 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{9} x^{8}+75113 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{6} x^{6}-240861 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}+1111320 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}-967680 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{7} x^{6}-178863 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+833490 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-725760 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+252497 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-833490 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}+725760 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+182307 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-1111320 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}+967680 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-101271 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x -74077 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+416745 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x -362880 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +138915 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}-120960 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{40320 a \sqrt {a^{2}}\, \left (a x +1\right )^{4} c^{4} \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x -1\right )^{4}}\) | \(766\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 231, normalized size = 0.71 \begin {gather*} \frac {1}{20160} \, a {\left (\frac {105 \, {\left (\frac {29 \, {\left (a x - 1\right )}}{a x + 1} - \frac {414 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {35 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 450 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 2961 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 14700 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 95445 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}} - \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} + \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 275, normalized size = 0.84 \begin {gather*} -\frac {945 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 945 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (315 \, a^{7} x^{7} + 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} - 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} - 339 \, a^{2} x^{2} + 4047 \, a x + 1664\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 224, normalized size = 0.69 \begin {gather*} \frac {303\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {\frac {29\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {138\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {35\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{48\,a\,c^4}+\frac {47\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{320\,a\,c^4}+\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{224\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{576\,a\,c^4}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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