3.1.81 \(\int x^3 \text {ArcCos}(a x)^{3/2} \, dx\) [81]

Optimal. Leaf size=157 \[ -\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{32 a}-\frac {3 \text {ArcCos}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {ArcCos}(a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{64 a^4} \]

[Out]

-3/32*arccos(a*x)^(3/2)/a^4+1/4*x^4*arccos(a*x)^(3/2)+3/1024*FresnelS(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^
(1/2)*Pi^(1/2)/a^4+3/64*FresnelS(2*arccos(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^4-9/64*x*(-a^2*x^2+1)^(1/2)*arccos(a
*x)^(1/2)/a^3-3/32*x^3*(-a^2*x^2+1)^(1/2)*arccos(a*x)^(1/2)/a

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Rubi [A]
time = 0.26, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4726, 4796, 4738, 4732, 4491, 12, 3386, 3432} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{64 a^4}-\frac {3 \text {ArcCos}(a x)^{3/2}}{32 a^4}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{32 a}-\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{64 a^3}+\frac {1}{4} x^4 \text {ArcCos}(a x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCos[a*x]^(3/2),x]

[Out]

(-9*x*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(64*a^3) - (3*x^3*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(32*a) - (3*
ArcCos[a*x]^(3/2))/(32*a^4) + (x^4*ArcCos[a*x]^(3/2))/4 + (3*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]
]])/(512*a^4) + (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(64*a^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3386

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 3432

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

Rule 4491

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 4726

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

Rule 4732

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

Rule 4738

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

Rule 4796

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

Rubi steps

\begin {align*} \int x^3 \cos ^{-1}(a x)^{3/2} \, dx &=\frac {1}{4} x^4 \cos ^{-1}(a x)^{3/2}+\frac {1}{8} (3 a) \int \frac {x^4 \sqrt {\cos ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{32 a}+\frac {1}{4} x^4 \cos ^{-1}(a x)^{3/2}-\frac {3}{64} \int \frac {x^3}{\sqrt {\cos ^{-1}(a x)}} \, dx+\frac {9 \int \frac {x^2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{32 a}\\ &=-\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{32 a}+\frac {1}{4} x^4 \cos ^{-1}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^4}+\frac {9 \int \frac {\sqrt {\cos ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{64 a^3}-\frac {9 \int \frac {x}{\sqrt {\cos ^{-1}(a x)}} \, dx}{128 a^2}\\ &=-\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{32 a}-\frac {3 \cos ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \cos ^{-1}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{64 a^4}+\frac {9 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{128 a^4}\\ &=-\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{32 a}-\frac {3 \cos ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \cos ^{-1}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{512 a^4}+\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{256 a^4}+\frac {9 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{128 a^4}\\ &=-\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{32 a}-\frac {3 \cos ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \cos ^{-1}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{256 a^4}+\frac {3 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{128 a^4}+\frac {9 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{256 a^4}\\ &=-\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{32 a}-\frac {3 \cos ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \cos ^{-1}(a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{256 a^4}+\frac {9 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{128 a^4}\\ &=-\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{32 a}-\frac {3 \cos ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \cos ^{-1}(a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{64 a^4}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.05, size = 128, normalized size = 0.82 \begin {gather*} -\frac {8 \sqrt {2} \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},-2 i \text {ArcCos}(a x)\right )+8 \sqrt {2} \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},2 i \text {ArcCos}(a x)\right )+\sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},-4 i \text {ArcCos}(a x)\right )+\sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},4 i \text {ArcCos}(a x)\right )}{512 a^4 \sqrt {\text {ArcCos}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCos[a*x]^(3/2),x]

[Out]

-1/512*(8*Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-2*I)*ArcCos[a*x]] + 8*Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma[
5/2, (2*I)*ArcCos[a*x]] + Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-4*I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/
2, (4*I)*ArcCos[a*x]])/(a^4*Sqrt[ArcCos[a*x]])

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Maple [A]
time = 0.25, size = 121, normalized size = 0.77

method	result	size
default	\(\frac {3 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+128 \arccos \left (a x \right )^{2} \cos \left (2 \arccos \left (a x \right )\right )+32 \arccos \left (a x \right )^{2} \cos \left (4 \arccos \left (a x \right )\right )+48 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-96 \arccos \left (a x \right ) \sin \left (2 \arccos \left (a x \right )\right )-12 \arccos \left (a x \right ) \sin \left (4 \arccos \left (a x \right )\right )}{1024 a^{4} \sqrt {\arccos \left (a x \right )}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccos(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/1024/a^4*(3*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))+128*arccos(a*x
)^2*cos(2*arccos(a*x))+32*arccos(a*x)^2*cos(4*arccos(a*x))+48*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*arccos(a*x
)^(1/2)/Pi^(1/2))-96*arccos(a*x)*sin(2*arccos(a*x))-12*arccos(a*x)*sin(4*arccos(a*x)))/arccos(a*x)^(1/2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acos(a*x)**(3/2),x)

[Out]

Integral(x**3*acos(a*x)**(3/2), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.46, size = 225, normalized size = 1.43 \begin {gather*} \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (4 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-4 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} + \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{4096 \, a^{4}} - \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{4096 \, a^{4}} + \frac {\left (3 i - 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{256 \, a^{4}} - \frac {\left (3 i + 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{256 \, a^{4}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (4 i \, \arccos \left (a x\right )\right )}}{512 \, a^{4}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-4 i \, \arccos \left (a x\right )\right )}}{512 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(a*x)^(3/2),x, algorithm="giac")

[Out]

1/64*arccos(a*x)^(3/2)*e^(4*I*arccos(a*x))/a^4 + 1/16*arccos(a*x)^(3/2)*e^(2*I*arccos(a*x))/a^4 + 1/16*arccos(
a*x)^(3/2)*e^(-2*I*arccos(a*x))/a^4 + 1/64*arccos(a*x)^(3/2)*e^(-4*I*arccos(a*x))/a^4 + (3/4096*I - 3/4096)*sq
rt(2)*sqrt(pi)*erf((I - 1)*sqrt(2)*sqrt(arccos(a*x)))/a^4 - (3/4096*I + 3/4096)*sqrt(2)*sqrt(pi)*erf(-(I + 1)*
sqrt(2)*sqrt(arccos(a*x)))/a^4 + (3/256*I - 3/256)*sqrt(pi)*erf((I - 1)*sqrt(arccos(a*x)))/a^4 - (3/256*I + 3/
256)*sqrt(pi)*erf(-(I + 1)*sqrt(arccos(a*x)))/a^4 + 3/512*I*sqrt(arccos(a*x))*e^(4*I*arccos(a*x))/a^4 + 3/64*I
*sqrt(arccos(a*x))*e^(2*I*arccos(a*x))/a^4 - 3/64*I*sqrt(arccos(a*x))*e^(-2*I*arccos(a*x))/a^4 - 3/512*I*sqrt(
arccos(a*x))*e^(-4*I*arccos(a*x))/a^4

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\mathrm {acos}\left (a\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*acos(a*x)^(3/2), x)

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