Optimal. Leaf size=343 \[ \frac {33 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}+\frac {27 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {11 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}{10 a}+\frac {17 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x+\frac {15 c^4 \csc ^{-1}(a x)}{16 a}-\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]
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Rubi [A]
time = 0.17, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6329, 99, 159,
163, 41, 222, 94, 214} \begin {gather*} c^4 x \left (\frac {1}{a x}+1\right )^{5/2} \left (1-\frac {1}{a x}\right )^{11/2}+\frac {8 c^4 \left (\frac {1}{a x}+1\right )^{5/2} \left (1-\frac {1}{a x}\right )^{9/2}}{7 a}+\frac {17 c^4 \left (\frac {1}{a x}+1\right )^{5/2} \left (1-\frac {1}{a x}\right )^{7/2}}{14 a}+\frac {11 c^4 \left (\frac {1}{a x}+1\right )^{5/2} \left (1-\frac {1}{a x}\right )^{5/2}}{10 a}+\frac {5 c^4 \left (\frac {1}{a x}+1\right )^{5/2} \left (1-\frac {1}{a x}\right )^{3/2}}{8 a}-\frac {3 c^4 \left (\frac {1}{a x}+1\right )^{5/2} \sqrt {1-\frac {1}{a x}}}{8 a}+\frac {27 c^4 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {1-\frac {1}{a x}}}{16 a}+\frac {33 c^4 \sqrt {\frac {1}{a x}+1} \sqrt {1-\frac {1}{a x}}}{16 a}+\frac {15 c^4 \csc ^{-1}(a x)}{16 a}-\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 94
Rule 99
Rule 159
Rule 163
Rule 214
Rule 222
Rule 6329
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=-\left (c^4 \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{11/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x-c^4 \text {Subst}\left (\int \frac {\left (-\frac {3}{a}-\frac {8 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x-\frac {1}{7} \left (a c^4\right ) \text {Subst}\left (\int \frac {\left (-\frac {21}{a^2}-\frac {51 x}{a^3}\right ) \left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {17 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x-\frac {1}{42} \left (a^2 c^4\right ) \text {Subst}\left (\int \frac {\left (-\frac {126}{a^3}-\frac {231 x}{a^4}\right ) \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {11 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}{10 a}+\frac {17 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x-\frac {1}{210} \left (a^3 c^4\right ) \text {Subst}\left (\int \frac {\left (-\frac {630}{a^4}-\frac {525 x}{a^5}\right ) \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {11 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}{10 a}+\frac {17 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x-\frac {1}{840} \left (a^4 c^4\right ) \text {Subst}\left (\int \frac {\left (-\frac {2520}{a^5}+\frac {945 x}{a^6}\right ) \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {11 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}{10 a}+\frac {17 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x-\frac {\left (a^5 c^4\right ) \text {Subst}\left (\int \frac {\left (-\frac {7560}{a^6}+\frac {8505 x}{a^7}\right ) \left (1+\frac {x}{a}\right )^{3/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2520}\\ &=\frac {27 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {11 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}{10 a}+\frac {17 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x+\frac {\left (a^6 c^4\right ) \text {Subst}\left (\int \frac {\left (\frac {15120}{a^7}-\frac {10395 x}{a^8}\right ) \sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5040}\\ &=\frac {33 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}+\frac {27 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {11 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}{10 a}+\frac {17 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x-\frac {\left (a^7 c^4\right ) \text {Subst}\left (\int \frac {-\frac {15120}{a^8}-\frac {4725 x}{a^9}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5040}\\ &=\frac {33 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}+\frac {27 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {11 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}{10 a}+\frac {17 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x+\frac {\left (15 c^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 a^2}+\frac {\left (3 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {33 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}+\frac {27 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {11 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}{10 a}+\frac {17 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x+\frac {\left (15 c^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{16 a^2}-\frac {\left (3 c^4\right ) \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}\\ &=\frac {33 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}+\frac {27 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}-\frac {3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {5 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}{8 a}+\frac {11 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}{10 a}+\frac {17 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}{14 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2} x+\frac {15 c^4 \csc ^{-1}(a x)}{16 a}-\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 126, normalized size = 0.37 \begin {gather*} \frac {c^4 \left (\sqrt {1-\frac {1}{a^2 x^2}} \left (80-280 a x+96 a^2 x^2+770 a^3 x^3-992 a^4 x^4-525 a^5 x^5+2496 a^6 x^6+560 a^7 x^7\right )+525 a^6 x^6 \text {ArcSin}\left (\frac {1}{a x}\right )-1680 a^6 x^6 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{560 a^7 x^6} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.08, size = 329, normalized size = 0.96
method | result | size |
risch | \(\frac {\left (a x +1\right ) \left (560 a^{7} x^{7}+2496 a^{6} x^{6}-525 a^{5} x^{5}-992 a^{4} x^{4}+770 a^{3} x^{3}+96 a^{2} x^{2}-280 a x +80\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}}{560 x^{7} a^{8}}+\frac {\left (-\frac {3 a^{8} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {15 a^{7} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{16}\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{8} \left (a x -1\right )}\) | \(180\) |
default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{2} c^{4} \left (-1680 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{8} x^{8}+1680 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{6} x^{6}-525 a^{7} x^{7} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}-525 a^{7} x^{7} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+1680 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{8} x^{7}-35 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}-816 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{4} x^{4}+490 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+176 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-280 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +80 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{560 \left (a x -1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{8} x^{7} \sqrt {a^{2}}}\) | \(329\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 379, normalized size = 1.10 \begin {gather*} -\frac {1}{280} \, {\left (\frac {525 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {1155 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 7665 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} + 20811 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - 12799 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 39071 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 33621 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 13615 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 2205 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {6 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {14 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {14 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {14 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {14 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} - \frac {6 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 201, normalized size = 0.59 \begin {gather*} -\frac {1050 \, a^{7} c^{4} x^{7} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (560 \, a^{8} c^{4} x^{8} + 3056 \, a^{7} c^{4} x^{7} + 1971 \, a^{6} c^{4} x^{6} - 1517 \, a^{5} c^{4} x^{5} - 222 \, a^{4} c^{4} x^{4} + 866 \, a^{3} c^{4} x^{3} - 184 \, a^{2} c^{4} x^{2} - 200 \, a c^{4} x + 80 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{560 \, a^{8} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {c^{4} \left (\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{9} + x^{8}}\right )\, dx + \int \frac {a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{8} + x^{7}}\, dx + \int \frac {4 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{7} + x^{6}}\, dx + \int \left (- \frac {4 a^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{6} + x^{5}}\right )\, dx + \int \left (- \frac {6 a^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{5} + x^{4}}\right )\, dx + \int \frac {6 a^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{4} + x^{3}}\, dx + \int \frac {4 a^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{3} + x^{2}}\, dx + \int \left (- \frac {4 a^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{2} + x}\right )\, dx + \int \left (- \frac {a^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{9} x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right )}{a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 525, normalized size = 1.53 \begin {gather*} -\frac {15 \, c^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{8 \, a} + \frac {3 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{4} \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {525 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{13} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 4480 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{12} a c^{4} \mathrm {sgn}\left (a x + 1\right ) - 980 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{11} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 20160 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{10} a c^{4} \mathrm {sgn}\left (a x + 1\right ) + 945 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{9} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 38080 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{8} a c^{4} \mathrm {sgn}\left (a x + 1\right ) + 49280 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{6} a c^{4} \mathrm {sgn}\left (a x + 1\right ) - 945 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 32256 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{4} \mathrm {sgn}\left (a x + 1\right ) + 980 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 12992 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{4} \mathrm {sgn}\left (a x + 1\right ) - 525 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 2496 \, a c^{4} \mathrm {sgn}\left (a x + 1\right )}{280 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{7} a {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 332, normalized size = 0.97 \begin {gather*} \frac {\frac {63\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}+\frac {389\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{8}+\frac {4803\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{40}+\frac {39071\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{280}+\frac {12799\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{280}-\frac {2973\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{40}-\frac {219\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{8}-\frac {33\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{8}}{a+\frac {6\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {14\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {14\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {14\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {14\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {6\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}-\frac {a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}}-\frac {15\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a}-\frac {6\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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