3.114 \(\int \frac{x^3}{\cos ^{-1}(a x)^{7/2}} \, dx\)

Optimal. Leaf size=190 \[ \frac{32 \sqrt{2 \pi } \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a^4}+\frac{16 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{15 a^4}-\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{15 \cos ^{-1}(a x)^{3/2}} \]

[Out]

(2*x^3*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (4*x^2)/(5*a^2*ArcCos[a*x]^(3/2)) + (16*x^4)/(15*ArcCos[a*
x]^(3/2)) + (16*x*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcCos[a*x]]) - (128*x^3*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcCos
[a*x]]) + (32*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^4) + (16*Sqrt[Pi]*FresnelC[(2*Sqrt[Ar
cCos[a*x]])/Sqrt[Pi]])/(15*a^4)

________________________________________________________________________________________

Rubi [A]  time = 0.323089, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{32 \sqrt{2 \pi } \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a^4}+\frac{16 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{15 a^4}-\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{15 \cos ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/ArcCos[a*x]^(7/2),x]

[Out]

(2*x^3*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (4*x^2)/(5*a^2*ArcCos[a*x]^(3/2)) + (16*x^4)/(15*ArcCos[a*
x]^(3/2)) + (16*x*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcCos[a*x]]) - (128*x^3*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcCos
[a*x]]) + (32*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^4) + (16*Sqrt[Pi]*FresnelC[(2*Sqrt[Ar
cCos[a*x]])/Sqrt[Pi]])/(15*a^4)

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^3}{\cos ^{-1}(a x)^{7/2}} \, dx &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{6 \int \frac{x^2}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac{1}{5} (8 a) \int \frac{x^4}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{16 x^4}{15 \cos ^{-1}(a x)^{3/2}}-\frac{64}{15} \int \frac{x^3}{\cos ^{-1}(a x)^{3/2}} \, dx+\frac{8 \int \frac{x}{\cos ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{16 x^4}{15 \cos ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^4}-\frac{128 \operatorname{Subst}\left (\int \left (-\frac{\cos (2 x)}{2 \sqrt{x}}-\frac{\cos (4 x)}{2 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{15 a^4}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{16 x^4}{15 \cos ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{64 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{15 a^4}+\frac{64 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{15 a^4}-\frac{32 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{5 a^4}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{16 x^4}{15 \cos ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}-\frac{16 \sqrt{\pi } C\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{5 a^4}+\frac{128 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{15 a^4}+\frac{128 \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{15 a^4}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{16 x^4}{15 \cos ^{-1}(a x)^{3/2}}+\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{32 \sqrt{2 \pi } C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a^4}+\frac{16 \sqrt{\pi } C\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{15 a^4}\\ \end{align*}

Mathematica [C]  time = 4.23985, size = 264, normalized size = 1.39 \[ -\frac{\frac{16 \sqrt{2} \cos ^{-1}(a x)^3 \text{Gamma}\left (\frac{1}{2},-2 i \cos ^{-1}(a x)\right )}{\sqrt{-i \cos ^{-1}(a x)}}-2 \cos ^{-1}(a x) \left (32 \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-4 i \cos ^{-1}(a x)\right )+2 e^{-4 i \cos ^{-1}(a x)} \left (16 e^{4 i \cos ^{-1}(a x)} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},4 i \cos ^{-1}(a x)\right )-8 i \cos ^{-1}(a x)+1\right )+2 e^{4 i \cos ^{-1}(a x)} \left (1+8 i \cos ^{-1}(a x)\right )\right )+16 i \sqrt{2} \left (i \cos ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},2 i \cos ^{-1}(a x)\right )-4 e^{-2 i \cos ^{-1}(a x)} \left (e^{4 i \cos ^{-1}(a x)} \left (1+4 i \cos ^{-1}(a x)\right )-4 i \cos ^{-1}(a x)+1\right ) \cos ^{-1}(a x)-6 \sin \left (2 \cos ^{-1}(a x)\right )-3 \sin \left (4 \cos ^{-1}(a x)\right )}{60 a^4 \cos ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/ArcCos[a*x]^(7/2),x]

[Out]

-((-4*(1 + E^((4*I)*ArcCos[a*x])*(1 + (4*I)*ArcCos[a*x]) - (4*I)*ArcCos[a*x])*ArcCos[a*x])/E^((2*I)*ArcCos[a*x
]) + (16*Sqrt[2]*ArcCos[a*x]^3*Gamma[1/2, (-2*I)*ArcCos[a*x]])/Sqrt[(-I)*ArcCos[a*x]] + (16*I)*Sqrt[2]*(I*ArcC
os[a*x])^(5/2)*Gamma[1/2, (2*I)*ArcCos[a*x]] - 2*ArcCos[a*x]*(2*E^((4*I)*ArcCos[a*x])*(1 + (8*I)*ArcCos[a*x])
+ 32*((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcCos[a*x]] + (2*(1 - (8*I)*ArcCos[a*x] + 16*E^((4*I)*ArcCos[
a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (4*I)*ArcCos[a*x]]))/E^((4*I)*ArcCos[a*x])) - 6*Sin[2*ArcCos[a*x]] - 3*
Sin[4*ArcCos[a*x]])/(60*a^4*ArcCos[a*x]^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.091, size = 139, normalized size = 0.7 \begin{align*} -{\frac{1}{60\,{a}^{4}} \left ( -128\,\sqrt{2}\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{5/2}-64\,\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{5/2}+32\,\sin \left ( 2\,\arccos \left ( ax \right ) \right ) \left ( \arccos \left ( ax \right ) \right ) ^{2}+64\,\sin \left ( 4\,\arccos \left ( ax \right ) \right ) \left ( \arccos \left ( ax \right ) \right ) ^{2}-8\,\arccos \left ( ax \right ) \cos \left ( 2\,\arccos \left ( ax \right ) \right ) -8\,\arccos \left ( ax \right ) \cos \left ( 4\,\arccos \left ( ax \right ) \right ) -6\,\sin \left ( 2\,\arccos \left ( ax \right ) \right ) -3\,\sin \left ( 4\,\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^3/arccos(a*x)^(7/2),x)

[Out]

-1/60/a^4*(-128*2^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)-64*Pi^(1/2)*
FresnelC(2*arccos(a*x)^(1/2)/Pi^(1/2))*arccos(a*x)^(5/2)+32*sin(2*arccos(a*x))*arccos(a*x)^2+64*sin(4*arccos(a
*x))*arccos(a*x)^2-8*arccos(a*x)*cos(2*arccos(a*x))-8*arccos(a*x)*cos(4*arccos(a*x))-6*sin(2*arccos(a*x))-3*si
n(4*arccos(a*x)))/arccos(a*x)^(5/2)

________________________________________________________________________________________

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)^(7/2),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^3/arccos(a*x)^(7/2),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**3/acos(a*x)**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\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^3/arccos(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^3/arccos(a*x)^(7/2), x)