Optimal. Leaf size=88 \[ -\frac {3}{8} a^4 \csc ^{-1}(a x)+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a+\frac {9}{x}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6169, 833, 780, 216} \[ \frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a+\frac {9}{x}\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}-\frac {3}{8} a^4 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 833
Rule 6169
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{x^5} \, dx &=-\operatorname {Subst}\left (\int \frac {x^3 \left (1+\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {x^2 \left (-\frac {3}{a}-\frac {4 x}{a^2}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{12} a^4 \operatorname {Subst}\left (\int \frac {x \left (\frac {8}{a^2}+\frac {9 x}{a^3}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a+\frac {9}{x}\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{8} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a+\frac {9}{x}\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {3}{8} a^4 \csc ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.10, size = 59, normalized size = 0.67 \[ \frac {1}{24} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (16 a^3 x^3+9 a^2 x^2+8 a x+6\right )}{x^3}-9 a^3 \sin ^{-1}\left (\frac {1}{a x}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 76, normalized size = 0.86 \[ \frac {18 \, a^{4} x^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (16 \, a^{4} x^{4} + 25 \, a^{3} x^{3} + 17 \, a^{2} x^{2} + 14 \, a x + 6\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 164, normalized size = 1.86 \[ \frac {1}{12} \, {\left (9 \, a^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {\frac {31 \, {\left (a x - 1\right )} a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {49 \, {\left (a x - 1\right )}^{2} a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac {9 \, {\left (a x - 1\right )}^{3} a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + 39 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{4}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 308, normalized size = 3.50 \[ -\frac {\left (a x -1\right ) \left (-24 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{5} a^{5}+24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}+9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{4} a^{4}+9 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}-24 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}-24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}+15 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+8 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{24 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 172, normalized size = 1.95 \[ \frac {1}{12} \, {\left (9 \, a^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {9 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 49 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 31 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 39 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )}}{a x + 1} + \frac {6 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 129, normalized size = 1.47 \[ \frac {2\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,x^4}+\frac {3\,a^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4}+\frac {17\,a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{24\,x^2}+\frac {25\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{24\,x}+\frac {7\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{12\,x^3} \]
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 \[ \int \frac {1}{x^{5} \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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