\(\int \frac {x}{\arccos (a x)^{3/2}} \, dx\) [104]

Optimal result

Integrand size = 10, antiderivative size = 55 \[ \int \frac {x}{\arccos (a x)^{3/2}} \, dx=\frac {2 x \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {2 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4728, 3385, 3433} \[ \int \frac {x}{\arccos (a x)^{3/2}} \, dx=\frac {2 x \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {2 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{a^2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\arccos (a x)^{3/2}} \, dx=\frac {-2 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )+\frac {\sin (2 \arccos (a x))}{\sqrt {\arccos (a x)}}}{a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76

method	result	size
default	\(\frac {-2 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+\sin \left (2 \arccos \left (a x \right )\right )}{a^{2} \sqrt {\arccos \left (a x \right )}}\)	\(42\)

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x/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]

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

[In]

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

[Out]

Integral(x/acos(a*x)**(3/2), x)

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(x/arccos(a*x)^(3/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x/acos(a*x)^(3/2), x)