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

Optimal. Leaf size=191 \[ \frac{2 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a^3}+\frac{6 \sqrt{6 \pi } \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{5 a^3}-\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \cos ^{-1}(a x)^{3/2}} \]

[Out]

(2*x^2*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (8*x)/(15*a^2*ArcCos[a*x]^(3/2)) + (4*x^3)/(5*ArcCos[a*x]^
(3/2)) + (16*Sqrt[1 - a^2*x^2])/(15*a^3*Sqrt[ArcCos[a*x]]) - (24*x^2*Sqrt[1 - a^2*x^2])/(5*a*Sqrt[ArcCos[a*x]]
) + (2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^3) + (6*Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[Ar
cCos[a*x]]])/(5*a^3)

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Rubi [A]  time = 0.340763, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4634, 4720, 4632, 3304, 3352, 4622, 4724} \[ \frac{2 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a^3}+\frac{6 \sqrt{6 \pi } \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{5 a^3}-\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \cos ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^2*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (8*x)/(15*a^2*ArcCos[a*x]^(3/2)) + (4*x^3)/(5*ArcCos[a*x]^
(3/2)) + (16*Sqrt[1 - a^2*x^2])/(15*a^3*Sqrt[ArcCos[a*x]]) - (24*x^2*Sqrt[1 - a^2*x^2])/(5*a*Sqrt[ArcCos[a*x]]
) + (2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^3) + (6*Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[Ar
cCos[a*x]]])/(5*a^3)

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]

Rule 4622

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

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])

Rubi steps

\begin{align*} \int \frac{x^2}{\cos ^{-1}(a x)^{7/2}} \, dx &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4 \int \frac{x}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac{1}{5} (6 a) \int \frac{x^3}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \cos ^{-1}(a x)^{3/2}}-\frac{12}{5} \int \frac{x^2}{\cos ^{-1}(a x)^{3/2}} \, dx+\frac{8 \int \frac{1}{\cos ^{-1}(a x)^{3/2}} \, dx}{15 a^2}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \cos ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\cos ^{-1}(a x)}}-\frac{24 \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^3}+\frac{16 \int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx}{15 a}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \cos ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\cos ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{15 a^3}+\frac{6 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^3}+\frac{18 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^3}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \cos ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\cos ^{-1}(a x)}}-\frac{32 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{15 a^3}+\frac{12 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{5 a^3}+\frac{36 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{5 a^3}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \cos ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\cos ^{-1}(a x)}}-\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a^3}+\frac{6 \sqrt{6 \pi } C\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{5 a^3}\\ \end{align*}

Mathematica [C]  time = 2.72618, size = 281, normalized size = 1.47 \[ -\frac{-4 \cos ^{-1}(a x) \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a x)\right )+e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left (-4 e^{i \cos ^{-1}(a x)} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a x)\right )+4 i \cos ^{-1}(a x)-2\right )-6 \cos ^{-1}(a x) \left (6 \sqrt{3} \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-3 i \cos ^{-1}(a x)\right )+e^{-3 i \cos ^{-1}(a x)} \left (6 \sqrt{3} e^{3 i \cos ^{-1}(a x)} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},3 i \cos ^{-1}(a x)\right )-6 i \cos ^{-1}(a x)+1\right )+e^{3 i \cos ^{-1}(a x)} \left (1+6 i \cos ^{-1}(a x)\right )\right )-6 \sqrt{1-a^2 x^2}-2 i e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left (2 \cos ^{-1}(a x)-i\right )-6 \sin \left (3 \cos ^{-1}(a x)\right )}{60 a^3 \cos ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.091, size = 154, normalized size = 0.8 \begin{align*} -{\frac{1}{30\,{a}^{3}} \left ( -36\,\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}-4\,\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}+36\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sin \left ( 3\,\arccos \left ( ax \right ) \right ) +4\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}-2\,ax\arccos \left ( ax \right ) -6\,\arccos \left ( ax \right ) \cos \left ( 3\,\arccos \left ( ax \right ) \right ) -3\,\sin \left ( 3\,\arccos \left ( ax \right ) \right ) -3\,\sqrt{-{a}^{2}{x}^{2}+1} \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^2/arccos(a*x)^(7/2),x)

[Out]

-1/30/a^3*(-36*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)
-4*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)+36*arccos(a*x)^2*sin(3*arcc
os(a*x))+4*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)-2*a*x*arccos(a*x)-6*arccos(a*x)*cos(3*arccos(a*x))-3*sin(3*arccos(
a*x))-3*(-a^2*x^2+1)^(1/2))/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^2/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^2/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**2/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^{2}}{\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^2/arccos(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^2/arccos(a*x)^(7/2), x)