Integrand size = 12, antiderivative size = 101 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{36 x^5}+\frac {5 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{144 x^3}+\frac {5 b c^5 \sqrt {1-\frac {1}{c^2 x^2}}}{96 x}-\frac {5}{96} b c^6 \csc ^{-1}(c x)-\frac {a+b \sec ^{-1}(c x)}{6 x^6} \]
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Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5328, 342, 327, 222} \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=-\frac {a+b \sec ^{-1}(c x)}{6 x^6}-\frac {5}{96} b c^6 \csc ^{-1}(c x)+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{36 x^5}+\frac {5 b c^5 \sqrt {1-\frac {1}{c^2 x^2}}}{96 x}+\frac {5 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{144 x^3} \]
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Rule 222
Rule 327
Rule 342
Rule 5328
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \sec ^{-1}(c x)}{6 x^6}+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^8} \, dx}{6 c} \\ & = -\frac {a+b \sec ^{-1}(c x)}{6 x^6}-\frac {b \text {Subst}\left (\int \frac {x^6}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c} \\ & = \frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{36 x^5}-\frac {a+b \sec ^{-1}(c x)}{6 x^6}-\frac {1}{36} (5 b c) \text {Subst}\left (\int \frac {x^4}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{36 x^5}+\frac {5 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{144 x^3}-\frac {a+b \sec ^{-1}(c x)}{6 x^6}-\frac {1}{48} \left (5 b c^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{36 x^5}+\frac {5 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{144 x^3}+\frac {5 b c^5 \sqrt {1-\frac {1}{c^2 x^2}}}{96 x}-\frac {a+b \sec ^{-1}(c x)}{6 x^6}-\frac {1}{96} \left (5 b c^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{36 x^5}+\frac {5 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{144 x^3}+\frac {5 b c^5 \sqrt {1-\frac {1}{c^2 x^2}}}{96 x}-\frac {5}{96} b c^6 \csc ^{-1}(c x)-\frac {a+b \sec ^{-1}(c x)}{6 x^6} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=-\frac {a}{6 x^6}+b \left (\frac {c}{36 x^5}+\frac {5 c^3}{144 x^3}+\frac {5 c^5}{96 x}\right ) \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \sec ^{-1}(c x)}{6 x^6}-\frac {5}{96} b c^6 \arcsin \left (\frac {1}{c x}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.72
method | result | size |
parts | \(-\frac {a}{6 x^{6}}-\frac {b \,\operatorname {arcsec}\left (c x \right )}{6 x^{6}}-\frac {5 b \,c^{5} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {5 b \,c^{3} \left (c^{2} x^{2}-1\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}+\frac {5 b c \left (c^{2} x^{2}-1\right )}{144 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{5}}+\frac {b \left (c^{2} x^{2}-1\right )}{36 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{7}}\) | \(174\) |
derivativedivides | \(c^{6} \left (-\frac {a}{6 c^{6} x^{6}}-\frac {b \,\operatorname {arcsec}\left (c x \right )}{6 c^{6} x^{6}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {5 b \left (c^{2} x^{2}-1\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {5 b \left (c^{2} x^{2}-1\right )}{144 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}+\frac {b \left (c^{2} x^{2}-1\right )}{36 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{7} x^{7}}\right )\) | \(186\) |
default | \(c^{6} \left (-\frac {a}{6 c^{6} x^{6}}-\frac {b \,\operatorname {arcsec}\left (c x \right )}{6 c^{6} x^{6}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {5 b \left (c^{2} x^{2}-1\right )}{96 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {5 b \left (c^{2} x^{2}-1\right )}{144 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}+\frac {b \left (c^{2} x^{2}-1\right )}{36 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{7} x^{7}}\right )\) | \(186\) |
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Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.61 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\frac {3 \, {\left (5 \, b c^{6} x^{6} - 16 \, b\right )} \operatorname {arcsec}\left (c x\right ) + {\left (15 \, b c^{4} x^{4} + 10 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {c^{2} x^{2} - 1} - 48 \, a}{288 \, x^{6}} \]
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Time = 8.61 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.39 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=- \frac {a}{6 x^{6}} - \frac {b \operatorname {asec}{\left (c x \right )}}{6 x^{6}} + \frac {b \left (\begin {cases} \frac {5 i c^{7} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{16} - \frac {5 i c^{6}}{16 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {5 i c^{4}}{48 x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {i c^{2}}{24 x^{5} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {i}{6 x^{7} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\- \frac {5 c^{7} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{16} + \frac {5 c^{6}}{16 x \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {5 c^{4}}{48 x^{3} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {c^{2}}{24 x^{5} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {1}{6 x^{7} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{6 c} \]
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Time = 0.27 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.63 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\frac {1}{288} \, b {\left (\frac {15 \, c^{7} \arctan \left (c x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right ) - \frac {15 \, c^{12} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 40 \, c^{10} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, c^{8} x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{6} x^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} - 3 \, c^{4} x^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 1}}{c} - \frac {48 \, \operatorname {arcsec}\left (c x\right )}{x^{6}}\right )} - \frac {a}{6 \, x^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\frac {1}{288} \, {\left (15 \, b c^{5} \arccos \left (\frac {1}{c x}\right ) + \frac {15 \, b c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} + \frac {10 \, b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{3}} + \frac {8 \, b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{5}} - \frac {48 \, b \arccos \left (\frac {1}{c x}\right )}{c x^{6}} - \frac {48 \, a}{c x^{6}}\right )} c \]
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Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^7} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^7} \,d x \]
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