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

Optimal. Leaf size=119 \[ \frac{32 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{15 a^2}-\frac{32 x \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{8 x^2}{15 \cos ^{-1}(a x)^{3/2}} \]

[Out]

(2*x*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - 4/(15*a^2*ArcCos[a*x]^(3/2)) + (8*x^2)/(15*ArcCos[a*x]^(3/2)
) - (32*x*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcCos[a*x]]) + (32*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])
/(15*a^2)

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Rubi [A]  time = 0.172477, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4634, 4720, 4632, 3304, 3352, 4642} \[ \frac{32 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{15 a^2}-\frac{32 x \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{8 x^2}{15 \cos ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - 4/(15*a^2*ArcCos[a*x]^(3/2)) + (8*x^2)/(15*ArcCos[a*x]^(3/2)
) - (32*x*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcCos[a*x]]) + (32*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])
/(15*a^2)

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 4642

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

Rubi steps

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

Mathematica [A]  time = 0.0989414, size = 75, normalized size = 0.63 \[ \frac{32 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )+\frac{4 \cos \left (2 \cos ^{-1}(a x)\right )}{\cos ^{-1}(a x)^{3/2}}-\frac{\left (16 \cos ^{-1}(a x)^2-3\right ) \sin \left (2 \cos ^{-1}(a x)\right )}{\cos ^{-1}(a x)^{5/2}}}{15 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*Cos[2*ArcCos[a*x]])/ArcCos[a*x]^(3/2) + 32*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]] - ((-3 + 16*A
rcCos[a*x]^2)*Sin[2*ArcCos[a*x]])/ArcCos[a*x]^(5/2))/(15*a^2)

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Maple [A]  time = 0.075, size = 73, normalized size = 0.6 \begin{align*} -{\frac{1}{15\,{a}^{2}} \left ( -32\,\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{5/2}+16\,\sin \left ( 2\,\arccos \left ( ax \right ) \right ) \left ( \arccos \left ( ax \right ) \right ) ^{2}-4\,\arccos \left ( ax \right ) \cos \left ( 2\,\arccos \left ( ax \right ) \right ) -3\,\sin \left ( 2\,\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/arccos(a*x)^(7/2),x)

[Out]

-1/15/a^2*(-32*Pi^(1/2)*FresnelC(2*arccos(a*x)^(1/2)/Pi^(1/2))*arccos(a*x)^(5/2)+16*sin(2*arccos(a*x))*arccos(
a*x)^2-4*arccos(a*x)*cos(2*arccos(a*x))-3*sin(2*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/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/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/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}{\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/arccos(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x/arccos(a*x)^(7/2), x)