3.1.95 \(\int \frac {x}{\sqrt {\text {ArcCos}(a x)}} \, dx\) [95]

Optimal. Leaf size=28 \[ -\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{2 a^2} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4732, 4491, 12, 3386, 3432} \begin {gather*} -\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[ArcCos[a*x]],x]

[Out]

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

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Maple [A]
time = 0.08, size = 21, normalized size = 0.75

method	result	size
default	\(-\frac {\mathrm {S}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\pi }}{2 a^{2}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(acos(a*x)), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.44, size = 35, normalized size = 1.25 \begin {gather*} -\frac {\left (i - 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{8 \, a^{2}} + \frac {\left (i + 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{8 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1/8*I - 1/8)*sqrt(pi)*erf((I - 1)*sqrt(arccos(a*x)))/a^2 + (1/8*I + 1/8)*sqrt(pi)*erf(-(I + 1)*sqrt(arccos(a
*x)))/a^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/acos(a*x)^(1/2), x)

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