Optimal. Leaf size=51 \[ \frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}-\frac {1}{4} b c^2 \csc ^{-1}(c x)-\frac {a+b \sec ^{-1}(c x)}{2 x^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 4, 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)}{2 x^2}+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}-\frac {1}{4} b c^2 \csc ^{-1}(c 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^3} \, dx &=-\frac {a+b \sec ^{-1}(c x)}{2 x^2}+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^4} \, dx}{2 c}\\ &=-\frac {a+b \sec ^{-1}(c x)}{2 x^2}-\frac {b \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}-\frac {a+b \sec ^{-1}(c x)}{2 x^2}-\frac {1}{4} (b c) \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}}}{4 x}-\frac {1}{4} b c^2 \csc ^{-1}(c x)-\frac {a+b \sec ^{-1}(c x)}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 66, normalized size = 1.29 \begin {gather*} -\frac {a}{2 x^2}+\frac {b c \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{4 x}-\frac {b \sec ^{-1}(c x)}{2 x^2}-\frac {1}{4} b c^2 \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. \(113\) vs.
\(2(46)=92\).
time = 0.09, size = 114, normalized size = 2.24
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a}{2 c^{2} x^{2}}-\frac {b \,\mathrm {arcsec}\left (c x \right )}{2 c^{2} x^{2}}-\frac {b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b \left (c^{2} x^{2}-1\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\) | \(114\) |
default | \(c^{2} \left (-\frac {a}{2 c^{2} x^{2}}-\frac {b \,\mathrm {arcsec}\left (c x \right )}{2 c^{2} x^{2}}-\frac {b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b \left (c^{2} x^{2}-1\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 83, normalized size = 1.63 \begin {gather*} -\frac {1}{4} \, b {\left (\frac {\frac {c^{4} x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 1} - c^{3} \arctan \left (c x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}{c} + \frac {2 \, \operatorname {arcsec}\left (c x\right )}{x^{2}}\right )} - \frac {a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.74, size = 39, normalized size = 0.76 \begin {gather*} \frac {{\left (b c^{2} x^{2} - 2 \, b\right )} \operatorname {arcsec}\left (c x\right ) + \sqrt {c^{2} x^{2} - 1} b - 2 \, a}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.34, size = 119, normalized size = 2.33 \begin {gather*} - \frac {a}{2 x^{2}} - \frac {b \operatorname {asec}{\left (c x \right )}}{2 x^{2}} + \frac {b \left (\begin {cases} \frac {i c^{3} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} - \frac {i c^{2}}{2 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {i}{2 x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\- \frac {c^{3} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{2} + \frac {c^{2} \sqrt {1 - \frac {1}{c^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{2 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 58, normalized size = 1.14 \begin {gather*} \frac {1}{4} \, {\left (b c \arccos \left (\frac {1}{c x}\right ) + \frac {b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} - \frac {2 \, b \arccos \left (\frac {1}{c x}\right )}{c x^{2}} - \frac {2 \, a}{c x^{2}}\right )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.73, size = 50, normalized size = 0.98 \begin {gather*} \frac {b\,c\,\sqrt {1-\frac {1}{c^2\,x^2}}}{4\,x}-\frac {b\,c^2\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\,\left (\frac {2}{c^2\,x^2}-1\right )}{4}-\frac {a}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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