Optimal. Leaf size=18 \[ -\frac {1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5675} \[ -\frac {1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5675
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {1}{b c \left (a+b \sinh ^{-1}(c x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 30, normalized size = 1.67 \[ -\frac {1}{b^{2} c \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
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 \[ -\frac {1}{b c \left (a +b \arcsinh \left (c x \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 18, normalized size = 1.00 \[ -\frac {1}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 18, normalized size = 1.00 \[ -\frac {1}{c\,\mathrm {asinh}\left (c\,x\right )\,b^2+a\,c\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.06, size = 36, normalized size = 2.00 \[ \begin {cases} \frac {x}{a^{2}} & \text {for}\: b = 0 \wedge c = 0 \\\frac {\operatorname {asinh}{\left (c x \right )}}{a^{2} c} & \text {for}\: b = 0 \\\frac {x}{a^{2}} & \text {for}\: c = 0 \\- \frac {1}{a b c + b^{2} c \operatorname {asinh}{\left (c x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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