3.89 \(\int \sqrt {a+b \cos ^{-1}(1+d x^2)} \, dx\)

Optimal. Leaf size=184 \[ -\frac {2 \sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}+\frac {2 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sin ^2\left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{d x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4812} \[ -\frac {2 \sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}+\frac {2 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sin ^2\left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{d x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*ArcCos[1 + d*x^2]],x]

[Out]

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

Rule 4812

Int[Sqrt[(a_.) + ArcCos[1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[(-2*Sqrt[a + b*ArcCos[1 + d*x^2]]*Sin[ArcC
os[1 + d*x^2]/2]^2)/(d*x), x] + (-Simp[(2*Sqrt[Pi]*Sin[a/(2*b)]*Sin[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]*Cos[a/(2*b)]*Sin[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*} \int \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )} \, dx &=\frac {2 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sqrt {\pi } C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )} \sin ^2\left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{d x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 157, normalized size = 0.85 \[ -\frac {2 \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \left (\sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )-\sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )+\sqrt {\frac {1}{b}} \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}\right )}{\sqrt {\frac {1}{b}} d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \arccos \left (d x^{2} + 1\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \sqrt {a +b \arccos \left (d \,x^{2}+1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found sqrt((-_SAGE_VAR_d*_SAGE_VAR_x^2)-2)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a+b\,\mathrm {acos}\left (d\,x^2+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acos(d*x^2 + 1))^(1/2),x)

[Out]

int((a + b*acos(d*x^2 + 1))^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________