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

Optimal. Leaf size=88 \[ -\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+\frac {15 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{4 a}+x \cos ^{-1}(a x)^{5/2}-\frac {15}{4} x \sqrt {\cos ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4620, 4678, 4724, 3304, 3352} \[ -\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+\frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{4 a}+x \cos ^{-1}(a x)^{5/2}-\frac {15}{4} x \sqrt {\cos ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*x*Sqrt[ArcCos[a*x]])/4 - (5*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(2*a) + x*ArcCos[a*x]^(5/2) + (15*Sqrt[P
i/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(4*a)

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 4620

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
(x*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

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 \cos ^{-1}(a x)^{5/2} \, dx &=x \cos ^{-1}(a x)^{5/2}+\frac {1}{2} (5 a) \int \frac {x \cos ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+x \cos ^{-1}(a x)^{5/2}-\frac {15}{4} \int \sqrt {\cos ^{-1}(a x)} \, dx\\ &=-\frac {15}{4} x \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+x \cos ^{-1}(a x)^{5/2}-\frac {1}{8} (15 a) \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx\\ &=-\frac {15}{4} x \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+x \cos ^{-1}(a x)^{5/2}+\frac {15 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a}\\ &=-\frac {15}{4} x \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+x \cos ^{-1}(a x)^{5/2}+\frac {15 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{4 a}\\ &=-\frac {15}{4} x \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+x \cos ^{-1}(a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{4 a}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 76, normalized size = 0.86 \[ -\frac {\sqrt {\cos ^{-1}(a x)} \left (\sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {7}{2},-i \cos ^{-1}(a x)\right )+\sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {7}{2},i \cos ^{-1}(a x)\right )\right )}{2 a \sqrt {\cos ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(Sqrt[ArcCos[a*x]]*(Sqrt[I*ArcCos[a*x]]*Gamma[7/2, (-I)*ArcCos[a*x]] + Sqrt[(-I)*ArcCos[a*x]]*Gamma[7/2,
I*ArcCos[a*x]]))/(a*Sqrt[ArcCos[a*x]^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(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 [B]  time = 2.68, size = 181, normalized size = 2.06 \[ \frac {5 \, i \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \arccos \left (a x\right )\right )}}{4 \, a} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (i \arccos \left (a x\right )\right )}}{2 \, a} - \frac {5 \, i \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \arccos \left (a x\right )\right )}}{4 \, a} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (-i \arccos \left (a x\right )\right )}}{2 \, a} - \frac {15 \, \sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{16 \, a {\left (i - 1\right )}} - \frac {15 \, \sqrt {\arccos \left (a x\right )} e^{\left (i \arccos \left (a x\right )\right )}}{8 \, a} - \frac {15 \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \arccos \left (a x\right )\right )}}{8 \, a} + \frac {15 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{16 \, a {\left (i - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*i*arccos(a*x)^(3/2)*e^(i*arccos(a*x))/a + 1/2*arccos(a*x)^(5/2)*e^(i*arccos(a*x))/a - 5/4*i*arccos(a*x)^(3
/2)*e^(-i*arccos(a*x))/a + 1/2*arccos(a*x)^(5/2)*e^(-i*arccos(a*x))/a - 15/16*sqrt(2)*sqrt(pi)*i*erf(sqrt(2)*s
qrt(arccos(a*x))/(i - 1))/(a*(i - 1)) - 15/8*sqrt(arccos(a*x))*e^(i*arccos(a*x))/a - 15/8*sqrt(arccos(a*x))*e^
(-i*arccos(a*x))/a + 15/16*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*i*sqrt(arccos(a*x))/(i - 1))/(a*(i - 1))

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maple [A]  time = 0.14, size = 88, normalized size = 1.00 \[ \frac {\sqrt {2}\, \left (4 \arccos \left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, x a -10 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}-15 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, x a +15 \pi \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{8 a \sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/a*2^(1/2)/Pi^(1/2)*(4*arccos(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*x*a-10*arccos(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*(-a^2*x
^2+1)^(1/2)-15*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*x*a+15*Pi*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2)))

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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(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 = -0.01 \[ \int {\mathrm {acos}\left (a\,x\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acos(a*x)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x)**(5/2), x)

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