Optimal. Leaf size=76 \[ \frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{16 x^3}+\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{32 x}-\frac {3}{32} b c^4 \csc ^{-1}(c x)-\frac {a+b \sec ^{-1}(c x)}{4 x^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5328, 342, 327,
222} \begin {gather*} -\frac {a+b \sec ^{-1}(c x)}{4 x^4}-\frac {3}{32} b c^4 \csc ^{-1}(c x)+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{16 x^3}+\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{32 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rule 342
Rule 5328
Rubi steps
\begin {align*} \int \frac {a+b \sec ^{-1}(c x)}{x^5} \, dx &=-\frac {a+b \sec ^{-1}(c x)}{4 x^4}+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^6} \, dx}{4 c}\\ &=-\frac {a+b \sec ^{-1}(c x)}{4 x^4}-\frac {b \text {Subst}\left (\int \frac {x^4}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c}\\ &=\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{16 x^3}-\frac {a+b \sec ^{-1}(c x)}{4 x^4}-\frac {1}{16} (3 b c) \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}}}{16 x^3}+\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{32 x}-\frac {a+b \sec ^{-1}(c x)}{4 x^4}-\frac {1}{32} \left (3 b c^3\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}}}{16 x^3}+\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{32 x}-\frac {3}{32} b c^4 \csc ^{-1}(c x)-\frac {a+b \sec ^{-1}(c x)}{4 x^4}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 78, normalized size = 1.03 \begin {gather*} -\frac {a}{4 x^4}+b \left (\frac {c}{16 x^3}+\frac {3 c^3}{32 x}\right ) \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \sec ^{-1}(c x)}{4 x^4}-\frac {3}{32} b c^4 \text {ArcSin}\left (\frac {1}{c x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(149\) vs.
\(2(67)=134\).
time = 0.09, size = 150, normalized size = 1.97
method | result | size |
derivativedivides | \(c^{4} \left (-\frac {a}{4 c^{4} x^{4}}-\frac {b \,\mathrm {arcsec}\left (c x \right )}{4 c^{4} x^{4}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(150\) |
default | \(c^{4} \left (-\frac {a}{4 c^{4} x^{4}}-\frac {b \,\mathrm {arcsec}\left (c x \right )}{4 c^{4} x^{4}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(150\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 125, normalized size = 1.64 \begin {gather*} \frac {1}{32} \, b {\left (\frac {3 \, c^{5} \arctan \left (c x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right ) + \frac {3 \, c^{8} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 5 \, c^{6} x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{4} x^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} - 2 \, c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 1}}{c} - \frac {8 \, \operatorname {arcsec}\left (c x\right )}{x^{4}}\right )} - \frac {a}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.75, size = 52, normalized size = 0.68 \begin {gather*} \frac {{\left (3 \, b c^{4} x^{4} - 8 \, b\right )} \operatorname {arcsec}\left (c x\right ) + {\left (3 \, b c^{2} x^{2} + 2 \, b\right )} \sqrt {c^{2} x^{2} - 1} - 8 \, a}{32 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.73, size = 192, normalized size = 2.53 \begin {gather*} - \frac {a}{4 x^{4}} - \frac {b \operatorname {asec}{\left (c x \right )}}{4 x^{4}} + \frac {b \left (\begin {cases} \frac {3 i c^{5} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{8} - \frac {3 i c^{4}}{8 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {i c^{2}}{8 x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {i}{4 x^{5} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\- \frac {3 c^{5} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{8} + \frac {3 c^{4}}{8 x \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {c^{2}}{8 x^{3} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {1}{4 x^{5} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{4 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 83, normalized size = 1.09 \begin {gather*} \frac {1}{32} \, {\left (3 \, b c^{3} \arccos \left (\frac {1}{c x}\right ) + \frac {3 \, b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} + \frac {2 \, b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{3}} - \frac {8 \, b \arccos \left (\frac {1}{c x}\right )}{c x^{4}} - \frac {8 \, a}{c x^{4}}\right )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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