Integrand size = 10, antiderivative size = 146 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}+\frac {4 a^3 \sqrt {1-a x}}{105 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {8 a^5 \sqrt {1-a x}}{105 x \sqrt {\frac {1}{1+a x}}} \]
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Time = 0.05 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6470, 30, 105, 12, 97} \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {8 a^5 \sqrt {1-a x}}{105 x \sqrt {\frac {1}{a x+1}}}+\frac {4 a^3 \sqrt {1-a x}}{105 x^3 \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{a x+1}}}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{a x+1}}} \]
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Rule 12
Rule 30
Rule 97
Rule 105
Rule 6470
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}-\frac {\int \frac {1}{x^8} \, dx}{6 a}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^8 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{6 a} \\ & = \frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {6 a^2}{x^6 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{42 a} \\ & = \frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}-\frac {1}{7} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^6 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}+\frac {1}{35} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {4 a^2}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}-\frac {1}{35} \left (4 a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}+\frac {4 a^3 \sqrt {1-a x}}{105 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {1}{105} \left (4 a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {2 a^2}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}+\frac {4 a^3 \sqrt {1-a x}}{105 x^3 \sqrt {\frac {1}{1+a x}}}-\frac {1}{105} \left (8 a^5 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}+\frac {4 a^3 \sqrt {1-a x}}{105 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {8 a^5 \sqrt {1-a x}}{105 x \sqrt {\frac {1}{1+a x}}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {-15+\sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (-15+15 a x-12 a^2 x^2+12 a^3 x^3-8 a^4 x^4+8 a^5 x^5\right )}{105 a x^7} \]
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Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.49
method | result | size |
default | \(\frac {\sqrt {\frac {a x +1}{a x}}\, \sqrt {-\frac {a x -1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (8 a^{4} x^{4}+12 a^{2} x^{2}+15\right )}{105 x^{6}}-\frac {1}{7 a \,x^{7}}\) | \(71\) |
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Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.47 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {{\left (8 \, a^{7} x^{7} + 4 \, a^{5} x^{5} + 3 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 15}{105 \, a x^{7}} \]
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\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {\int \frac {1}{x^{8}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{7}}\, dx}{a} \]
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Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.41 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {{\left (8 \, a^{6} x^{7} + 4 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - 15 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{105 \, a x^{8}} - \frac {1}{7 \, a x^{7}} \]
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\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{7}} \,d x } \]
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Time = 4.86 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {a^2\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{35}-\frac {\sqrt {\frac {1}{a\,x}+1}}{7}+\frac {4\,a^4\,x^4\,\sqrt {\frac {1}{a\,x}+1}}{105}+\frac {8\,a^6\,x^6\,\sqrt {\frac {1}{a\,x}+1}}{105}\right )}{x^6}-\frac {1}{7\,a\,x^7} \]
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