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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.367786, antiderivative size = 157, normalized size of antiderivative = 1., 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 = {4630, 4708, 4642, 4636, 4406, 12, 3305, 3351} \[ \frac{3 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{512 a^4}+\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{64 a^4}-\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{32 a}-\frac{9 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{64 a^3}-\frac{3 \cos ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{3/2} \]

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 4630

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 4708

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

Rule 4642

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

Rule 4636

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

Rule 4406

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 12

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

Rule 3305

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 3351

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

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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{512 a^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{256 a^4}+\frac{9 \operatorname{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 \operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{256 a^4}+\frac{3 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{128 a^4}+\frac{9 \operatorname{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 \operatorname{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*}

Mathematica [C]  time = 0.0724389, size = 128, normalized size = 0.82 \[ -\frac{8 \sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-2 i \cos ^{-1}(a x)\right )+8 \sqrt{2} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},2 i \cos ^{-1}(a x)\right )+\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-4 i \cos ^{-1}(a x)\right )+\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},4 i \cos ^{-1}(a x)\right )}{512 a^4 \sqrt{\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(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]])/(512*a^4*Sqrt[ArcCos[a*x]])

________________________________________________________________________________________

Maple [A]  time = 0.089, size = 121, normalized size = 0.8 \begin{align*}{\frac{1}{1024\,{a}^{4}} \left ( 3\,\sqrt{2}\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +128\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\cos \left ( 2\,\arccos \left ( ax \right ) \right ) +32\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\cos \left ( 4\,\arccos \left ( ax \right ) \right ) +48\,\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -12\,\arccos \left ( ax \right ) \sin \left ( 4\,\arccos \left ( ax \right ) \right ) -96\,\arccos \left ( ax \right ) \sin \left ( 2\,\arccos \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1024/a^4*(3*2^(1/2)*Pi^(1/2)*arccos(a*x)^(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*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelS(2*arccos(a*x
)^(1/2)/Pi^(1/2))-12*arccos(a*x)*sin(4*arccos(a*x))-96*arccos(a*x)*sin(2*arccos(a*x)))/arccos(a*x)^(1/2)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [B]  time = 1.32685, size = 363, normalized size = 2.31 \begin{align*} \frac{3 \, i \sqrt{\arccos \left (a x\right )} e^{\left (4 \, i \arccos \left (a x\right )\right )}}{512 \, 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{3 \, i \sqrt{\arccos \left (a x\right )} e^{\left (2 \, 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{3 \, i \sqrt{\arccos \left (a x\right )} e^{\left (-2 \, 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{3 \, i \sqrt{\arccos \left (a x\right )} e^{\left (-4 \, i \arccos \left (a x\right )\right )}}{512 \, 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{3 \, \sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (\sqrt{2}{\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{2048 \, a^{4}{\left (i - 1\right )}} - \frac{3 \, \sqrt{\pi } i \operatorname{erf}\left ({\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{128 \, a^{4}{\left (i - 1\right )}} + \frac{3 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{2}{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{2048 \, a^{4}{\left (i - 1\right )}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{128 \, a^{4}{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/512*i*sqrt(arccos(a*x))*e^(4*i*arccos(a*x))/a^4 + 1/64*arccos(a*x)^(3/2)*e^(4*i*arccos(a*x))/a^4 + 3/64*i*sq
rt(arccos(a*x))*e^(2*i*arccos(a*x))/a^4 + 1/16*arccos(a*x)^(3/2)*e^(2*i*arccos(a*x))/a^4 - 3/64*i*sqrt(arccos(
a*x))*e^(-2*i*arccos(a*x))/a^4 + 1/16*arccos(a*x)^(3/2)*e^(-2*i*arccos(a*x))/a^4 - 3/512*i*sqrt(arccos(a*x))*e
^(-4*i*arccos(a*x))/a^4 + 1/64*arccos(a*x)^(3/2)*e^(-4*i*arccos(a*x))/a^4 - 3/2048*sqrt(2)*sqrt(pi)*i*erf(sqrt
(2)*(i - 1)*sqrt(arccos(a*x)))/(a^4*(i - 1)) - 3/128*sqrt(pi)*i*erf((i - 1)*sqrt(arccos(a*x)))/(a^4*(i - 1)) +
3/2048*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*(i + 1)*sqrt(arccos(a*x)))/(a^4*(i - 1)) + 3/128*sqrt(pi)*erf(-(i + 1)*s
qrt(arccos(a*x)))/(a^4*(i - 1))