3.67 \(\int \frac{x^4}{\cos ^{-1}(a x)^4} \, dx\)

Optimal. Leaf size=158 \[ \frac{\text{CosIntegral}\left (\cos ^{-1}(a x)\right )}{48 a^5}+\frac{27 \text{CosIntegral}\left (3 \cos ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{CosIntegral}\left (5 \cos ^{-1}(a x)\right )}{96 a^5}-\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}+\frac{5 x^5}{6 \cos ^{-1}(a x)^2} \]

[Out]

(x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) - (2*x^3)/(3*a^2*ArcCos[a*x]^2) + (5*x^5)/(6*ArcCos[a*x]^2) + (2*x
^2*Sqrt[1 - a^2*x^2])/(a^3*ArcCos[a*x]) - (25*x^4*Sqrt[1 - a^2*x^2])/(6*a*ArcCos[a*x]) + CosIntegral[ArcCos[a*
x]]/(48*a^5) + (27*CosIntegral[3*ArcCos[a*x]])/(32*a^5) + (125*CosIntegral[5*ArcCos[a*x]])/(96*a^5)

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Rubi [A]  time = 0.337903, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4634, 4720, 4632, 3302} \[ \frac{\text{CosIntegral}\left (\cos ^{-1}(a x)\right )}{48 a^5}+\frac{27 \text{CosIntegral}\left (3 \cos ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{CosIntegral}\left (5 \cos ^{-1}(a x)\right )}{96 a^5}-\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}+\frac{5 x^5}{6 \cos ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCos[a*x]^4,x]

[Out]

(x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) - (2*x^3)/(3*a^2*ArcCos[a*x]^2) + (5*x^5)/(6*ArcCos[a*x]^2) + (2*x
^2*Sqrt[1 - a^2*x^2])/(a^3*ArcCos[a*x]) - (25*x^4*Sqrt[1 - a^2*x^2])/(6*a*ArcCos[a*x]) + CosIntegral[ArcCos[a*
x]]/(48*a^5) + (27*CosIntegral[3*ArcCos[a*x]])/(32*a^5) + (125*CosIntegral[5*ArcCos[a*x]])/(96*a^5)

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 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x^4}{\cos ^{-1}(a x)^4} \, dx &=\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{4 \int \frac{x^3}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx}{3 a}+\frac{1}{3} (5 a) \int \frac{x^5}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx\\ &=\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac{5 x^5}{6 \cos ^{-1}(a x)^2}-\frac{25}{6} \int \frac{x^4}{\cos ^{-1}(a x)^2} \, dx+\frac{2 \int \frac{x^2}{\cos ^{-1}(a x)^2} \, dx}{a^2}\\ &=\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac{5 x^5}{6 \cos ^{-1}(a x)^2}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{\cos (x)}{4 x}-\frac{3 \cos (3 x)}{4 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^5}-\frac{25 \operatorname{Subst}\left (\int \left (-\frac{\cos (x)}{8 x}-\frac{9 \cos (3 x)}{16 x}-\frac{5 \cos (5 x)}{16 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{6 a^5}\\ &=\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac{5 x^5}{6 \cos ^{-1}(a x)^2}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{48 a^5}+\frac{125 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{96 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^5}+\frac{75 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^5}\\ &=\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \cos ^{-1}(a x)^2}+\frac{5 x^5}{6 \cos ^{-1}(a x)^2}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}-\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac{\text{Ci}\left (\cos ^{-1}(a x)\right )}{48 a^5}+\frac{27 \text{Ci}\left (3 \cos ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Ci}\left (5 \cos ^{-1}(a x)\right )}{96 a^5}\\ \end{align*}

Mathematica [A]  time = 0.169638, size = 159, normalized size = 1.01 \[ \frac{32 a^4 x^4 \sqrt{1-a^2 x^2}+80 a^5 x^5 \cos ^{-1}(a x)-400 a^4 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2-64 a^3 x^3 \cos ^{-1}(a x)+192 a^2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2+2 \cos ^{-1}(a x)^3 \text{CosIntegral}\left (\cos ^{-1}(a x)\right )+81 \cos ^{-1}(a x)^3 \text{CosIntegral}\left (3 \cos ^{-1}(a x)\right )+125 \cos ^{-1}(a x)^3 \text{CosIntegral}\left (5 \cos ^{-1}(a x)\right )}{96 a^5 \cos ^{-1}(a x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/ArcCos[a*x]^4,x]

[Out]

(32*a^4*x^4*Sqrt[1 - a^2*x^2] - 64*a^3*x^3*ArcCos[a*x] + 80*a^5*x^5*ArcCos[a*x] + 192*a^2*x^2*Sqrt[1 - a^2*x^2
]*ArcCos[a*x]^2 - 400*a^4*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2 + 2*ArcCos[a*x]^3*CosIntegral[ArcCos[a*x]] + 81*
ArcCos[a*x]^3*CosIntegral[3*ArcCos[a*x]] + 125*ArcCos[a*x]^3*CosIntegral[5*ArcCos[a*x]])/(96*a^5*ArcCos[a*x]^3
)

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Maple [A]  time = 0.059, size = 171, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{1}{24\, \left ( \arccos \left ( ax \right ) \right ) ^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{ax}{48\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}}-{\frac{1}{48\,\arccos \left ( ax \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{\it Ci} \left ( \arccos \left ( ax \right ) \right ) }{48}}+{\frac{\sin \left ( 3\,\arccos \left ( ax \right ) \right ) }{16\, \left ( \arccos \left ( ax \right ) \right ) ^{3}}}+{\frac{3\,\cos \left ( 3\,\arccos \left ( ax \right ) \right ) }{32\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}}-{\frac{9\,\sin \left ( 3\,\arccos \left ( ax \right ) \right ) }{32\,\arccos \left ( ax \right ) }}+{\frac{27\,{\it Ci} \left ( 3\,\arccos \left ( ax \right ) \right ) }{32}}+{\frac{\sin \left ( 5\,\arccos \left ( ax \right ) \right ) }{48\, \left ( \arccos \left ( ax \right ) \right ) ^{3}}}+{\frac{5\,\cos \left ( 5\,\arccos \left ( ax \right ) \right ) }{96\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}}-{\frac{25\,\sin \left ( 5\,\arccos \left ( ax \right ) \right ) }{96\,\arccos \left ( ax \right ) }}+{\frac{125\,{\it Ci} \left ( 5\,\arccos \left ( ax \right ) \right ) }{96}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccos(a*x)^4,x)

[Out]

1/a^5*(1/24/arccos(a*x)^3*(-a^2*x^2+1)^(1/2)+1/48*a*x/arccos(a*x)^2-1/48*(-a^2*x^2+1)^(1/2)/arccos(a*x)+1/48*C
i(arccos(a*x))+1/16/arccos(a*x)^3*sin(3*arccos(a*x))+3/32/arccos(a*x)^2*cos(3*arccos(a*x))-9/32/arccos(a*x)*si
n(3*arccos(a*x))+27/32*Ci(3*arccos(a*x))+1/48/arccos(a*x)^3*sin(5*arccos(a*x))+5/96/arccos(a*x)^2*cos(5*arccos
(a*x))-25/96/arccos(a*x)*sin(5*arccos(a*x))+125/96*Ci(5*arccos(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{3} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{3} \int \frac{{\left (125 \, a^{4} x^{5} - 136 \, a^{2} x^{3} + 24 \, x\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{{\left (a^{5} x^{2} - a^{3}\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )}\,{d x} +{\left (2 \, a^{2} x^{4} -{\left (25 \, a^{2} x^{4} - 12 \, x^{2}\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{2}\right )} \sqrt{a x + 1} \sqrt{-a x + 1} +{\left (5 \, a^{3} x^{5} - 4 \, a x^{3}\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )}{6 \, a^{3} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x)^4,x, algorithm="maxima")

[Out]

1/6*(6*a^3*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3*integrate(1/6*(125*a^4*x^5 - 136*a^2*x^3 + 24*x)*sqrt(
a*x + 1)*sqrt(-a*x + 1)/((a^5*x^2 - a^3)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)), x) + (2*a^2*x^4 - (25*a^
2*x^4 - 12*x^2)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2)*sqrt(a*x + 1)*sqrt(-a*x + 1) + (5*a^3*x^5 - 4*a*
x^3)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x))/(a^3*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4}}{\arccos \left (a x\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x)^4,x, algorithm="fricas")

[Out]

integral(x^4/arccos(a*x)^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{acos}^{4}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acos(a*x)**4,x)

[Out]

Integral(x**4/acos(a*x)**4, x)

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Giac [A]  time = 1.23685, size = 186, normalized size = 1.18 \begin{align*} \frac{5 \, x^{5}}{6 \, \arccos \left (a x\right )^{2}} - \frac{25 \, \sqrt{-a^{2} x^{2} + 1} x^{4}}{6 \, a \arccos \left (a x\right )} + \frac{\sqrt{-a^{2} x^{2} + 1} x^{4}}{3 \, a \arccos \left (a x\right )^{3}} - \frac{2 \, x^{3}}{3 \, a^{2} \arccos \left (a x\right )^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{3} \arccos \left (a x\right )} + \frac{125 \, \operatorname{Ci}\left (5 \, \arccos \left (a x\right )\right )}{96 \, a^{5}} + \frac{27 \, \operatorname{Ci}\left (3 \, \arccos \left (a x\right )\right )}{32 \, a^{5}} + \frac{\operatorname{Ci}\left (\arccos \left (a x\right )\right )}{48 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x)^4,x, algorithm="giac")

[Out]

5/6*x^5/arccos(a*x)^2 - 25/6*sqrt(-a^2*x^2 + 1)*x^4/(a*arccos(a*x)) + 1/3*sqrt(-a^2*x^2 + 1)*x^4/(a*arccos(a*x
)^3) - 2/3*x^3/(a^2*arccos(a*x)^2) + 2*sqrt(-a^2*x^2 + 1)*x^2/(a^3*arccos(a*x)) + 125/96*cos_integral(5*arccos
(a*x))/a^5 + 27/32*cos_integral(3*arccos(a*x))/a^5 + 1/48*cos_integral(arccos(a*x))/a^5