Optimal. Leaf size=31 \[ a x-\frac {b \sqrt {1-c^2 x^2}}{c}+b x \cos ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, 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 = {4620, 261} \[ a x-\frac {b \sqrt {1-c^2 x^2}}{c}+b x \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 4620
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ a x-\frac {b \sqrt {1-c^2 x^2}}{c}+b x \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 32, normalized size = 1.03 \[ \frac {b c x \arccos \left (c x\right ) + a c x - \sqrt {-c^{2} x^{2} + 1} b}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 31, normalized size = 1.00 \[ a x + \frac {{\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} b}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 32, normalized size = 1.03 \[ a x +\frac {b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 31, normalized size = 1.00 \[ a x + \frac {{\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} b}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.33, size = 29, normalized size = 0.94 \[ a\,x-\frac {b\,\sqrt {1-c^2\,x^2}}{c}+b\,x\,\mathrm {acos}\left (c\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 29, normalized size = 0.94 \[ a x + b \left (\begin {cases} x \operatorname {acos}{\left (c x \right )} - \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {for}\: c \neq 0 \\\frac {\pi x}{2} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________