Optimal. Leaf size=313 \[ \frac {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7+\frac {9 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a} \]
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Rubi [A]
time = 0.18, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6326, 6330,
96, 94, 214} \begin {gather*} -\frac {1}{7} a^6 c^3 x^7 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {3}{14} a^5 c^3 x^6 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {3}{10} a^4 c^3 x^5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {3}{8} a^3 c^3 x^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {3}{8} a^2 c^3 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {3}{16} a c^3 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {9}{16} c^3 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}+\frac {9 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 94
Rule 96
Rule 214
Rule 6326
Rule 6330
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=-\left (\left (a^6 c^3\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^3 x^6 \, dx\right )\\ &=\left (a^6 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^8} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7-\frac {1}{7} \left (9 a^5 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^7} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7+\frac {1}{2} \left (3 a^4 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^6} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7-\frac {1}{2} \left (3 a^3 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7+\frac {1}{8} \left (9 a^2 c^3\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7-\frac {1}{8} \left (3 a c^3\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^3 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7-\frac {1}{16} \left (9 c^3\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7-\frac {\left (9 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 a}\\ &=\frac {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7+\frac {\left (9 c^3\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 )}{16 a^2}\\ &=\frac {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{8} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {3}{8} a^3 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4-\frac {3}{10} a^4 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {3}{14} a^5 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2} x^7+\frac {9 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 95, normalized size = 0.30 \begin {gather*} -\frac {c^3 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (368+245 a x-656 a^2 x^2+350 a^3 x^3+208 a^4 x^4-280 a^5 x^5+80 a^6 x^6\right )-315 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{560 a} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 240, normalized size = 0.77
method | result | size |
risch | \(-\frac {\left (80 a^{6} x^{6}-280 a^{5} x^{5}+208 a^{4} x^{4}+350 a^{3} x^{3}-656 a^{2} x^{2}+245 a x +368\right ) \left (a x +1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{560 a}+\frac {9 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{16 \sqrt {a^{2}}\, \left (a x -1\right )}\) | \(144\) |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{2} c^{3} \left (-80 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{4} x^{4}+280 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-288 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-70 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +560 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-192 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-315 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +315 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{560 a \left (a x -1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}}\) | \(240\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 337, normalized size = 1.08 \begin {gather*} \frac {1}{560} \, {\left (\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (315 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} - 2100 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - 8393 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 9216 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 5943 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2100 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 315 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {7 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {21 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {35 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {35 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {21 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {7 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 148, normalized size = 0.47 \begin {gather*} \frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (80 \, a^{7} c^{3} x^{7} - 200 \, a^{6} c^{3} x^{6} - 72 \, a^{5} c^{3} x^{5} + 558 \, a^{4} c^{3} x^{4} - 306 \, a^{3} c^{3} x^{3} - 411 \, a^{2} c^{3} x^{2} + 613 \, a c^{3} x + 368 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{560 \, a} \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*} - c^{3} \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \left (- \frac {3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {3 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \frac {3 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {3 a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \left (- \frac {a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{7} x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 162, normalized size = 0.52 \begin {gather*} -\frac {9 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{16 \, {\left | a \right |}} - \frac {1}{560} \, \sqrt {a^{2} x^{2} - 1} {\left (\frac {368 \, c^{3} \mathrm {sgn}\left (a x + 1\right )}{a} + {\left (245 \, c^{3} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (328 \, a c^{3} \mathrm {sgn}\left (a x + 1\right ) - {\left (175 \, a^{2} c^{3} \mathrm {sgn}\left (a x + 1\right ) + 4 \, {\left (26 \, a^{3} c^{3} \mathrm {sgn}\left (a x + 1\right ) + 5 \, {\left (2 \, a^{5} c^{3} x \mathrm {sgn}\left (a x + 1\right ) - 7 \, a^{4} c^{3} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 289, normalized size = 0.92 \begin {gather*} \frac {9\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a}-\frac {\frac {9\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}-\frac {15\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{2}+\frac {849\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{40}-\frac {1152\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{35}+\frac {1199\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{40}+\frac {15\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{2}-\frac {9\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{8}}{a-\frac {7\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {21\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {35\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {35\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {21\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {7\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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