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

Optimal. Leaf size=105 \[ -\frac{8 \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac{8 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a}+\frac{4 x}{15 \cos ^{-1}(a x)^{3/2}} \]

[Out]

(2*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) + (4*x)/(15*ArcCos[a*x]^(3/2)) - (8*Sqrt[1 - a^2*x^2])/(15*a*Sqr
t[ArcCos[a*x]]) + (8*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a)

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Rubi [A]  time = 0.165503, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4622, 4720, 4724, 3304, 3352} \[ -\frac{8 \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac{8 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{15 a}+\frac{4 x}{15 \cos ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) + (4*x)/(15*ArcCos[a*x]^(3/2)) - (8*Sqrt[1 - a^2*x^2])/(15*a*Sqr
t[ArcCos[a*x]]) + (8*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a)

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 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 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]

Rubi steps

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

Mathematica [C]  time = 1.04488, size = 151, normalized size = 1.44 \[ -\frac{-4 \cos ^{-1}(a x) \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a x)\right )+e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left (-4 e^{i \cos ^{-1}(a x)} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a x)\right )+4 i \cos ^{-1}(a x)-2\right )-6 \sqrt{1-a^2 x^2}-2 i e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left (2 \cos ^{-1}(a x)-i\right )}{15 a \cos ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[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]))/(15*a*ArcCos[a*x]^(5/2))

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Maple [A]  time = 0.075, size = 110, normalized size = 1.1 \begin{align*}{\frac{\sqrt{2}}{15\,a\sqrt{\pi } \left ( \arccos \left ( ax \right ) \right ) ^{3}} \left ( 8\, \left ( \arccos \left ( ax \right ) \right ) ^{3}\pi \,{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -4\, \left ( \arccos \left ( ax \right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }\sqrt{-{a}^{2}{x}^{2}+1}+2\, \left ( \arccos \left ( ax \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }xa+3\,\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccos(a*x)^(7/2),x)

[Out]

1/15/a*2^(1/2)/Pi^(1/2)*(8*arccos(a*x)^3*Pi*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))-4*arccos(a*x)^(5/2)*2
^(1/2)*Pi^(1/2)*(-a^2*x^2+1)^(1/2)+2*arccos(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*x*a+3*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/
2)*(-a^2*x^2+1)^(1/2))/arccos(a*x)^3

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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(1/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(1/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(1/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{1}{\arccos \left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccos(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(arccos(a*x)^(-7/2), x)