Integrand size = 10, antiderivative size = 163 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^6} \, dx=\frac {1}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{24 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {a^3 \sqrt {1-a x}}{16 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {1}{16} a^5 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6470, 30, 105, 12, 94, 214} \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^6} \, dx=\frac {1}{16} a^5 \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {a^3 \sqrt {1-a x}}{16 x^2 \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{30 a x^6 \sqrt {\frac {1}{a x+1}}}+\frac {1}{30 a x^6}-\frac {e^{\text {sech}^{-1}(a x)}}{5 x^5}+\frac {a \sqrt {1-a x}}{24 x^4 \sqrt {\frac {1}{a x+1}}} \]
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Rule 12
Rule 30
Rule 94
Rule 105
Rule 214
Rule 6470
Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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} \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^6} \, dx=\frac {-8+\sqrt {\frac {1-a x}{1+a x}} \left (-8-8 a x+2 a^2 x^2+2 a^3 x^3+3 a^4 x^4+3 a^5 x^5\right )-3 a^6 x^6 \log (x)+3 a^6 x^6 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{48 a x^6} \]
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Time = 0.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {\sqrt {\frac {a x +1}{a x}}\, \sqrt {-\frac {a x -1}{a x}}\, \left (3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{6} x^{6}+3 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-8 \sqrt {-a^{2} x^{2}+1}\right )}{48 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{6 a \,x^{6}}\) | \(132\) |
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Time = 0.26 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^6} \, dx=\frac {3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - 3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, {\left (3 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - 8 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 16}{96 \, a x^{6}} \]
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\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^6} \, dx=\frac {\int \frac {1}{x^{7}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{6}}\, dx}{a} \]
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\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^6} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{6}} \,d x } \]
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\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^6} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{6}} \,d x } \]
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Time = 41.12 (sec) , antiderivative size = 878, normalized size of antiderivative = 5.39 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^6} \, dx=\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} \]
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