3.94 \(\int \frac {x^2}{\sqrt {\cos ^{-1}(a x)}} \, dx\)

Optimal. Leaf size=71 \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3} \]

[Out]

-1/12*FresnelS(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3-1/4*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a
*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3

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Rubi [A]  time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4636, 4406, 3305, 3351} \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[ArcCos[a*x]],x]

[Out]

-(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(2*a^3) - (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x
]]])/(2*a^3)

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(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 3351

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

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4636

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Cos[x]^m*Sin[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {\cos ^{-1}(a x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {\sin (x)}{4 \sqrt {x}}+\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}-\frac {\operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac {\operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}-\frac {\operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}\\ &=-\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^3}\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 126, normalized size = 1.77 \[ -\frac {-3 \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \cos ^{-1}(a x)\right )-3 \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \cos ^{-1}(a x)\right )-\sqrt {3} \left (\sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 i \cos ^{-1}(a x)\right )+\sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 i \cos ^{-1}(a x)\right )\right )}{24 a^3 \sqrt {\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/Sqrt[ArcCos[a*x]],x]

[Out]

-1/24*(-3*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - 3*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]
] - Sqrt[3]*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcC
os[a*x]]))/(a^3*Sqrt[ArcCos[a*x]])

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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(x^2/arccos(a*x)^(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 = 0.34, size = 135, normalized size = 1.90 \[ \frac {\sqrt {6} \sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {6} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{24 \, a^{3} {\left (i - 1\right )}} + \frac {\sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{8 \, a^{3} {\left (i - 1\right )}} - \frac {\sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {6} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{24 \, a^{3} {\left (i - 1\right )}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{8 \, a^{3} {\left (i - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(6)*sqrt(pi)*i*erf(-sqrt(6)*i*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1)) + 1/8*sqrt(2)*sqrt(pi)*i*erf(-
sqrt(2)*i*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1)) - 1/24*sqrt(6)*sqrt(pi)*erf(sqrt(6)*sqrt(arccos(a*x))/(i -
1))/(a^3*(i - 1)) - 1/8*sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1))

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maple [A]  time = 0.17, size = 50, normalized size = 0.70 \[ -\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\sqrt {3}\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+3 \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{12 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/a^3*2^(1/2)*Pi^(1/2)*(3^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))+3*FresnelS(2^(1/2)/Pi
^(1/2)*arccos(a*x)^(1/2)))

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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(x^2/arccos(a*x)^(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 = -0.01 \[ \int \frac {x^2}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/acos(a*x)^(1/2),x)

[Out]

int(x^2/acos(a*x)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/sqrt(acos(a*x)), x)

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