\(\int x^2 \sqrt {\arccos (a x)} \, dx\) [76]

Optimal result

Integrand size = 12, antiderivative size = 86 \[ \int x^2 \sqrt {\arccos (a x)} \, dx=\frac {1}{3} x^3 \sqrt {\arccos (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{4 a^3}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{12 a^3} \]

[Out]

-1/72*FresnelC(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3-1/8*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a
*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3+1/3*x^3*arccos(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4726, 4810, 3393, 3385, 3433} \[ \int x^2 \sqrt {\arccos (a x)} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{4 a^3}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{12 a^3}+\frac {1}{3} x^3 \sqrt {\arccos (a x)} \]

[In]

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

[Out]

(x^3*Sqrt[ArcCos[a*x]])/3 - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(4*a^3) - (Sqrt[Pi/6]*FresnelC
[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(12*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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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 4726

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

Rule 4810

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \sqrt {\arccos (a x)}+\frac {1}{6} a \int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}} \, dx \\ & = \frac {1}{3} x^3 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{6 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 \sqrt {x}}+\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{6 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{24 a^3}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{8 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{12 a^3}-\frac {\text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{4 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\arccos (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{4 a^3}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{12 a^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.49 \[ \int x^2 \sqrt {\arccos (a x)} \, dx=\frac {i \left (9 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {3}{2},-i \arccos (a x)\right )-9 \sqrt {i \arccos (a x)} \Gamma \left (\frac {3}{2},i \arccos (a x)\right )+\sqrt {3} \left (\sqrt {-i \arccos (a x)} \Gamma \left (\frac {3}{2},-3 i \arccos (a x)\right )-\sqrt {i \arccos (a x)} \Gamma \left (\frac {3}{2},3 i \arccos (a x)\right )\right )\right )}{72 a^3 \sqrt {\arccos (a x)}} \]

[In]

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

[Out]

((I/72)*(9*Sqrt[(-I)*ArcCos[a*x]]*Gamma[3/2, (-I)*ArcCos[a*x]] - 9*Sqrt[I*ArcCos[a*x]]*Gamma[3/2, I*ArcCos[a*x
]] + Sqrt[3]*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[3/2, (-3*I)*ArcCos[a*x]] - Sqrt[I*ArcCos[a*x]]*Gamma[3/2, (3*I)*Arc
Cos[a*x]])))/(a^3*Sqrt[ArcCos[a*x]])

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {-\sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-9 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+18 \arccos \left (a x \right ) a x +6 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )}{72 a^{3} \sqrt {\arccos \left (a x \right )}}\)	\(96\)

[In]

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

[Out]

1/72/a^3/arccos(a*x)^(1/2)*(-3^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arcc
os(a*x)^(1/2))-9*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))+18*arccos(a*x
)*a*x+6*arccos(a*x)*cos(3*arccos(a*x)))

Fricas [F(-2)]

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

[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]

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

[In]

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

[Out]

Integral(x**2*sqrt(acos(a*x)), x)

Maxima [F(-2)]

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

[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 = 0.33 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.92 \[ \int x^2 \sqrt {\arccos (a x)} \, dx=\frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{288 \, a^{3}} - \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{288 \, a^{3}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{3}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{3}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} \]

[In]

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

[Out]

(1/288*I + 1/288)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^3 - (1/288*I - 1/288)*sqrt(6
)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^3 + (1/32*I + 1/32)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1
/2)*sqrt(2)*sqrt(arccos(a*x)))/a^3 - (1/32*I - 1/32)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arccos(a
*x)))/a^3 + 1/24*sqrt(arccos(a*x))*e^(3*I*arccos(a*x))/a^3 + 1/8*sqrt(arccos(a*x))*e^(I*arccos(a*x))/a^3 + 1/8
*sqrt(arccos(a*x))*e^(-I*arccos(a*x))/a^3 + 1/24*sqrt(arccos(a*x))*e^(-3*I*arccos(a*x))/a^3

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2*acos(a*x)^(1/2), x)