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

Optimal. Leaf size=55 \[ \frac {(a+b x) \sqrt {\cos ^{-1}(a+b x)}}{b}-\frac {\sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a+b x)}\right )}{b} \]

[Out]

-1/2*FresnelC(2^(1/2)/Pi^(1/2)*arccos(b*x+a)^(1/2))*2^(1/2)*Pi^(1/2)/b+(b*x+a)*arccos(b*x+a)^(1/2)/b

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Rubi [A]  time = 0.08, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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]

Int[Sqrt[ArcCos[a + b*x]],x]

[Out]

((a + b*x)*Sqrt[ArcCos[a + b*x]])/b - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a + b*x]]])/b

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4620

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
(x*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 4804

Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCos[x])^n, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

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*}

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Mathematica [C]  time = 0.04, size = 90, normalized size = 1.64 \[ -\frac {-\frac {\sqrt {\cos ^{-1}(a+b x)} \Gamma \left (\frac {3}{2},-i \cos ^{-1}(a+b x)\right )}{2 \sqrt {-i \cos ^{-1}(a+b x)}}-\frac {\sqrt {\cos ^{-1}(a+b x)} \Gamma \left (\frac {3}{2},i \cos ^{-1}(a+b x)\right )}{2 \sqrt {i \cos ^{-1}(a+b x)}}}{b} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[ArcCos[a + b*x]],x]

[Out]

-((-1/2*(Sqrt[ArcCos[a + b*x]]*Gamma[3/2, (-I)*ArcCos[a + b*x]])/Sqrt[(-I)*ArcCos[a + b*x]] - (Sqrt[ArcCos[a +
b*x]]*Gamma[3/2, I*ArcCos[a + b*x]])/(2*Sqrt[I*ArcCos[a + b*x]]))/b)

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

[Out]

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

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giac [B]  time = 0.27, size = 117, normalized size = 2.13 \[ \frac {\sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \left (b x + a\right )}}{i - 1}\right )}{4 \, b {\left (i - 1\right )}} + \frac {\sqrt {\arccos \left (b x + a\right )} e^{\left (i \arccos \left (b x + a\right )\right )}}{2 \, b} + \frac {\sqrt {\arccos \left (b x + a\right )} e^{\left (-i \arccos \left (b x + a\right )\right )}}{2 \, b} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \left (b x + a\right )}}{i - 1}\right )}{4 \, b {\left (i - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*sqrt(pi)*i*erf(sqrt(2)*sqrt(arccos(b*x + a))/(i - 1))/(b*(i - 1)) + 1/2*sqrt(arccos(b*x + a))*e^(i
*arccos(b*x + a))/b + 1/2*sqrt(arccos(b*x + a))*e^(-i*arccos(b*x + a))/b - 1/4*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*i
*sqrt(arccos(b*x + a))/(i - 1))/(b*(i - 1))

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maple [A]  time = 0.15, size = 66, normalized size = 1.20 \[ \frac {-\FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (b x +a \right )}+2 \arccos \left (b x +a \right ) x b +2 \arccos \left (b x +a \right ) a}{2 b \sqrt {\arccos \left (b x +a \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/b/arccos(b*x+a)^(1/2)*(-FresnelC(2^(1/2)/Pi^(1/2)*arccos(b*x+a)^(1/2))*2^(1/2)*Pi^(1/2)*arccos(b*x+a)^(1/2
)+2*arccos(b*x+a)*x*b+2*arccos(b*x+a)*a)

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\mathrm {acos}\left (a+b\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(a + b*x)^(1/2),x)

[Out]

int(acos(a + b*x)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(acos(a + b*x)), x)

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