\(\int \frac {1}{\arccos (a x)^{3/2}} \, dx\) [105]

Optimal result

Integrand size = 8, antiderivative size = 59 \[ \int \frac {1}{\arccos (a x)^{3/2}} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4718, 4810, 3385, 3433} \[ \int \frac {1}{\arccos (a x)^{3/2}} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a} \]

[In]

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

[Out]

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

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 4718

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-Sqrt[1 - c^2*x^2])*((a + b*ArcCos[c*x])^(n +
1)/(b*c*(n + 1))), x] - Dist[c/(b*(n + 1)), Int[x*((a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\arccos (a x)^{3/2}} \, dx=-\frac {-2 \sqrt {1-a^2 x^2}-i \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+i \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )}{a \sqrt {\arccos (a x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12

method	result	size
default	\(-\frac {\sqrt {2}\, \left (2 \arccos \left (a x \right ) \pi \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-\sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{a \sqrt {\pi }\, \arccos \left (a x \right )}\)	\(66\)

[In]

int(1/arccos(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/a*2^(1/2)/Pi^(1/2)/arccos(a*x)*(2*arccos(a*x)*Pi*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))-2^(1/2)*arcco
s(a*x)^(1/2)*Pi^(1/2)*(-a^2*x^2+1)^(1/2))

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/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]

\[ \int \frac {1}{\arccos (a x)^{3/2}} \, dx=\int \frac {1}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(1/acos(a*x)**(3/2),x)

[Out]

Integral(acos(a*x)**(-3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/arccos(a*x)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{\arccos (a x)^{3/2}} \, dx=\int { \frac {1}{\arccos \left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/arccos(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(arccos(a*x)^(-3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\arccos (a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \]

[In]

int(1/acos(a*x)^(3/2),x)

[Out]

int(1/acos(a*x)^(3/2), x)