Optimal. Leaf size=194 \[ \frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}+\frac {\sqrt {1-a x}}{56 a x^8 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{48 x^6 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^3 \sqrt {1-a x}}{192 x^4 \sqrt {\frac {1}{1+a x}}}+\frac {5 a^5 \sqrt {1-a x}}{128 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {5}{128} a^7 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \]
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Rubi [A]
time = 0.07, 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 = {6470, 30, 105,
12, 94, 214} \begin {gather*} \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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 94
Rule 105
Rule 214
Rule 6470
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 ) \text {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 = 0.08, size = 145, normalized size = 0.75 \begin {gather*} \frac {-48+\sqrt {\frac {1-a x}{1+a x}} \left (-48-48 a x+8 a^2 x^2+8 a^3 x^3+10 a^4 x^4+10 a^5 x^5+15 a^6 x^6+15 a^7 x^7\right )-15 a^8 x^8 \log (x)+15 a^8 x^8 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{384 a x^8} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.05, size = 152, normalized size = 0.78
method | result | size |
default | \(\frac {\sqrt {\frac {a x +1}{a x}}\, \sqrt {-\frac {a x -1}{a x}}\, \left (15 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{8} x^{8}+15 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+10 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+8 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-48 \sqrt {-a^{2} x^{2}+1}\right )}{384 x^{7} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{8 a \,x^{8}}\) | \(152\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 156, normalized size = 0.80 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\frac {-\frac {235\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{96\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {1723\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{96\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {72283\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {848801\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^9}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^9}+\frac {4181067\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{11}}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{11}}+\frac {10994181\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{13}}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{13}}+\frac {17457599\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{15}}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{15}}+\frac {17457599\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{17}}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{17}}+\frac {10994181\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{19}}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{19}}+\frac {4181067\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{21}}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{21}}+\frac {848801\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{23}}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{23}}+\frac {72283\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{25}}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{25}}+\frac {1723\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{27}}{96\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{27}}-\frac {235\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{29}}{96\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{29}}+\frac {5\,a^7\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{31}}{32\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{31}}+\frac {5\,a^7\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{32\,\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}}{1+\frac {120\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {560\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {1820\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {4368\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {8008\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}-\frac {11440\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{14}}+\frac {12870\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{16}}-\frac {11440\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{18}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{18}}+\frac {8008\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{20}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{20}}-\frac {4368\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{22}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{22}}+\frac {1820\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{24}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{24}}-\frac {560\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{26}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{26}}+\frac {120\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{28}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{28}}-\frac {16\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{30}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{30}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{32}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{32}}-\frac {16\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}+\frac {5\,a^7\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )}{32}-\frac {1}{8\,a\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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