Optimal. Leaf size=32 \[ -b c \sqrt {1-\frac {1}{c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{x} \]
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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5329, 267}
\begin {gather*} -\frac {a+b \csc ^{-1}(c x)}{x}-b c \sqrt {1-\frac {1}{c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5329
Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x^2} \, dx &=-\frac {a+b \csc ^{-1}(c x)}{x}-\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^3} \, dx}{c}\\ &=-b c \sqrt {1-\frac {1}{c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 41, normalized size = 1.28 \begin {gather*} -\frac {a}{x}-b c \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \csc ^{-1}(c x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 63, normalized size = 1.97
method | result | size |
derivativedivides | \(c \left (-\frac {a}{c x}+b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\) | \(63\) |
default | \(c \left (-\frac {a}{c x}+b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 33, normalized size = 1.03 \begin {gather*} -{\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} b - \frac {a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 26, normalized size = 0.81 \begin {gather*} -\frac {b \operatorname {arccsc}\left (c x\right ) + \sqrt {c^{2} x^{2} - 1} b + a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.48, size = 37, normalized size = 1.16 \begin {gather*} \begin {cases} - \frac {a}{x} - b c \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b \operatorname {acsc}{\left (c x \right )}}{x} & \text {for}\: c \neq 0 \\- \frac {a + \tilde {\infty } b}{x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 42, normalized size = 1.31 \begin {gather*} -{\left (b \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {b \arcsin \left (\frac {1}{c x}\right )}{c x} + \frac {a}{c x}\right )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.64, size = 37, normalized size = 1.16 \begin {gather*} -\frac {a}{x}-b\,c\,\sqrt {1-\frac {1}{c^2\,x^2}}-\frac {b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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