Optimal. Leaf size=18 \[ -\frac {1}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {4641} \[ -\frac {1}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 4641
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {1}{b c \left (a+b \sin ^{-1}(c x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {1}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 18, normalized size = 1.00 \[ -\frac {1}{b^{2} c \arcsin \left (c x\right ) + a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 18, normalized size = 1.00 \[ -\frac {1}{b^{2} c \arcsin \left (c x\right ) + a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 19, normalized size = 1.06 \[ -\frac {1}{b c \left (a +b \arcsin \left (c x \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 18, normalized size = 1.00 \[ -\frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )} b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 18, normalized size = 1.00 \[ -\frac {1}{c\,\mathrm {asin}\left (c\,x\right )\,b^2+a\,c\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.11, size = 53, normalized size = 2.94 \[ \begin {cases} \frac {x}{a^{2}} & \text {for}\: b = 0 \wedge c = 0 \\\frac {\begin {cases} - \frac {i \operatorname {acosh}{\left (c x \right )}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {\operatorname {asin}{\left (c x \right )}}{c} & \text {otherwise} \end {cases}}{a^{2}} & \text {for}\: b = 0 \\\frac {x}{a^{2}} & \text {for}\: c = 0 \\- \frac {1}{a b c + b^{2} c \operatorname {asin}{\left (c x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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