3.113 \(\int \frac{x^4}{\cos ^{-1}(a x)^{7/2}} \, dx\)

Optimal. Leaf size=264 \[ \frac{\sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a^5}-\frac{8 \sqrt{6 \pi } \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{5 a^5}+\frac{5 \sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^5}-\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^5}{3 \cos ^{-1}(a x)^{3/2}} \]

[Out]

(2*x^4*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (16*x^3)/(15*a^2*ArcCos[a*x]^(3/2)) + (4*x^5)/(3*ArcCos[a*
x]^(3/2)) + (32*x^2*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcCos[a*x]]) - (40*x^4*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[ArcCos
[a*x]]) + (Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^5) + (5*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]
*Sqrt[ArcCos[a*x]]])/a^5 - (8*Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(5*a^5) + (5*Sqrt[(5*Pi)/2]*F
resnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^5)

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Rubi [A]  time = 0.38294, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4634, 4720, 4632, 3304, 3352} \[ \frac{\sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a^5}-\frac{8 \sqrt{6 \pi } \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{5 a^5}+\frac{5 \sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^5}-\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^5}{3 \cos ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCos[a*x]^(7/2),x]

[Out]

(2*x^4*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (16*x^3)/(15*a^2*ArcCos[a*x]^(3/2)) + (4*x^5)/(3*ArcCos[a*
x]^(3/2)) + (32*x^2*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcCos[a*x]]) - (40*x^4*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[ArcCos
[a*x]]) + (Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^5) + (5*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]
*Sqrt[ArcCos[a*x]]])/a^5 - (8*Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(5*a^5) + (5*Sqrt[(5*Pi)/2]*F
resnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^5)

Rule 4634

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

Rule 4720

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

Rule 4632

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcCo
s[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1
), Cos[x]^(m - 1)*(m - (m + 1)*Cos[x]^2), x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] &&
GeQ[n, -2] && LtQ[n, -1]

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]

Rubi steps

\begin{align*} \int \frac{x^4}{\cos ^{-1}(a x)^{7/2}} \, dx &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{8 \int \frac{x^3}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx}{5 a}+(2 a) \int \frac{x^5}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^5}{3 \cos ^{-1}(a x)^{3/2}}-\frac{20}{3} \int \frac{x^4}{\cos ^{-1}(a x)^{3/2}} \, dx+\frac{16 \int \frac{x^2}{\cos ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^5}{3 \cos ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\cos ^{-1}(a x)}}+\frac{32 \operatorname{Subst}\left (\int \left (-\frac{\cos (x)}{4 \sqrt{x}}-\frac{3 \cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{5 a^5}-\frac{40 \operatorname{Subst}\left (\int \left (-\frac{\cos (x)}{8 \sqrt{x}}-\frac{9 \cos (3 x)}{16 \sqrt{x}}-\frac{5 \cos (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{3 a^5}\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^5}{3 \cos ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\cos ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{6 a^5}-\frac{24 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^5}+\frac{15 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^5}\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^5}{3 \cos ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\cos ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{5 a^5}+\frac{10 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{3 a^5}-\frac{48 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{5 a^5}+\frac{15 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{a^5}\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^5}{3 \cos ^{-1}(a x)^{3/2}}+\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\cos ^{-1}(a x)}}+\frac{\sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a^5}+\frac{5 \sqrt{\frac{3 \pi }{2}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^5}-\frac{8 \sqrt{6 \pi } C\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{5 a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} C\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^5}\\ \end{align*}

Mathematica [C]  time = 7.73285, size = 418, normalized size = 1.58 \[ -\frac{2 \left (-4 \cos ^{-1}(a x) \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a x)\right )+e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left (-4 e^{i \cos ^{-1}(a x)} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a x)\right )+4 i \cos ^{-1}(a x)-2\right )-6 \sqrt{1-a^2 x^2}-2 i e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left (2 \cos ^{-1}(a x)-i\right )\right )-5 \cos ^{-1}(a x) \left (20 \sqrt{5} \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-5 i \cos ^{-1}(a x)\right )+e^{-5 i \cos ^{-1}(a x)} \left (20 \sqrt{5} e^{5 i \cos ^{-1}(a x)} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},5 i \cos ^{-1}(a x)\right )-20 i \cos ^{-1}(a x)+2\right )+2 e^{5 i \cos ^{-1}(a x)} \left (1+10 i \cos ^{-1}(a x)\right )\right )+9 \left (-2 \sin \left (3 \cos ^{-1}(a x)\right )-2 \cos ^{-1}(a x) \left (6 \sqrt{3} \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-3 i \cos ^{-1}(a x)\right )+e^{-3 i \cos ^{-1}(a x)} \left (6 \sqrt{3} e^{3 i \cos ^{-1}(a x)} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},3 i \cos ^{-1}(a x)\right )-6 i \cos ^{-1}(a x)+1\right )+e^{3 i \cos ^{-1}(a x)} \left (1+6 i \cos ^{-1}(a x)\right )\right )\right )-6 \sin \left (5 \cos ^{-1}(a x)\right )}{240 a^5 \cos ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/ArcCos[a*x]^(7/2),x]

[Out]

-(2*(-6*Sqrt[1 - a^2*x^2] - (2*I)*E^(I*ArcCos[a*x])*ArcCos[a*x]*(-I + 2*ArcCos[a*x]) - 4*((-I)*ArcCos[a*x])^(3
/2)*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + (ArcCos[a*x]*(-2 + (4*I)*ArcCos[a*x] - 4*E^(I*ArcCos[a*x])*(I*A
rcCos[a*x])^(3/2)*Gamma[1/2, I*ArcCos[a*x]]))/E^(I*ArcCos[a*x])) - 5*ArcCos[a*x]*(2*E^((5*I)*ArcCos[a*x])*(1 +
(10*I)*ArcCos[a*x]) + 20*Sqrt[5]*((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-5*I)*ArcCos[a*x]] + (2 - (20*I)*ArcCos
[a*x] + 20*Sqrt[5]*E^((5*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (5*I)*ArcCos[a*x]])/E^((5*I)*ArcCos[
a*x])) + 9*(-2*ArcCos[a*x]*(E^((3*I)*ArcCos[a*x])*(1 + (6*I)*ArcCos[a*x]) + 6*Sqrt[3]*((-I)*ArcCos[a*x])^(3/2)
*Gamma[1/2, (-3*I)*ArcCos[a*x]] + (1 - (6*I)*ArcCos[a*x] + 6*Sqrt[3]*E^((3*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/
2)*Gamma[1/2, (3*I)*ArcCos[a*x]])/E^((3*I)*ArcCos[a*x])) - 2*Sin[3*ArcCos[a*x]]) - 6*Sin[5*ArcCos[a*x]])/(240*
a^5*ArcCos[a*x]^(5/2))

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Maple [A]  time = 0.11, size = 225, normalized size = 0.9 \begin{align*} -{\frac{1}{120\,{a}^{5}} \left ( -100\,\sqrt{2}\sqrt{\pi }\sqrt{5}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{5}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{5/2}-108\,\sqrt{2}\sqrt{\pi }\sqrt{3}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{5/2}-8\,\sqrt{2}\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{5/2}+8\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}+108\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sin \left ( 3\,\arccos \left ( ax \right ) \right ) +100\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sin \left ( 5\,\arccos \left ( ax \right ) \right ) -4\,ax\arccos \left ( ax \right ) -18\,\arccos \left ( ax \right ) \cos \left ( 3\,\arccos \left ( ax \right ) \right ) -10\,\arccos \left ( ax \right ) \cos \left ( 5\,\arccos \left ( ax \right ) \right ) -6\,\sqrt{-{a}^{2}{x}^{2}+1}-9\,\sin \left ( 3\,\arccos \left ( ax \right ) \right ) -3\,\sin \left ( 5\,\arccos \left ( ax \right ) \right ) \right ) \left ( \arccos \left ( ax \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccos(a*x)^(7/2),x)

[Out]

-1/120/a^5*(-100*2^(1/2)*Pi^(1/2)*5^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/
2)-108*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)-8*2^(1/
2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)+8*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)+
108*arccos(a*x)^2*sin(3*arccos(a*x))+100*arccos(a*x)^2*sin(5*arccos(a*x))-4*a*x*arccos(a*x)-18*arccos(a*x)*cos
(3*arccos(a*x))-10*arccos(a*x)*cos(5*arccos(a*x))-6*(-a^2*x^2+1)^(1/2)-9*sin(3*arccos(a*x))-3*sin(5*arccos(a*x
)))/arccos(a*x)^(5/2)

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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)^(7/2),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)^(7/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\arccos \left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^4/arccos(a*x)^(7/2), x)