Optimal. Leaf size=106 \[ \frac{\sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d}-\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.130424, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4804, 4624, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d}-\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4804
Rule 4624
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \cos ^{-1}(c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \cos ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cos ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{\cos \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cos ^{-1}(c+d x)\right )}{b d}+\frac{\sin \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cos ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{\left (2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c+d x)}\right )}{b d}+\frac{\left (2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c+d x)}\right )}{b d}\\ &=-\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d}+\frac{\sqrt{2 \pi } C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{\sqrt{b} d}\\ \end{align*}
Mathematica [C] time = 0.104934, size = 128, normalized size = 1.21 \[ \frac{e^{-\frac{i a}{b}} \left (\sqrt{-\frac{i \left (a+b \cos ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \cos ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \cos ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \cos ^{-1}(c+d x)\right )}{b}\right )\right )}{2 d \sqrt{a+b \cos ^{-1}(c+d x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.071, size = 89, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{d}\sqrt{{b}^{-1}} \left ( \cos \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arccos \left ( dx+c \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) -\sin \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arccos \left ( dx+c \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \arccos \left (d x + c\right ) + a}}\,{d x} \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 \frac{1}{\sqrt{a + b \operatorname{acos}{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.34393, size = 231, normalized size = 2.18 \begin{align*} \frac{\sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b \arccos \left (d x + c\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arccos \left (d x + c\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac{a i}{b}\right )}}{{\left (\frac{\sqrt{2} b i}{\sqrt{{\left | b \right |}}} + \sqrt{2} \sqrt{{\left | b \right |}}\right )} d} + \frac{\sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{b \arccos \left (d x + c\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arccos \left (d x + c\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac{a i}{b}\right )}}{{\left (\frac{\sqrt{2} b i}{\sqrt{{\left | b \right |}}} - \sqrt{2} \sqrt{{\left | b \right |}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]