Optimal. Leaf size=18 \[ -\frac {1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A]
time = 0.03, 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 = {5783}
\begin {gather*} -\frac {1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 5783
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 \begin {gather*} -\frac {1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 19, normalized size = 1.06
method | result | size |
derivativedivides | \(-\frac {1}{b c \left (a +b \arcsinh \left (c x \right )\right )}\) | \(19\) |
default | \(-\frac {1}{b c \left (a +b \arcsinh \left (c x \right )\right )}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 30, normalized size = 1.67 \begin {gather*} -\frac {1}{b^{2} c \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + a b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs.
\(2 (14) = 28\).
time = 1.79, size = 36, normalized size = 2.00 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\frac {1}{c\,\mathrm {asinh}\left (c\,x\right )\,b^2+a\,c\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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