3.2.3 \(\int \frac {x^2}{\text {ArcCos}(a x)^{3/2}} \, dx\) [103]

Optimal. Leaf size=97 \[ \frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\text {ArcCos}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a^3}-\frac {\sqrt {\frac {3 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a^3} \]

[Out]

-1/2*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3-1/2*FresnelC(6^(1/2)/Pi^(1/2)*arccos(a*
x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3+2*x^2*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4728, 3385, 3433} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a^3}-\frac {\sqrt {\frac {3 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a^3}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\text {ArcCos}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/ArcCos[a*x]^(3/2),x]

[Out]

(2*x^2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^3 - (S
qrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/a^3

Rule 3385

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 3433

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

Rule 4728

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(-x^m)*Sqrt[1 - c^2*x^2]*((a + b*Arc
Cos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), C
os[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cos[-a/b + x/b]^2), x], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c},
x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 159, normalized size = 1.64 \begin {gather*} \frac {i \left (-2 i \sqrt {1-a^2 x^2}+\sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-i \text {ArcCos}(a x)\right )-\sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},i \text {ArcCos}(a x)\right )+\sqrt {3} \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-3 i \text {ArcCos}(a x)\right )-\sqrt {3} \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},3 i \text {ArcCos}(a x)\right )-2 i \sin (3 \text {ArcCos}(a x))\right )}{4 a^3 \sqrt {\text {ArcCos}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/ArcCos[a*x]^(3/2),x]

[Out]

((I/4)*((-2*I)*Sqrt[1 - a^2*x^2] + Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - Sqrt[I*ArcCos[a*x]]*G
amma[1/2, I*ArcCos[a*x]] + Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - Sqrt[3]*Sqrt[I*ArcC
os[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]] - (2*I)*Sin[3*ArcCos[a*x]]))/(a^3*Sqrt[ArcCos[a*x]])

________________________________________________________________________________________

Maple [A]
time = 0.21, size = 94, normalized size = 0.97

method	result	size
default	\(\frac {-\sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-\FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }+\sqrt {-a^{2} x^{2}+1}+\sin \left (3 \arccos \left (a x \right )\right )}{2 a^{3} \sqrt {\arccos \left (a x \right )}}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/arccos(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2/a^3*(-3^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))-Fres
nelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)+(-a^2*x^2+1)^(1/2)+sin(3*arccos(a*
x)))/arccos(a*x)^(1/2)

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arccos(a*x)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arccos(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/acos(a*x)**(3/2),x)

[Out]

Integral(x**2/acos(a*x)**(3/2), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arccos(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(x^2/arccos(a*x)^(3/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/acos(a*x)^(3/2),x)

[Out]

int(x^2/acos(a*x)^(3/2), x)

________________________________________________________________________________________