\(\int x \arccos (a x)^{3/2} \, dx\) [83]

Optimal result

Integrand size = 10, antiderivative size = 89 \[ \int x \arccos (a x)^{3/2} \, dx=-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{32 a^2} \]

[Out]

-1/4*arccos(a*x)^(3/2)/a^2+1/2*x^2*arccos(a*x)^(3/2)+3/32*FresnelS(2*arccos(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^2-
3/8*x*(-a^2*x^2+1)^(1/2)*arccos(a*x)^(1/2)/a

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4726, 4796, 4738, 4732, 4491, 12, 3386, 3432} \[ \int x \arccos (a x)^{3/2} \, dx=\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{32 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2} \]

[In]

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

[Out]

(-3*x*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(8*a) - ArcCos[a*x]^(3/2)/(4*a^2) + (x^2*ArcCos[a*x]^(3/2))/2 + (3*
Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(32*a^2)

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*} \text {integral}& = \frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {1}{4} (3 a) \int \frac {x^2 \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}+\frac {1}{2} x^2 \arccos (a x)^{3/2}-\frac {3}{16} \int \frac {x}{\sqrt {\arccos (a x)}} \, dx+\frac {3 \int \frac {\sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}} \, dx}{8 a} \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{16 a^2} \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arccos (a x)\right )}{16 a^2} \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{32 a^2} \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{16 a^2} \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{32 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72 \[ \int x \arccos (a x)^{3/2} \, dx=\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )-2 \sqrt {\arccos (a x)} (-4 \arccos (a x) \cos (2 \arccos (a x))+3 \sin (2 \arccos (a x)))}{32 a^2} \]

[In]

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

[Out]

(3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]] - 2*Sqrt[ArcCos[a*x]]*(-4*ArcCos[a*x]*Cos[2*ArcCos[a*x]]
+ 3*Sin[2*ArcCos[a*x]]))/(32*a^2)

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72

method	result	size
default	\(\frac {8 \arccos \left (a x \right )^{2} \cos \left (2 \arccos \left (a x \right )\right )+3 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-6 \arccos \left (a x \right ) \sin \left (2 \arccos \left (a x \right )\right )}{32 a^{2} \sqrt {\arccos \left (a x \right )}}\)	\(64\)

[In]

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

[Out]

1/32/a^2*(8*arccos(a*x)^2*cos(2*arccos(a*x))+3*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*arccos(a*x)^(1/2)/Pi^(1/2
))-6*arccos(a*x)*sin(2*arccos(a*x)))/arccos(a*x)^(1/2)

Fricas [F(-2)]

Exception generated. \[ \int x \arccos (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int x \arccos (a x)^{3/2} \, dx=\int x \operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}\, dx \]

[In]

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

[Out]

Integral(x*acos(a*x)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int x \arccos (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.20 \[ \int x \arccos (a x)^{3/2} \, dx=\frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{8 \, a^{2}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{8 \, a^{2}} + \frac {\left (3 i - 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{128 \, a^{2}} - \frac {\left (3 i + 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{128 \, a^{2}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{32 \, a^{2}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{32 \, a^{2}} \]

[In]

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

[Out]

1/8*arccos(a*x)^(3/2)*e^(2*I*arccos(a*x))/a^2 + 1/8*arccos(a*x)^(3/2)*e^(-2*I*arccos(a*x))/a^2 + (3/128*I - 3/
128)*sqrt(pi)*erf((I - 1)*sqrt(arccos(a*x)))/a^2 - (3/128*I + 3/128)*sqrt(pi)*erf(-(I + 1)*sqrt(arccos(a*x)))/
a^2 + 3/32*I*sqrt(arccos(a*x))*e^(2*I*arccos(a*x))/a^2 - 3/32*I*sqrt(arccos(a*x))*e^(-2*I*arccos(a*x))/a^2

Mupad [F(-1)]

Timed out. \[ \int x \arccos (a x)^{3/2} \, dx=\int x\,{\mathrm {acos}\left (a\,x\right )}^{3/2} \,d x \]

[In]

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

[Out]

int(x*acos(a*x)^(3/2), x)