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

Optimal. Leaf size=191 \[ \frac {2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{15 a^3}+\frac {6 \sqrt {6 \pi } C\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{5 a^3}-\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\cos ^{-1}(a x)}}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\cos ^{-1}(a x)}}+\frac {4 x^3}{5 \cos ^{-1}(a x)^{3/2}} \]

[Out]

-8/15*x/a^2/arccos(a*x)^(3/2)+4/5*x^3/arccos(a*x)^(3/2)+2/15*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1
/2)*Pi^(1/2)/a^3+6/5*FresnelC(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3+2/5*x^2*(-a^2*x^2+1)^(1
/2)/a/arccos(a*x)^(5/2)+16/15*(-a^2*x^2+1)^(1/2)/a^3/arccos(a*x)^(1/2)-24/5*x^2*(-a^2*x^2+1)^(1/2)/a/arccos(a*
x)^(1/2)

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Rubi [A]  time = 0.34, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4634, 4720, 4632, 3304, 3352, 4622, 4724} \[ \frac {2 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{15 a^3}+\frac {6 \sqrt {6 \pi } \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{5 a^3}-\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\cos ^{-1}(a x)}}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \cos ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^2*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (8*x)/(15*a^2*ArcCos[a*x]^(3/2)) + (4*x^3)/(5*ArcCos[a*x]^
(3/2)) + (16*Sqrt[1 - a^2*x^2])/(15*a^3*Sqrt[ArcCos[a*x]]) - (24*x^2*Sqrt[1 - a^2*x^2])/(5*a*Sqrt[ArcCos[a*x]]
) + (2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^3) + (6*Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[Ar
cCos[a*x]]])/(5*a^3)

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 4622

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1)
)/(b*c*(n + 1)), x] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcCos[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && LtQ[n, -1]

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 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 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rubi steps

\begin {align*} \int \frac {x^2}{\cos ^{-1}(a x)^{7/2}} \, dx &=\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4 \int \frac {x}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (6 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx\\ &=\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \cos ^{-1}(a x)^{3/2}}-\frac {12}{5} \int \frac {x^2}{\cos ^{-1}(a x)^{3/2}} \, dx+\frac {8 \int \frac {1}{\cos ^{-1}(a x)^{3/2}} \, dx}{15 a^2}\\ &=\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \cos ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\cos ^{-1}(a x)}}-\frac {24 \operatorname {Subst}\left (\int \left (-\frac {\cos (x)}{4 \sqrt {x}}-\frac {3 \cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{5 a^3}+\frac {16 \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx}{15 a}\\ &=\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \cos ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\cos ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{15 a^3}+\frac {6 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^3}+\frac {18 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^3}\\ &=\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \cos ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\cos ^{-1}(a x)}}-\frac {32 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{15 a^3}+\frac {12 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{5 a^3}+\frac {36 \operatorname {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{5 a^3}\\ &=\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \cos ^{-1}(a x)^{3/2}}+\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\cos ^{-1}(a x)}}+\frac {2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{15 a^3}+\frac {6 \sqrt {6 \pi } C\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{5 a^3}\\ \end {align*}

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Mathematica [C]  time = 2.86, size = 281, normalized size = 1.47 \[ -\frac {-6 \sqrt {1-a^2 x^2}-2 i e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left (2 \cos ^{-1}(a x)-i\right )-6 \sin \left (3 \cos ^{-1}(a x)\right )-4 \cos ^{-1}(a x) \left (-i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-i \cos ^{-1}(a x)\right )+e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left (4 i \cos ^{-1}(a x)-4 e^{i \cos ^{-1}(a x)} \left (i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},i \cos ^{-1}(a x)\right )-2\right )-6 \cos ^{-1}(a x) \left (e^{3 i \cos ^{-1}(a x)} \left (1+6 i \cos ^{-1}(a x)\right )+6 \sqrt {3} \left (-i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-3 i \cos ^{-1}(a x)\right )+e^{-3 i \cos ^{-1}(a x)} \left (-6 i \cos ^{-1}(a x)+6 \sqrt {3} e^{3 i \cos ^{-1}(a x)} \left (i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},3 i \cos ^{-1}(a x)\right )+1\right )\right )}{60 a^3 \cos ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/60*(-6*Sqrt[1 - a^2*x^2] - (2*I)*E^(I*ArcCos[a*x])*ArcCos[a*x]*(-I + 2*ArcCos[a*x]) - 4*((-I)*ArcCos[a*x])^
(3/2)*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + (ArcCos[a*x]*(-2 + (4*I)*ArcCos[a*x] - 4*E^(I*ArcCos[a*x])*(I
*ArcCos[a*x])^(3/2)*Gamma[1/2, I*ArcCos[a*x]]))/E^(I*ArcCos[a*x]) - 6*ArcCos[a*x]*(E^((3*I)*ArcCos[a*x])*(1 +
(6*I)*ArcCos[a*x]) + 6*Sqrt[3]*((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-3*I)*ArcCos[a*x]] + (1 - (6*I)*ArcCos[a*x
] + 6*Sqrt[3]*E^((3*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcCos[a*x]])/E^((3*I)*ArcCos[a*x])
) - 6*Sin[3*ArcCos[a*x]])/(a^3*ArcCos[a*x]^(5/2))

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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^2/arccos(a*x)^(7/2),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 \[ \int \frac {x^{2}}{\arccos \left (a x\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arccos(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^2/arccos(a*x)^(7/2), x)

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maple [A]  time = 0.18, size = 154, normalized size = 0.81 \[ -\frac {-36 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-4 \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}+4 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+36 \arccos \left (a x \right )^{2} \sin \left (3 \arccos \left (a x \right )\right )-2 a x \arccos \left (a x \right )-6 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )-3 \sqrt {-a^{2} x^{2}+1}-3 \sin \left (3 \arccos \left (a x \right )\right )}{30 a^{3} \arccos \left (a x \right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/arccos(a*x)^(7/2),x)

[Out]

-1/30/a^3*(-36*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)
-4*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)+4*arccos(a*x)^2*(-a^2*x^2+1
)^(1/2)+36*arccos(a*x)^2*sin(3*arccos(a*x))-2*a*x*arccos(a*x)-6*arccos(a*x)*cos(3*arccos(a*x))-3*(-a^2*x^2+1)^
(1/2)-3*sin(3*arccos(a*x)))/arccos(a*x)^(5/2)

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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^2/arccos(a*x)^(7/2),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 = -0.01 \[ \int \frac {x^2}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/acos(a*x)^(7/2),x)

[Out]

int(x^2/acos(a*x)^(7/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/acos(a*x)**(7/2),x)

[Out]

Integral(x**2/acos(a*x)**(7/2), x)

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