Optimal. Leaf size=133 \[ -\frac {51}{8} a^4 \csc ^{-1}(a x)-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {27}{4} a^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {9}{8} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {3}{x}\right ) \]
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Rubi [A] time = 0.84, 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 = {6169, 1633, 1593, 12, 852, 1635, 1815, 27, 743, 641, 216} \[ -\frac {27}{4} a^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {9}{8} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {3}{x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{x^2}-\frac {a \left (a-\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {51}{8} a^4 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 216
Rule 641
Rule 743
Rule 852
Rule 1593
Rule 1633
Rule 1635
Rule 1815
Rule 6169
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx &=-\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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}\\ &=-\operatorname {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 )\\ &=-\operatorname {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}}}+\operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.06, size = 75, normalized size = 0.56 \[ -\frac {51}{8} a^4 \sin ^{-1}\left (\frac {1}{a x}\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}} \left (80 a^4 x^4+29 a^3 x^3-11 a^2 x^2+6 a x-2\right )}{8 x^3 (a x+1)} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.67, size = 77, normalized size = 0.58 \[ \frac {102 \, a^{4} x^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (80 \, a^{4} x^{4} + 29 \, a^{3} x^{3} - 11 \, a^{2} x^{2} + 6 \, a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8 \, x^{4}} \]
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 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 690, normalized size = 5.19 \[ \frac {\left (-56 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{7} a^{7}+56 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}-163 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{6} a^{6}-51 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, x^{6} a^{6}+56 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{6} a^{7}+56 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{6} a^{6}-56 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{6} a^{7}+91 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}-158 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{5} a^{5}-102 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, x^{5} a^{5}+112 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}+16 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{4} a^{4}+112 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}-112 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}+22 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}-51 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{4} a^{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 ) x^{4} a^{5}+56 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}-56 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}-7 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+4 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a -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 \sqrt {a^{2}}\, x^{4} \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 193, normalized size = 1.45 \[ \frac {1}{4} \, {\left (51 \, a^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 16 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}} - \frac {77 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 149 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 123 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 35 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )}}{a x + 1} + \frac {6 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}\right )} a \]
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 \[ \frac {51\,a^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4}-4\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {\frac {35\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {123\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}+\frac {149\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{4}+\frac {77\,a^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{\frac {6\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {4\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {4\,\left (a\,x-1\right )}{a\,x+1}+1} \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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