Optimal. Leaf size=163 \[ \frac {1}{16} a^5 \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {a^3 \sqrt {1-a x}}{16 x^2 \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{a x+1}}}+\frac {1}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {a \sqrt {1-a x}}{24 x^4 \sqrt {\frac {1}{a x+1}}} \]
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Rubi [A] time = 0.08, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {a^3 \sqrt {1-a x}}{16 x^2 \sqrt {\frac {1}{a x+1}}}+\frac {1}{16} a^5 \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {a \sqrt {1-a x}}{24 x^4 \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{a x+1}}}+\frac {1}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5} \]
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^6} \, dx &=-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}-\frac {\int \frac {1}{x^7} \, dx}{5 a}-\frac {\left (\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}{5 a}\\ &=\frac {1}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\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}{30 a}\\ &=\frac {1}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{1+a x}}}-\frac {1}{6} \left (a \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}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{24 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {1}{24} \left (a \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}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{24 x^4 \sqrt {\frac {1}{1+a x}}}-\frac {1}{8} \left (a^3 \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}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{24 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {a^3 \sqrt {1-a x}}{16 x^2 \sqrt {\frac {1}{1+a x}}}-\frac {1}{16} \left (a^3 \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}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{24 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {a^3 \sqrt {1-a x}}{16 x^2 \sqrt {\frac {1}{1+a x}}}-\frac {1}{16} \left (a^5 \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}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{24 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {a^3 \sqrt {1-a x}}{16 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {1}{16} \left (a^6 \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}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{24 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {a^3 \sqrt {1-a x}}{16 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {1}{16} a^5 \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 = 0.10, size = 129, normalized size = 0.79 \[ \frac {-3 a^6 x^6 \log (x)+3 a^6 x^6 \log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )+\sqrt {\frac {1-a x}{a x+1}} \left (3 a^5 x^5+3 a^4 x^4+2 a^3 x^3+2 a^2 x^2-8 a x-8\right )-8}{48 a x^6} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.86, size = 148, normalized size = 0.91 \[ \frac {3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - 3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, {\left (3 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - 8 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 16}{96 \, a x^{6}} \]
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 \[ \int \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 132, normalized size = 0.81 \[ \frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (3 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) x^{6} a^{6}+3 \sqrt {-a^{2} x^{2}+1}\, x^{4} a^{4}+2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-8 \sqrt {-a^{2} x^{2}+1}\right )}{48 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{6 x^{6} a} \]
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 \[ \frac {\frac {1}{16} \, a^{6} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {1}{16} \, \sqrt {-a^{2} x^{2} + 1} a^{6} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{4}}{16 \, x^{2}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{8 \, x^{4}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}}}{a} - \frac {1}{6 \, a x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 34.08, size = 878, normalized size = 5.39 \[ \frac {\frac {35\,a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{12\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {757\,a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {7339\,a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {41929\,a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^9}{6\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^9}+\frac {25661\,a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{11}}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{11}}+\frac {25661\,a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{13}}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{13}}+\frac {41929\,a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{15}}{6\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{15}}+\frac {7339\,a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{17}}{4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{17}}+\frac {757\,a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{19}}{4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{19}}+\frac {35\,a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{21}}{12\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{21}}-\frac {a^5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{23}}{4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{23}}-\frac {a^5\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{4\,\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}}{1+\frac {66\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {220\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {495\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {792\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {924\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}-\frac {792\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{14}}+\frac {495\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{16}}-\frac {220\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{18}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{18}}+\frac {66\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{20}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{20}}-\frac {12\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{22}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{22}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{24}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{24}}-\frac {12\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}+\frac {a^5\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )}{4}-\frac {1}{6\,a\,x^6} \]
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 \[ \frac {\int \frac {1}{x^{7}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{6}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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