3.93 \(\int \frac{x^3}{\sqrt{\cos ^{-1}(a x)}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^4}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4} \]

[Out]

-(Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^4) - (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt
[Pi]])/(4*a^4)

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Rubi [A]  time = 0.0777967, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4636, 4406, 3305, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^4}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[ArcCos[a*x]],x]

[Out]

-(Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^4) - (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt
[Pi]])/(4*a^4)

Rule 4636

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Cos[x]^m*Sin[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

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 3305

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 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

\begin{align*} \int \frac{x^3}{\sqrt{\cos ^{-1}(a x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}+\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{4 a^4}-\frac{\operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{2 a^4}\\ &=-\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^4}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4}\\ \end{align*}

Mathematica [C]  time = 0.0678617, size = 130, normalized size = 2. \[ -\frac{-2 \sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \cos ^{-1}(a x)\right )-2 \sqrt{2} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \cos ^{-1}(a x)\right )-\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \cos ^{-1}(a x)\right )-\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \cos ^{-1}(a x)\right )}{32 a^4 \sqrt{\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/Sqrt[ArcCos[a*x]],x]

[Out]

-(-2*Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a*x]] - 2*Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2,
(2*I)*ArcCos[a*x]] - Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-4*I)*ArcCos[a*x]] - Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (4
*I)*ArcCos[a*x]])/(32*a^4*Sqrt[ArcCos[a*x]])

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Maple [A]  time = 0.07, size = 43, normalized size = 0.7 \begin{align*} -{\frac{\sqrt{\pi }}{16\,{a}^{4}} \left ( \sqrt{2}{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +4\,{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16/a^4*Pi^(1/2)*(2^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))+4*FresnelS(2*arccos(a*x)^(1/2)/Pi^(
1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{\operatorname{acos}{\left (a x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/sqrt(acos(a*x)), x)

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Giac [B]  time = 1.3581, size = 153, normalized size = 2.35 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (\sqrt{2}{\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{32 \, a^{4}{\left (i - 1\right )}} + \frac{\sqrt{\pi } i \operatorname{erf}\left ({\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{8 \, a^{4}{\left (i - 1\right )}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{2}{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{32 \, a^{4}{\left (i - 1\right )}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{8 \, a^{4}{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*sqrt(2)*sqrt(pi)*i*erf(sqrt(2)*(i - 1)*sqrt(arccos(a*x)))/(a^4*(i - 1)) + 1/8*sqrt(pi)*i*erf((i - 1)*sqrt
(arccos(a*x)))/(a^4*(i - 1)) - 1/32*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*(i + 1)*sqrt(arccos(a*x)))/(a^4*(i - 1)) - 1
/8*sqrt(pi)*erf(-(i + 1)*sqrt(arccos(a*x)))/(a^4*(i - 1))