3.92 \(\int \frac{x^4}{\sqrt{\cos ^{-1}(a x)}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{4 a^5}-\frac{\sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^5}-\frac{\sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^5} \]

[Out]

-(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(4*a^5) - (Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos
[a*x]]])/(8*a^5) - (Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^5)

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Rubi [A]  time = 0.104841, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4636, 4406, 3305, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{4 a^5}-\frac{\sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^5}-\frac{\sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[ArcCos[a*x]],x]

[Out]

-(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(4*a^5) - (Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos
[a*x]]])/(8*a^5) - (Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^5)

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^4}{\sqrt{\cos ^{-1}(a x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^4(x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{\sin (x)}{8 \sqrt{x}}+\frac{3 \sin (3 x)}{16 \sqrt{x}}+\frac{\sin (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^5}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac{\operatorname{Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{8 a^5}-\frac{\operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{4 a^5}-\frac{3 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{8 a^5}\\ &=-\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{4 a^5}-\frac{\sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^5}-\frac{\sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^5}\\ \end{align*}

Mathematica [C]  time = 0.0999738, size = 192, normalized size = 1.81 \[ -\frac{-10 \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a x)\right )-10 \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a x)\right )-5 \sqrt{3} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \cos ^{-1}(a x)\right )-5 \sqrt{3} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \cos ^{-1}(a x)\right )-\sqrt{5} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-5 i \cos ^{-1}(a x)\right )-\sqrt{5} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},5 i \cos ^{-1}(a x)\right )}{160 a^5 \sqrt{\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/Sqrt[ArcCos[a*x]],x]

[Out]

-(-10*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - 10*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]] -
5*Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - 5*Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3
*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-5*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[I*ArcCos[a*x]]
*Gamma[1/2, (5*I)*ArcCos[a*x]])/(160*a^5*Sqrt[ArcCos[a*x]])

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Maple [A]  time = 0.078, size = 72, normalized size = 0.7 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{80\,{a}^{5}} \left ( \sqrt{5}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{5}}{\sqrt{\pi }}\sqrt{\arccos \left ( ax \right ) }} \right ) +5\,\sqrt{3}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +10\,{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccos(a*x)^(1/2),x)

[Out]

-1/80/a^5*2^(1/2)*Pi^(1/2)*(5^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2))+5*3^(1/2)*FresnelS(2^
(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))+10*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(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^4/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^4/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^{4}}{\sqrt{\operatorname{acos}{\left (a x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acos(a*x)**(1/2),x)

[Out]

Integral(x**4/sqrt(acos(a*x)), x)

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Giac [B]  time = 1.41067, size = 273, normalized size = 2.58 \begin{align*} \frac{\sqrt{10} \sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{10} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{160 \, a^{5}{\left (i - 1\right )}} + \frac{\sqrt{6} \sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{6} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{32 \, a^{5}{\left (i - 1\right )}} + \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{16 \, a^{5}{\left (i - 1\right )}} - \frac{\sqrt{10} \sqrt{\pi } \operatorname{erf}\left (\frac{\sqrt{10} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{160 \, a^{5}{\left (i - 1\right )}} - \frac{\sqrt{6} \sqrt{\pi } \operatorname{erf}\left (\frac{\sqrt{6} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{32 \, a^{5}{\left (i - 1\right )}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{16 \, a^{5}{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x)^(1/2),x, algorithm="giac")

[Out]

1/160*sqrt(10)*sqrt(pi)*i*erf(-sqrt(10)*i*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) + 1/32*sqrt(6)*sqrt(pi)*i*e
rf(-sqrt(6)*i*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) + 1/16*sqrt(2)*sqrt(pi)*i*erf(-sqrt(2)*i*sqrt(arccos(a*
x))/(i - 1))/(a^5*(i - 1)) - 1/160*sqrt(10)*sqrt(pi)*erf(sqrt(10)*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) - 1
/32*sqrt(6)*sqrt(pi)*erf(sqrt(6)*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) - 1/16*sqrt(2)*sqrt(pi)*erf(sqrt(2)*
sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1))