3.107 \(\int \frac {x^4}{\cos ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=235 \[ -\frac {4 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{3 a^5}-\frac {4 \sqrt {\frac {2 \pi }{3}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{6 a^5}-\frac {16 x^3}{3 a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}+\frac {20 x^5}{3 \sqrt {\cos ^{-1}(a x)}} \]

[Out]

3/4*FresnelS(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^5+1/6*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x
)^(1/2))*2^(1/2)*Pi^(1/2)/a^5+5/12*FresnelS(10^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5+2/3*x^4
*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(3/2)-16/3*x^3/a^2/arccos(a*x)^(1/2)+20/3*x^5/arccos(a*x)^(1/2)

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Rubi [A]  time = 0.41, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4634, 4720, 4636, 4406, 3305, 3351} \[ -\frac {4 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{3 a^5}-\frac {4 \sqrt {\frac {2 \pi }{3}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{6 a^5}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\cos ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCos[a*x]^(5/2),x]

[Out]

(2*x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - (16*x^3)/(3*a^2*Sqrt[ArcCos[a*x]]) + (20*x^5)/(3*Sqrt[ArcC
os[a*x]]) + (25*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^5) - (4*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]
*Sqrt[ArcCos[a*x]]])/a^5 + (25*Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(2*a^5) - (4*Sqrt[(2*Pi)/3]*
FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/a^5 + (5*Sqrt[(5*Pi)/2]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(6*a^
5)

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(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 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

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 4636

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Cos[x]^m*Sin[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

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]

Rubi steps

\begin {align*} \int \frac {x^4}{\cos ^{-1}(a x)^{5/2}} \, dx &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}} \, dx}{3 a}+\frac {1}{3} (10 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}} \, dx\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\cos ^{-1}(a x)}}-\frac {100}{3} \int \frac {x^4}{\sqrt {\cos ^{-1}(a x)}} \, dx+\frac {16 \int \frac {x^2}{\sqrt {\cos ^{-1}(a x)}} \, dx}{a^2}\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\cos ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}+\frac {100 \operatorname {Subst}\left (\int \frac {\cos ^4(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^5}\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\cos ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \left (\frac {\sin (x)}{4 \sqrt {x}}+\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^5}+\frac {100 \operatorname {Subst}\left (\int \left (\frac {\sin (x)}{8 \sqrt {x}}+\frac {3 \sin (3 x)}{16 \sqrt {x}}+\frac {\sin (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{3 a^5}\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\cos ^{-1}(a x)}}+\frac {25 \operatorname {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{12 a^5}-\frac {4 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}-\frac {4 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{6 a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^5}\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\cos ^{-1}(a x)}}+\frac {25 \operatorname {Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{6 a^5}-\frac {8 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{a^5}-\frac {8 \operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{a^5}+\frac {25 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{3 a^5}+\frac {25 \operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{2 a^5}\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\cos ^{-1}(a x)}}+\frac {25 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{3 a^5}-\frac {4 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a^5}-\frac {4 \sqrt {\frac {2 \pi }{3}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{6 a^5}\\ \end {align*}

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Mathematica [C]  time = 1.82, size = 322, normalized size = 1.37 \[ -\frac {2 \left (-\sqrt {1-a^2 x^2}-e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x)-e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x)+\sqrt {-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \Gamma \left (\frac {1}{2},-i \cos ^{-1}(a x)\right )+\sqrt {i \cos ^{-1}(a x)} \cos ^{-1}(a x) \Gamma \left (\frac {1}{2},i \cos ^{-1}(a x)\right )\right )-\sin \left (5 \cos ^{-1}(a x)\right )-5 \cos ^{-1}(a x) \left (e^{-5 i \cos ^{-1}(a x)}+e^{5 i \cos ^{-1}(a x)}-\sqrt {5} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-5 i \cos ^{-1}(a x)\right )-\sqrt {5} \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},5 i \cos ^{-1}(a x)\right )\right )-3 \left (\sin \left (3 \cos ^{-1}(a x)\right )+3 \cos ^{-1}(a x) \left (e^{-3 i \cos ^{-1}(a x)}+e^{3 i \cos ^{-1}(a x)}-\sqrt {3} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 i \cos ^{-1}(a x)\right )-\sqrt {3} \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 i \cos ^{-1}(a x)\right )\right )\right )}{24 a^5 \cos ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/ArcCos[a*x]^(5/2),x]

[Out]

-1/24*(2*(-Sqrt[1 - a^2*x^2] - ArcCos[a*x]/E^(I*ArcCos[a*x]) - E^(I*ArcCos[a*x])*ArcCos[a*x] + Sqrt[(-I)*ArcCo
s[a*x]]*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, I*ArcCos[a*x]])
- 5*ArcCos[a*x]*(E^((-5*I)*ArcCos[a*x]) + E^((5*I)*ArcCos[a*x]) - Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-
5*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (5*I)*ArcCos[a*x]]) - 3*(3*ArcCos[a*x]*(E^((-3*I)*A
rcCos[a*x]) + E^((3*I)*ArcCos[a*x]) - Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - Sqrt[3]*
Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]]) + Sin[3*ArcCos[a*x]]) - Sin[5*ArcCos[a*x]])/(a^5*ArcCos[a*x
]^(3/2))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\arccos \left (a x\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(x^4/arccos(a*x)^(5/2), x)

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maple [A]  time = 0.24, size = 173, normalized size = 0.74 \[ \frac {10 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+18 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+4 \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+4 a x \arccos \left (a x \right )+18 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )+10 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )+2 \sqrt {-a^{2} x^{2}+1}+3 \sin \left (3 \arccos \left (a x \right )\right )+\sin \left (5 \arccos \left (a x \right )\right )}{24 a^{5} \arccos \left (a x \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccos(a*x)^(5/2),x)

[Out]

1/24/a^5*(10*2^(1/2)*Pi^(1/2)*5^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(3/2)+1
8*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(3/2)+4*2^(1/2)*Pi
^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(3/2)+4*a*x*arccos(a*x)+18*arccos(a*x)*cos(3*a
rccos(a*x))+10*arccos(a*x)*cos(5*arccos(a*x))+2*(-a^2*x^2+1)^(1/2)+3*sin(3*arccos(a*x))+sin(5*arccos(a*x)))/ar
ccos(a*x)^(3/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccos(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/acos(a*x)^(5/2),x)

[Out]

int(x^4/acos(a*x)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acos(a*x)**(5/2),x)

[Out]

Integral(x**4/acos(a*x)**(5/2), x)

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