Optimal. Leaf size=194 \[ \frac {5}{128} a^7 \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{a x+1}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{a x+1}}}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{a x+1}}} \]
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Rubi [A] time = 0.10, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6335, 30, 103, 12, 92, 208} \[ \frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{a x+1}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{a x+1}}}+\frac {5}{128} a^7 \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{a x+1}}}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 92
Rule 103
Rule 208
Rule 6335
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx &=-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}-\frac {\int \frac {1}{x^9} \, dx}{7 a}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^9 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{7 a}\\ &=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {7 a^2}{x^7 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{56 a}\\ &=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}-\frac {1}{8} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^7 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {1}{48} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {5 a^2}{x^5 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}-\frac {1}{48} \left (5 a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^5 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {1}{192} \left (5 a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {3 a^2}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}-\frac {1}{64} \left (5 a^5 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{1+a x}}}-\frac {1}{128} \left (5 a^5 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {a^2}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{1+a x}}}-\frac {1}{128} \left (5 a^7 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {1}{128} \left (5 a^8 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )\\ &=\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {5}{128} a^7 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 145, normalized size = 0.75 \[ \frac {-15 a^8 x^8 \log (x)+15 a^8 x^8 \log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )+\sqrt {\frac {1-a x}{a x+1}} \left (15 a^7 x^7+15 a^6 x^6+10 a^5 x^5+10 a^4 x^4+8 a^3 x^3+8 a^2 x^2-48 a x-48\right )-48}{384 a x^8} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 156, normalized size = 0.80 \[ \frac {15 \, a^{8} x^{8} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - 15 \, a^{8} x^{8} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, {\left (15 \, a^{7} x^{7} + 10 \, a^{5} x^{5} + 8 \, a^{3} x^{3} - 48 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 96}{768 \, a x^{8}} \]
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 \[ \int \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 152, normalized size = 0.78 \[ \frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (15 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) x^{8} a^{8}+15 \sqrt {-a^{2} x^{2}+1}\, x^{6} a^{6}+10 \sqrt {-a^{2} x^{2}+1}\, x^{4} a^{4}+8 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-48 \sqrt {-a^{2} x^{2}+1}\right )}{384 x^{7} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{8 a \,x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {5}{128} \, a^{8} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {5}{128} \, \sqrt {-a^{2} x^{2} + 1} a^{8} - \frac {5 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{6}}{128 \, x^{2}} - \frac {5 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{4}}{64 \, x^{4}} - \frac {5 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{48 \, x^{6}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{8 \, x^{8}}}{a} - \frac {1}{8 \, a x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 38.56, size = 1155, normalized size = 5.95 \[ \text {result too large to display} \]
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 \[ \frac {\int \frac {1}{x^{9}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{8}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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