3.100 \(\int \frac{x^5}{\cos ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=127 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^6}-\frac{\sqrt{3 \pi } \text{FresnelC}\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^6}-\frac{5 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a^6}+\frac{2 x^5 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}} \]

[Out]

(2*x^5*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^6 -
(Sqrt[3*Pi]*FresnelC[2*Sqrt[3/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^6) - (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqr
t[Pi]])/(8*a^6)

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Rubi [A]  time = 0.103803, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4632, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^6}-\frac{\sqrt{3 \pi } \text{FresnelC}\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^6}-\frac{5 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a^6}+\frac{2 x^5 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/ArcCos[a*x]^(3/2),x]

[Out]

(2*x^5*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^6 -
(Sqrt[3*Pi]*FresnelC[2*Sqrt[3/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^6) - (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqr
t[Pi]])/(8*a^6)

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

Mathematica [C]  time = 0.482975, size = 226, normalized size = 1.78 \[ \frac{i \left (5 \sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \cos ^{-1}(a x)\right )-5 \sqrt{2} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \cos ^{-1}(a x)\right )+8 \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \cos ^{-1}(a x)\right )-8 \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \cos ^{-1}(a x)\right )+\sqrt{6} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-6 i \cos ^{-1}(a x)\right )-\sqrt{6} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},6 i \cos ^{-1}(a x)\right )-10 i \sin \left (2 \cos ^{-1}(a x)\right )-8 i \sin \left (4 \cos ^{-1}(a x)\right )-2 i \sin \left (6 \cos ^{-1}(a x)\right )\right )}{32 a^6 \sqrt{\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^5/ArcCos[a*x]^(3/2),x]

[Out]

((I/32)*(5*Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a*x]] - 5*Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma
[1/2, (2*I)*ArcCos[a*x]] + 8*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-4*I)*ArcCos[a*x]] - 8*Sqrt[I*ArcCos[a*x]]*Gam
ma[1/2, (4*I)*ArcCos[a*x]] + Sqrt[6]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-6*I)*ArcCos[a*x]] - Sqrt[6]*Sqrt[I*Ar
cCos[a*x]]*Gamma[1/2, (6*I)*ArcCos[a*x]] - (10*I)*Sin[2*ArcCos[a*x]] - (8*I)*Sin[4*ArcCos[a*x]] - (2*I)*Sin[6*
ArcCos[a*x]]))/(a^6*Sqrt[ArcCos[a*x]])

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Maple [A]  time = 0.092, size = 121, normalized size = 1. \begin{align*}{\frac{1}{16\,{a}^{6}} \left ( -2\,\sqrt{\pi }\sqrt{3}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{6}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \sqrt{\arccos \left ( ax \right ) }-8\,\sqrt{2}\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -10\,\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +5\,\sin \left ( 2\,\arccos \left ( ax \right ) \right ) +4\,\sin \left ( 4\,\arccos \left ( ax \right ) \right ) +\sin \left ( 6\,\arccos \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/arccos(a*x)^(3/2),x)

[Out]

1/16/a^6/arccos(a*x)^(1/2)*(-2*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*6^(1/2)*arccos(a*x)^(1/2))*arccos(a*
x)^(1/2)-8*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))-10*Pi^(1/2)*arcco
s(a*x)^(1/2)*FresnelC(2*arccos(a*x)^(1/2)/Pi^(1/2))+5*sin(2*arccos(a*x))+4*sin(4*arccos(a*x))+sin(6*arccos(a*x
)))

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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^5/arccos(a*x)^(3/2),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^5/arccos(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\operatorname{acos}^{\frac{3}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/acos(a*x)**(3/2),x)

[Out]

Integral(x**5/acos(a*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\arccos \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arccos(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(x^5/arccos(a*x)^(3/2), x)