Optimal. Leaf size=30 \[ a x-\frac {b \sqrt {1+c^2 x^2}}{c}+b x \sinh ^{-1}(c x) \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5772, 267}
\begin {gather*} a x-\frac {b \sqrt {c^2 x^2+1}}{c}+b x \sinh ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5772
Rubi steps
\begin {align*} \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=a x+b \int \sinh ^{-1}(c x) \, dx\\ &=a x+b x \sinh ^{-1}(c x)-(b c) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx\\ &=a x-\frac {b \sqrt {1+c^2 x^2}}{c}+b x \sinh ^{-1}(c x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} a x-\frac {b \sqrt {1+c^2 x^2}}{c}+b x \sinh ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 31, normalized size = 1.03
method | result | size |
default | \(a x +\frac {b \left (\arcsinh \left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c}\) | \(31\) |
derivativedivides | \(\frac {a c x +b \left (\arcsinh \left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 30, normalized size = 1.00 \begin {gather*} a x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\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 = 43, normalized size = 1.43 \begin {gather*} \frac {b c x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + a c x - \sqrt {c^{2} x^{2} + 1} b}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 26, normalized size = 0.87 \begin {gather*} a x + b \left (\begin {cases} x \operatorname {asinh}{\left (c x \right )} - \frac {\sqrt {c^{2} x^{2} + 1}}{c} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 41, normalized size = 1.37 \begin {gather*} {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{c}\right )} b + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 28, normalized size = 0.93 \begin {gather*} a\,x-\frac {b\,\sqrt {c^2\,x^2+1}}{c}+b\,x\,\mathrm {asinh}\left (c\,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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