3.99 \(\int \frac{x^6}{\cos ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=171 \[ -\frac{5 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{9 \sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{5 \sqrt{\frac{5 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{\sqrt{\frac{7 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{14}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}+\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}} \]

[Out]

(2*x^6*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (5*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^
7) - (9*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^7) - (5*Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/P
i]*Sqrt[ArcCos[a*x]]])/(16*a^7) - (Sqrt[(7*Pi)/2]*FresnelC[Sqrt[14/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^7)

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Rubi [A]  time = 0.145915, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4632, 3304, 3352} \[ -\frac{5 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{9 \sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{5 \sqrt{\frac{5 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{\sqrt{\frac{7 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{14}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}+\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^6/ArcCos[a*x]^(3/2),x]

[Out]

(2*x^6*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (5*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^
7) - (9*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^7) - (5*Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/P
i]*Sqrt[ArcCos[a*x]]])/(16*a^7) - (Sqrt[(7*Pi)/2]*FresnelC[Sqrt[14/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^7)

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^6}{\cos ^{-1}(a x)^{3/2}} \, dx &=\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{5 \cos (x)}{64 \sqrt{x}}-\frac{27 \cos (3 x)}{64 \sqrt{x}}-\frac{25 \cos (5 x)}{64 \sqrt{x}}-\frac{7 \cos (7 x)}{64 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^7}\\ &=\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}-\frac{5 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^7}-\frac{7 \operatorname{Subst}\left (\int \frac{\cos (7 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^7}-\frac{25 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^7}-\frac{27 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^7}\\ &=\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}-\frac{5 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{7 \operatorname{Subst}\left (\int \cos \left (7 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{25 \operatorname{Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{27 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}\\ &=\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}-\frac{5 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{9 \sqrt{\frac{3 \pi }{2}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{5 \sqrt{\frac{5 \pi }{2}} C\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}-\frac{\sqrt{\frac{7 \pi }{2}} C\left (\sqrt{\frac{14}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^7}\\ \end{align*}

Mathematica [C]  time = 0.286505, size = 306, normalized size = 1.79 \[ \frac{i \left (5 \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a x)\right )-5 \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a x)\right )+9 \sqrt{3} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \cos ^{-1}(a x)\right )-9 \sqrt{3} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \cos ^{-1}(a x)\right )+5 \sqrt{5} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-5 i \cos ^{-1}(a x)\right )-5 \sqrt{5} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},5 i \cos ^{-1}(a x)\right )+\sqrt{7} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-7 i \cos ^{-1}(a x)\right )-\sqrt{7} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},7 i \cos ^{-1}(a x)\right )-10 i \sqrt{1-a^2 x^2}-18 i \sin \left (3 \cos ^{-1}(a x)\right )-10 i \sin \left (5 \cos ^{-1}(a x)\right )-2 i \sin \left (7 \cos ^{-1}(a x)\right )\right )}{64 a^7 \sqrt{\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^6/ArcCos[a*x]^(3/2),x]

[Out]

((I/64)*((-10*I)*Sqrt[1 - a^2*x^2] + 5*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - 5*Sqrt[I*ArcCos[a
*x]]*Gamma[1/2, I*ArcCos[a*x]] + 9*Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - 9*Sqrt[3]*S
qrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]] + 5*Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-5*I)*ArcCos[a
*x]] - 5*Sqrt[5]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (5*I)*ArcCos[a*x]] + Sqrt[7]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2,
(-7*I)*ArcCos[a*x]] - Sqrt[7]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (7*I)*ArcCos[a*x]] - (18*I)*Sin[3*ArcCos[a*x]] -
(10*I)*Sin[5*ArcCos[a*x]] - (2*I)*Sin[7*ArcCos[a*x]]))/(a^7*Sqrt[ArcCos[a*x]])

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Maple [A]  time = 0.109, size = 182, normalized size = 1.1 \begin{align*}{\frac{1}{32\,{a}^{7}} \left ( -5\,\sqrt{5}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{5}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -\sqrt{2}\sqrt{\pi }\sqrt{7}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{7}}{\sqrt{\pi }}\sqrt{\arccos \left ( ax \right ) }} \right ) \sqrt{\arccos \left ( ax \right ) }-9\,\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -5\,\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +5\,\sqrt{-{a}^{2}{x}^{2}+1}+9\,\sin \left ( 3\,\arccos \left ( ax \right ) \right ) +5\,\sin \left ( 5\,\arccos \left ( ax \right ) \right ) +\sin \left ( 7\,\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^6/arccos(a*x)^(3/2),x)

[Out]

1/32/a^7*(-5*5^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2))-2
^(1/2)*Pi^(1/2)*7^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*7^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(1/2)-9*3^(1/2)*2^(1/
2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))-5*2^(1/2)*arccos(a*x)^(1/2)
*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))+5*(-a^2*x^2+1)^(1/2)+9*sin(3*arccos(a*x))+5*sin(5*arcco
s(a*x))+sin(7*arccos(a*x)))/arccos(a*x)^(1/2)

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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^6/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^6/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^{6}}{\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**6/acos(a*x)**(3/2),x)

[Out]

Integral(x**6/acos(a*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\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^6/arccos(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(x^6/arccos(a*x)^(3/2), x)