\(\int \frac {x^2}{\sqrt {\arccos (a x)}} \, dx\) [94]

Optimal result

Integrand size = 12, antiderivative size = 71 \[ \int \frac {x^2}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a^3} \]

[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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4732, 4491, 3386, 3432} \[ \int \frac {x^2}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a^3} \]

[In]

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

[Out]

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

Rule 3386

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 3432

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

Rule 4491

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 4732

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.07 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.77 \[ \int \frac {x^2}{\sqrt {\arccos (a x)}} \, dx=-\frac {-3 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )-3 \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )-\sqrt {3} \left (\sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )\right )}{24 a^3 \sqrt {\arccos (a x)}} \]

[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]])

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.70

method	result	size
default	\(-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+3 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{12 a^{3}}\)	\(50\)

[In]

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

[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)))

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^2}{\sqrt {\arccos (a x)}} \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F]

\[ \int \frac {x^2}{\sqrt {\arccos (a x)}} \, dx=\int \frac {x^{2}}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \]

[In]

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

[Out]

Integral(x**2/sqrt(acos(a*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2}{\sqrt {\arccos (a x)}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.31 \[ \int \frac {x^2}{\sqrt {\arccos (a x)}} \, dx=-\frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{48 \, a^{3}} + \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{48 \, a^{3}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{16 \, a^{3}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{16 \, a^{3}} \]

[In]

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

[Out]

-(1/48*I - 1/48)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^3 + (1/48*I + 1/48)*sqrt(6)*s
qrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^3 - (1/16*I - 1/16)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)
*sqrt(2)*sqrt(arccos(a*x)))/a^3 + (1/16*I + 1/16)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arccos(a*x)
))/a^3

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2/acos(a*x)^(1/2), x)