\(\int \sqrt {a+b \arccos (-1+d x^2)} \, dx\) [96]

Optimal result

Integrand size = 16, antiderivative size = 184 \[ \int \sqrt {a+b \arccos \left (-1+d x^2\right )} \, dx=\frac {2 \sqrt {a+b \arccos \left (-1+d x^2\right )} \cos ^2\left (\frac {1}{2} \arccos \left (-1+d x^2\right )\right )}{d x}-\frac {2 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \cos \left (\frac {1}{2} \arccos \left (-1+d x^2\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (-1+d x^2\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sqrt {\pi } \cos \left (\frac {1}{2} \arccos \left (-1+d x^2\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (-1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right )}{\sqrt {\frac {1}{b}} d x} \]

[Out]

-2*cos(1/2*a/b)*cos(1/2*arccos(d*x^2-1))*FresnelC((1/b)^(1/2)*(a+b*arccos(d*x^2-1))^(1/2)/Pi^(1/2))*Pi^(1/2)/d
/x/(1/b)^(1/2)-2*cos(1/2*arccos(d*x^2-1))*FresnelS((1/b)^(1/2)*(a+b*arccos(d*x^2-1))^(1/2)/Pi^(1/2))*sin(1/2*a
/b)*Pi^(1/2)/d/x/(1/b)^(1/2)+2*cos(1/2*arccos(d*x^2-1))^2*(a+b*arccos(d*x^2-1))^(1/2)/d/x

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4897} \[ \int \sqrt {a+b \arccos \left (-1+d x^2\right )} \, dx=-\frac {2 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \cos \left (\frac {1}{2} \arccos \left (d x^2-1\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (d x^2-1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) \cos \left (\frac {1}{2} \arccos \left (d x^2-1\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (d x^2-1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}+\frac {2 \cos ^2\left (\frac {1}{2} \arccos \left (d x^2-1\right )\right ) \sqrt {a+b \arccos \left (d x^2-1\right )}}{d x} \]

[In]

Int[Sqrt[a + b*ArcCos[-1 + d*x^2]],x]

[Out]

(2*Sqrt[a + b*ArcCos[-1 + d*x^2]]*Cos[ArcCos[-1 + d*x^2]/2]^2)/(d*x) - (2*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1
+ d*x^2]/2]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/(Sqrt[b^(-1)]*d*x) - (2*Sqrt[Pi]
*Cos[ArcCos[-1 + d*x^2]/2]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])/(Sqr
t[b^(-1)]*d*x)

Rule 4897

Int[Sqrt[(a_.) + ArcCos[-1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[2*Sqrt[a + b*ArcCos[-1 + d*x^2]]*(Cos[(1/
2)*ArcCos[-1 + d*x^2]]^2/(d*x)), x] + (-Simp[2*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]*(FresnelC[Sqrt[
1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(Sqrt[1/b]*d*x)), x] - Simp[2*Sqrt[Pi]*Sin[a/(2*b)]*Cos[ArcCos[-1 +
d*x^2]/2]*(FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(Sqrt[1/b]*d*x)), x]) /; FreeQ[{a, b, d}, x
]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {a+b \arccos \left (-1+d x^2\right )} \cos ^2\left (\frac {1}{2} \arccos \left (-1+d x^2\right )\right )}{d x}-\frac {2 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \cos \left (\frac {1}{2} \arccos \left (-1+d x^2\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (-1+d x^2\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sqrt {\pi } \cos \left (\frac {1}{2} \arccos \left (-1+d x^2\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (-1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right )}{\sqrt {\frac {1}{b}} d x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.81 \[ \int \sqrt {a+b \arccos \left (-1+d x^2\right )} \, dx=-\frac {2 \cos \left (\frac {1}{2} \arccos \left (-1+d x^2\right )\right ) \left (-\sqrt {a+b \arccos \left (-1+d x^2\right )} \cos \left (\frac {1}{2} \arccos \left (-1+d x^2\right )\right )+\sqrt {b} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {a+b \arccos \left (-1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {\sqrt {a+b \arccos \left (-1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right )\right )}{d x} \]

[In]

Integrate[Sqrt[a + b*ArcCos[-1 + d*x^2]],x]

[Out]

(-2*Cos[ArcCos[-1 + d*x^2]/2]*(-(Sqrt[a + b*ArcCos[-1 + d*x^2]]*Cos[ArcCos[-1 + d*x^2]/2]) + Sqrt[b]*Sqrt[Pi]*
Cos[a/(2*b)]*FresnelC[Sqrt[a + b*ArcCos[-1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])] + Sqrt[b]*Sqrt[Pi]*FresnelS[Sqrt[a +
b*ArcCos[-1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*Sin[a/(2*b)]))/(d*x)

Maple [F]

\[\int \sqrt {a +b \arccos \left (d \,x^{2}-1\right )}d x\]

[In]

int((a+b*arccos(d*x^2-1))^(1/2),x)

[Out]

int((a+b*arccos(d*x^2-1))^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \sqrt {a+b \arccos \left (-1+d x^2\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arccos(d*x^2-1))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \sqrt {a+b \arccos \left (-1+d x^2\right )} \, dx=\int \sqrt {a + b \operatorname {acos}{\left (d x^{2} - 1 \right )}}\, dx \]

[In]

integrate((a+b*acos(d*x**2-1))**(1/2),x)

[Out]

Integral(sqrt(a + b*acos(d*x**2 - 1)), x)

Maxima [F]

\[ \int \sqrt {a+b \arccos \left (-1+d x^2\right )} \, dx=\int { \sqrt {b \arccos \left (d x^{2} - 1\right ) + a} \,d x } \]

[In]

integrate((a+b*arccos(d*x^2-1))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*arccos(d*x^2 - 1) + a), x)

Giac [F]

\[ \int \sqrt {a+b \arccos \left (-1+d x^2\right )} \, dx=\int { \sqrt {b \arccos \left (d x^{2} - 1\right ) + a} \,d x } \]

[In]

integrate((a+b*arccos(d*x^2-1))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*arccos(d*x^2 - 1) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \arccos \left (-1+d x^2\right )} \, dx=\int \sqrt {a+b\,\mathrm {acos}\left (d\,x^2-1\right )} \,d x \]

[In]

int((a + b*acos(d*x^2 - 1))^(1/2),x)

[Out]

int((a + b*acos(d*x^2 - 1))^(1/2), x)