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

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

[Out]

-5/32*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^7-9/32*FresnelC(6^(1/2)/Pi^(1/2)*arccos(
a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^7-5/32*FresnelC(10^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^7-1/32
*FresnelC(14^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*14^(1/2)*Pi^(1/2)/a^7+2*x^6*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(1
/2)

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Rubi [A]  time = 0.15, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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 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]

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*}

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Mathematica [C]  time = 0.35, size = 306, normalized size = 1.79 \[ \frac {i \left (-10 i \sqrt {1-a^2 x^2}-18 i \sin \left (3 \cos ^{-1}(a x)\right )-10 i \sin \left (5 \cos ^{-1}(a x)\right )-2 i \sin \left (7 \cos ^{-1}(a x)\right )+5 \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \cos ^{-1}(a x)\right )-5 \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \cos ^{-1}(a x)\right )+9 \sqrt {3} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 i \cos ^{-1}(a x)\right )-9 \sqrt {3} \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 i \cos ^{-1}(a x)\right )+5 \sqrt {5} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-5 i \cos ^{-1}(a x)\right )-5 \sqrt {5} \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},5 i \cos ^{-1}(a x)\right )+\sqrt {7} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-7 i \cos ^{-1}(a x)\right )-\sqrt {7} \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},7 i \cos ^{-1}(a x)\right )\right )}{64 a^7 \sqrt {\cos ^{-1}(a x)}} \]

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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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^6/arccos(a*x)^(3/2),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 \[ \int \frac {x^{6}}{\arccos \left (a x\right )^{\frac {3}{2}}}\,{d x} \]

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)

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maple [A]  time = 0.33, size = 182, normalized size = 1.06 \[ \frac {-\sqrt {2}\, \sqrt {\pi }\, \sqrt {7}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {7}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arccos \left (a x \right )}-5 \sqrt {5}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-9 \sqrt {3}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-5 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+5 \sqrt {-a^{2} x^{2}+1}+\sin \left (7 \arccos \left (a x \right )\right )+9 \sin \left (3 \arccos \left (a x \right )\right )+5 \sin \left (5 \arccos \left (a x \right )\right )}{32 a^{7} \sqrt {\arccos \left (a x \right )}} \]

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/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))*arc
cos(a*x)^(1/2)-5*5^(1/2)*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2
))-9*3^(1/2)*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))-5*2^(1/2)
*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))+5*(-a^2*x^2+1)^(1/2)+sin(7*arccos(a*x
))+9*sin(3*arccos(a*x))+5*sin(5*arccos(a*x)))

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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^6/arccos(a*x)^(3/2),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 \[ \int \frac {x^6}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/acos(a*x)^(3/2),x)

[Out]

int(x^6/acos(a*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

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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