Optimal. Leaf size=31 \[ a x-\frac {b \sqrt {1-c^2 x^2}}{c}+b x \text {ArcCos}(c x) \]
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Rubi [A]
time = 0.01, antiderivative size = 31, 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 = {4716, 267}
\begin {gather*} a x+b x \text {ArcCos}(c x)-\frac {b \sqrt {1-c^2 x^2}}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 4716
Rubi steps
\begin {align*} \int \left (a+b \cos ^{-1}(c x)\right ) \, dx &=a x+b \int \cos ^{-1}(c x) \, dx\\ &=a x+b x \cos ^{-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 \cos ^{-1}(c x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} a x-\frac {b \sqrt {1-c^2 x^2}}{c}+b x \text {ArcCos}(c x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 32, normalized size = 1.03
method | result | size |
default | \(a x +\frac {b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(32\) |
derivativedivides | \(\frac {c x a +b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 31, normalized size = 1.00 \begin {gather*} a x + \frac {{\left (c x \arccos \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 = 2.33, size = 32, normalized size = 1.03 \begin {gather*} \frac {b c x \arccos \left (c x\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 = 29, normalized size = 0.94 \begin {gather*} a x + b \left (\begin {cases} x \operatorname {acos}{\left (c x \right )} - \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {for}\: c \neq 0 \\\frac {\pi x}{2} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 31, normalized size = 1.00 \begin {gather*} a x + \frac {{\left (c x \arccos \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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Mupad [B]
time = 0.33, size = 29, normalized size = 0.94 \begin {gather*} a\,x-\frac {b\,\sqrt {1-c^2\,x^2}}{c}+b\,x\,\mathrm {acos}\left (c\,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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