Optimal. Leaf size=33 \[ -\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{b} \]
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Rubi [A] time = 0.0297642, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4804, 4624, 3305, 3351} \[ -\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 4804
Rule 4624
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\cos ^{-1}(a+b x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\cos ^{-1}(x)}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a+b x)\right )}{b}\\ &=-\frac{2 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a+b x)}\right )}{b}\\ &=-\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{b}\\ \end{align*}
Mathematica [C] time = 0.0316839, size = 78, normalized size = 2.36 \[ -\frac{-\sqrt{-i \cos ^{-1}(a+b x)} \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a+b x)\right )-\sqrt{i \cos ^{-1}(a+b x)} \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a+b x)\right )}{2 b \sqrt{\cos ^{-1}(a+b x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.055, size = 28, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{b}{\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{\arccos \left ( bx+a \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\operatorname{acos}{\left (a + b x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.57481, size = 97, normalized size = 2.94 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (b x + a\right )}}{i - 1}\right )}{2 \, b{\left (i - 1\right )}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (b x + a\right )}}{i - 1}\right )}{2 \, b{\left (i - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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