3.86 \(\int x^4 \cos ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=298 \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{32 a^5}+\frac{\sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{320 a^5}+\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{60 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{1600 a^5}-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}-\frac{x^3 \sqrt{\cos ^{-1}(a x)}}{15 a^2}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^3}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x \sqrt{\cos ^{-1}(a x)}}{5 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}-\frac{3}{100} x^5 \sqrt{\cos ^{-1}(a x)} \]

[Out]

(-2*x*Sqrt[ArcCos[a*x]])/(5*a^4) - (x^3*Sqrt[ArcCos[a*x]])/(15*a^2) - (3*x^5*Sqrt[ArcCos[a*x]])/100 - (4*Sqrt[
1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(15*a^5) - (2*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(15*a^3) - (x^4*Sqrt[1
- a^2*x^2]*ArcCos[a*x]^(3/2))/(10*a) + (x^5*ArcCos[a*x]^(5/2))/5 + (15*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[Arc
Cos[a*x]]])/(32*a^5) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(60*a^5) + (Sqrt[(3*Pi)/2]*FresnelC
[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(320*a^5) + (3*Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(1600*a^5)

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Rubi [A]  time = 0.789701, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4630, 4708, 4678, 4620, 4724, 3304, 3352, 3312} \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{32 a^5}+\frac{\sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{320 a^5}+\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{60 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{1600 a^5}-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}-\frac{x^3 \sqrt{\cos ^{-1}(a x)}}{15 a^2}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^3}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x \sqrt{\cos ^{-1}(a x)}}{5 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}-\frac{3}{100} x^5 \sqrt{\cos ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCos[a*x]^(5/2),x]

[Out]

(-2*x*Sqrt[ArcCos[a*x]])/(5*a^4) - (x^3*Sqrt[ArcCos[a*x]])/(15*a^2) - (3*x^5*Sqrt[ArcCos[a*x]])/100 - (4*Sqrt[
1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(15*a^5) - (2*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(15*a^3) - (x^4*Sqrt[1
- a^2*x^2]*ArcCos[a*x]^(3/2))/(10*a) + (x^5*ArcCos[a*x]^(5/2))/5 + (15*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[Arc
Cos[a*x]]])/(32*a^5) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(60*a^5) + (Sqrt[(3*Pi)/2]*FresnelC
[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(320*a^5) + (3*Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(1600*a^5)

Rule 4630

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcCos[c*x])^n)/(m
+ 1), x] + Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4708

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcCos[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4620

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
(x*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3304

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 3352

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

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int x^4 \cos ^{-1}(a x)^{5/2} \, dx &=\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}+\frac{1}{2} a \int \frac{x^5 \cos ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}-\frac{3}{20} \int x^4 \sqrt{\cos ^{-1}(a x)} \, dx+\frac{2 \int \frac{x^3 \cos ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{5 a}\\ &=-\frac{3}{100} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}+\frac{4 \int \frac{x \cos ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{15 a^3}-\frac{\int x^2 \sqrt{\cos ^{-1}(a x)} \, dx}{5 a^2}-\frac{1}{200} (3 a) \int \frac{x^5}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx\\ &=-\frac{x^3 \sqrt{\cos ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos ^5(x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{200 a^5}-\frac{2 \int \sqrt{\cos ^{-1}(a x)} \, dx}{5 a^4}-\frac{\int \frac{x^3}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx}{30 a}\\ &=-\frac{2 x \sqrt{\cos ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\cos ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}+\frac{3 \operatorname{Subst}\left (\int \left (\frac{5 \cos (x)}{8 \sqrt{x}}+\frac{5 \cos (3 x)}{16 \sqrt{x}}+\frac{\cos (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{200 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cos ^3(x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{30 a^5}-\frac{\int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx}{5 a^3}\\ &=-\frac{2 x \sqrt{\cos ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\cos ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{3200 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{640 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{320 a^5}+\frac{\operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{4 \sqrt{x}}+\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{30 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^5}\\ &=-\frac{2 x \sqrt{\cos ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\cos ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}+\frac{3 \operatorname{Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{1600 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{120 a^5}+\frac{3 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{320 a^5}+\frac{3 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{160 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{40 a^5}+\frac{2 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{5 a^5}\\ &=-\frac{2 x \sqrt{\cos ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\cos ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}+\frac{3 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{160 a^5}+\frac{\sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{5 a^5}+\frac{\sqrt{\frac{3 \pi }{2}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{320 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} C\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{1600 a^5}+\frac{\operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{60 a^5}+\frac{\operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{20 a^5}\\ &=-\frac{2 x \sqrt{\cos ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\cos ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}+\frac{11 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{160 a^5}+\frac{\sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{5 a^5}+\frac{\sqrt{\frac{\pi }{6}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{60 a^5}+\frac{\sqrt{\frac{3 \pi }{2}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{320 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} C\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{1600 a^5}\\ \end{align*}

Mathematica [C]  time = 0.191514, size = 212, normalized size = 0.71 \[ -\frac{-625 \sqrt{3} \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{7}{2},-3 i \cos ^{-1}(a x)\right )-27 \sqrt{5} \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{7}{2},-5 i \cos ^{-1}(a x)\right )+33750 \sqrt{\cos ^{-1}(a x)^2} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},i \cos ^{-1}(a x)\right )+33750 \sqrt{i \cos ^{-1}(a x)} \sqrt{\cos ^{-1}(a x)^2} \text{Gamma}\left (\frac{7}{2},-i \cos ^{-1}(a x)\right )-625 \sqrt{3} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{7}{2},3 i \cos ^{-1}(a x)\right )-27 \sqrt{5} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{7}{2},5 i \cos ^{-1}(a x)\right )}{540000 a^5 \cos ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4*ArcCos[a*x]^(5/2),x]

[Out]

-(33750*Sqrt[I*ArcCos[a*x]]*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, (-I)*ArcCos[a*x]] + 33750*Sqrt[(-I)*ArcCos[a*x]]*Sq
rt[ArcCos[a*x]^2]*Gamma[7/2, I*ArcCos[a*x]] - 625*Sqrt[3]*((-I)*ArcCos[a*x])^(3/2)*Gamma[7/2, (-3*I)*ArcCos[a*
x]] - 625*Sqrt[3]*(I*ArcCos[a*x])^(3/2)*Gamma[7/2, (3*I)*ArcCos[a*x]] - 27*Sqrt[5]*((-I)*ArcCos[a*x])^(3/2)*Ga
mma[7/2, (-5*I)*ArcCos[a*x]] - 27*Sqrt[5]*(I*ArcCos[a*x])^(3/2)*Gamma[7/2, (5*I)*ArcCos[a*x]])/(540000*a^5*Arc
Cos[a*x]^(3/2))

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Maple [A]  time = 0.109, size = 233, normalized size = 0.8 \begin{align*}{\frac{1}{144000\,{a}^{5}} \left ( 18000\,ax \left ( \arccos \left ( ax \right ) \right ) ^{3}+9000\, \left ( \arccos \left ( ax \right ) \right ) ^{3}\cos \left ( 3\,\arccos \left ( ax \right ) \right ) +1800\, \left ( \arccos \left ( ax \right ) \right ) ^{3}\cos \left ( 5\,\arccos \left ( ax \right ) \right ) +27\,\sqrt{5}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{5}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +625\,\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -45000\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}-7500\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sin \left ( 3\,\arccos \left ( ax \right ) \right ) -900\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sin \left ( 5\,\arccos \left ( ax \right ) \right ) +33750\,\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -67500\,ax\arccos \left ( ax \right ) -3750\,\arccos \left ( ax \right ) \cos \left ( 3\,\arccos \left ( ax \right ) \right ) -270\,\arccos \left ( ax \right ) \cos \left ( 5\,\arccos \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccos(a*x)^(5/2),x)

[Out]

1/144000/a^5*(18000*a*x*arccos(a*x)^3+9000*arccos(a*x)^3*cos(3*arccos(a*x))+1800*arccos(a*x)^3*cos(5*arccos(a*
x))+27*5^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2))+625*3^(
1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))-45000*arccos(a*x)
^2*(-a^2*x^2+1)^(1/2)-7500*arccos(a*x)^2*sin(3*arccos(a*x))-900*arccos(a*x)^2*sin(5*arccos(a*x))+33750*2^(1/2)
*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))-67500*a*x*arccos(a*x)-3750*arccos(a*x
)*cos(3*arccos(a*x))-270*arccos(a*x)*cos(5*arccos(a*x)))/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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acos(a*x)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.45306, size = 738, normalized size = 2.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccos(a*x)^(5/2),x, algorithm="giac")

[Out]

1/320*i*arccos(a*x)^(3/2)*e^(5*i*arccos(a*x))/a^5 + 1/160*arccos(a*x)^(5/2)*e^(5*i*arccos(a*x))/a^5 + 5/192*i*
arccos(a*x)^(3/2)*e^(3*i*arccos(a*x))/a^5 + 1/32*arccos(a*x)^(5/2)*e^(3*i*arccos(a*x))/a^5 + 5/32*i*arccos(a*x
)^(3/2)*e^(i*arccos(a*x))/a^5 + 1/16*arccos(a*x)^(5/2)*e^(i*arccos(a*x))/a^5 - 5/32*i*arccos(a*x)^(3/2)*e^(-i*
arccos(a*x))/a^5 + 1/16*arccos(a*x)^(5/2)*e^(-i*arccos(a*x))/a^5 - 5/192*i*arccos(a*x)^(3/2)*e^(-3*i*arccos(a*
x))/a^5 + 1/32*arccos(a*x)^(5/2)*e^(-3*i*arccos(a*x))/a^5 - 1/320*i*arccos(a*x)^(3/2)*e^(-5*i*arccos(a*x))/a^5
+ 1/160*arccos(a*x)^(5/2)*e^(-5*i*arccos(a*x))/a^5 - 3/32000*sqrt(10)*sqrt(pi)*i*erf(sqrt(10)*sqrt(arccos(a*x
))/(i - 1))/(a^5*(i - 1)) - 5/2304*sqrt(6)*sqrt(pi)*i*erf(sqrt(6)*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) - 1
5/128*sqrt(2)*sqrt(pi)*i*erf(sqrt(2)*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) - 3/3200*sqrt(arccos(a*x))*e^(5*
i*arccos(a*x))/a^5 - 5/384*sqrt(arccos(a*x))*e^(3*i*arccos(a*x))/a^5 - 15/64*sqrt(arccos(a*x))*e^(i*arccos(a*x
))/a^5 - 15/64*sqrt(arccos(a*x))*e^(-i*arccos(a*x))/a^5 - 5/384*sqrt(arccos(a*x))*e^(-3*i*arccos(a*x))/a^5 - 3
/3200*sqrt(arccos(a*x))*e^(-5*i*arccos(a*x))/a^5 + 3/32000*sqrt(10)*sqrt(pi)*erf(-sqrt(10)*i*sqrt(arccos(a*x))
/(i - 1))/(a^5*(i - 1)) + 5/2304*sqrt(6)*sqrt(pi)*erf(-sqrt(6)*i*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) + 15
/128*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*i*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1))