Optimal. Leaf size=133 \[ -\frac {27}{4} a^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {9}{8} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {3}{x}\right )-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}-\frac {51}{8} a^4 \csc ^{-1}(a x) \]
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Rubi [A]
time = 0.54, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6304, 1647,
1607, 12, 866, 1649, 1829, 27, 757, 655, 222} \begin {gather*} -\frac {51}{8} a^4 \csc ^{-1}(a x)-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {27}{4} a^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {9}{8} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {3}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 222
Rule 655
Rule 757
Rule 866
Rule 1607
Rule 1647
Rule 1649
Rule 1829
Rule 6304
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx &=-\text {Subst}\left (\int \frac {x^3 \left (1-\frac {x}{a}\right )^2}{\left (1+\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}} \left (a x^3-x^4\right )}{\left (1+\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {\text {Subst}\left (\int \frac {(a-x) x^3 \sqrt {1-\frac {x^2}{a^2}}}{\left (1+\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {\text {Subst}\left (\int \frac {a^2 x^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1+\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=-\text {Subst}\left (\int \frac {x^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1+\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \frac {x^3 \left (1-\frac {x}{a}\right )^3}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2 \left (-3 a^3+a^2 x-a x^2\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {1}{4} a^2 \text {Subst}\left (\int \frac {12 a-28 x+\frac {27 x^2}{a}-\frac {12 x^3}{a^2}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}+\frac {1}{12} a^4 \text {Subst}\left (\int \frac {-\frac {36}{a}+\frac {108 x}{a^2}-\frac {81 x^2}{a^3}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}+\frac {1}{12} a^4 \text {Subst}\left (\int -\frac {9 (2 a-3 x)^2}{a^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}-\frac {1}{4} (3 a) \text {Subst}\left (\int \frac {(2 a-3 x)^2}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {9}{8} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {3}{x}\right )-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}+\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {-17+\frac {18 x}{a}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {27}{4} a^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {9}{8} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {3}{x}\right )-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}-\frac {1}{8} \left (51 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {27}{4} a^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {9}{8} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {3}{x}\right )-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}-\frac {51}{8} a^4 \csc ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 75, normalized size = 0.56 \begin {gather*} -\frac {a \sqrt {1-\frac {1}{a^2 x^2}} \left (-2+6 a x-11 a^2 x^2+29 a^3 x^3+80 a^4 x^4\right )}{8 x^3 (1+a x)}-\frac {51}{8} a^4 \text {ArcSin}\left (\frac {1}{a x}\right ) \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. \(689\) vs.
\(2(115)=230\).
time = 0.12, size = 690, normalized size = 5.19
method | result | size |
risch | \(-\frac {\left (a x +1\right ) \left (48 a^{3} x^{3}-19 a^{2} x^{2}+8 a x -2\right ) \sqrt {\frac {a x -1}{a x +1}}}{8 x^{4}}+\frac {\left (-\frac {4 a^{3} \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}-\frac {51 a^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{8}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a x -1}\) | \(137\) |
default | \(\frac {\left (-56 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{7} x^{7}+56 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}-56 \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}+56 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{5} x^{5}-163 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{6} x^{6}-51 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{6} x^{6}+56 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{7} x^{6}+16 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+112 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-112 \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}+91 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{4} x^{4}-158 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}-102 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{5} x^{5}+112 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+56 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}-56 \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}+22 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-51 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}-51 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+56 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-7 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+4 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{8 x^{4} \sqrt {a^{2}}\, \left (a x -1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}}\) | \(690\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 193, normalized size = 1.45 \begin {gather*} \frac {1}{4} \, {\left (51 \, a^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 16 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}} - \frac {77 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 149 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 123 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 35 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )}}{a x + 1} + \frac {6 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 77, normalized size = 0.58 \begin {gather*} \frac {102 \, a^{4} x^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (80 \, a^{4} x^{4} + 29 \, a^{3} x^{3} - 11 \, a^{2} x^{2} + 6 \, a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8 \, x^{4}} \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*} \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}}\, dx \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 = 1.24, size = 190, normalized size = 1.43 \begin {gather*} \frac {51\,a^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4}-4\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {\frac {35\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {123\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}+\frac {149\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{4}+\frac {77\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{\frac {6\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {4\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {4\,\left (a\,x-1\right )}{a\,x+1}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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