\(\int \frac {x^3}{\arccos (a x)^{3/2}} \, dx\) [102]

Optimal result

Integrand size = 12, antiderivative size = 91 \[ \int \frac {x^3}{\arccos (a x)^{3/2}} \, dx=\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^4}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{a^4} \]

[Out]

-1/2*FresnelC(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-FresnelC(2*arccos(a*x)^(1/2)/Pi^(1/2)
)*Pi^(1/2)/a^4+2*x^3*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4728, 3385, 3433} \[ \int \frac {x^3}{\arccos (a x)^{3/2}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^4}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{a^4}+\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}} \]

[In]

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

[Out]

(2*x^3*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^4 -
(Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/a^4

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.69 \[ \int \frac {x^3}{\arccos (a x)^{3/2}} \, dx=\frac {i \sqrt {2} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-2 i \arccos (a x)\right )-i \sqrt {2} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},2 i \arccos (a x)\right )+i \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-4 i \arccos (a x)\right )-i \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},4 i \arccos (a x)\right )+2 \sin (2 \arccos (a x))+\sin (4 \arccos (a x))}{4 a^4 \sqrt {\arccos (a x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {-2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-4 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+2 \sin \left (2 \arccos \left (a x \right )\right )+\sin \left (4 \arccos \left (a x \right )\right )}{4 a^{4} \sqrt {\arccos \left (a x \right )}}\)	\(81\)

[In]

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

[Out]

1/4/a^4/arccos(a*x)^(1/2)*(-2*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2)
)-4*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*arccos(a*x)^(1/2)/Pi^(1/2))+2*sin(2*arccos(a*x))+sin(4*arccos(a*x)))

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3/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]

\[ \int \frac {x^3}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^{3}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral(x**3/acos(a*x)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^3}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

int(x^3/acos(a*x)^(3/2), x)