Optimal. Leaf size=55 \[ \frac{(a+b x) \sqrt{\cos ^{-1}(a+b x)}}{b}-\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0806047, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4804, 4620, 4724, 3304, 3352} \[ \frac{(a+b x) \sqrt{\cos ^{-1}(a+b x)}}{b}-\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4804
Rule 4620
Rule 4724
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \sqrt{\cos ^{-1}(a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{\cos ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \sqrt{\cos ^{-1}(a+b x)}}{b}+\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \sqrt{\cos ^{-1}(x)}} \, dx,x,a+b x\right )}{2 b}\\ &=\frac{(a+b x) \sqrt{\cos ^{-1}(a+b x)}}{b}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a+b x)\right )}{2 b}\\ &=\frac{(a+b x) \sqrt{\cos ^{-1}(a+b x)}}{b}-\frac{\operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a+b x)}\right )}{b}\\ &=\frac{(a+b x) \sqrt{\cos ^{-1}(a+b x)}}{b}-\frac{\sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{b}\\ \end{align*}
Mathematica [C] time = 0.0390328, size = 90, normalized size = 1.64 \[ -\frac{-\frac{\sqrt{\cos ^{-1}(a+b x)} \text{Gamma}\left (\frac{3}{2},-i \cos ^{-1}(a+b x)\right )}{2 \sqrt{-i \cos ^{-1}(a+b x)}}-\frac{\sqrt{\cos ^{-1}(a+b x)} \text{Gamma}\left (\frac{3}{2},i \cos ^{-1}(a+b x)\right )}{2 \sqrt{i \cos ^{-1}(a+b x)}}}{b} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.073, size = 66, normalized size = 1.2 \begin{align*}{\frac{1}{2\,b} \left ( -\sqrt{2}\sqrt{\arccos \left ( bx+a \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{\arccos \left ( bx+a \right ) }} \right ) +2\,\arccos \left ( bx+a \right ) xb+2\,\arccos \left ( bx+a \right ) a \right ){\frac{1}{\sqrt{\arccos \left ( bx+a \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\operatorname{acos}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.37766, size = 158, normalized size = 2.87 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (b x + a\right )}}{i - 1}\right )}{4 \, b{\left (i - 1\right )}} + \frac{\sqrt{\arccos \left (b x + a\right )} e^{\left (i \arccos \left (b x + a\right )\right )}}{2 \, b} + \frac{\sqrt{\arccos \left (b x + a\right )} e^{\left (-i \arccos \left (b x + a\right )\right )}}{2 \, b} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (b x + a\right )}}{i - 1}\right )}{4 \, b{\left (i - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]