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

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

[Out]

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

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Rubi [A]  time = 0.38, 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 = {4629, 4707, 4641, 4635, 4406, 12, 3305, 3351} \[ \frac {3 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{512 a^4}-\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{64 a^4}+\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{32 a}+\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{64 a^3}-\frac {3 \sin ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4629

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcSin[c*x])^n)/(m
+ 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSin[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 4635

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

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[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 4707

Int[(((a_.) + ArcSin[(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*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[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*ArcSin[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]

Rubi steps

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

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Mathematica [C]  time = 0.03, size = 130, normalized size = 0.83 \[ \frac {8 \sqrt {2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},-2 i \sin ^{-1}(a x)\right )+8 \sqrt {2} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},2 i \sin ^{-1}(a x)\right )-\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},-4 i \sin ^{-1}(a x)\right )-\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},4 i \sin ^{-1}(a x)\right )}{512 a^4 \sqrt {\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(8*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, (-2*I)*ArcSin[a*x]] + 8*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (2
*I)*ArcSin[a*x]] - Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, (-4*I)*ArcSin[a*x]] - Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (4*I
)*ArcSin[a*x]])/(512*a^4*Sqrt[ArcSin[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^3*arcsin(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 [C]  time = 0.31, size = 225, normalized size = 1.43 \[ \frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} - \frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} - \frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} + \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{4096 \, a^{4}} - \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{4096 \, a^{4}} - \frac {\left (3 i - 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{256 \, a^{4}} + \frac {\left (3 i + 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{256 \, a^{4}} + \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{512 \, a^{4}} - \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} + \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} - \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{512 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 121, normalized size = 0.77 \[ -\frac {128 \arcsin \left (a x \right )^{2} \cos \left (2 \arcsin \left (a x \right )\right )-32 \arcsin \left (a x \right )^{2} \cos \left (4 \arcsin \left (a x \right )\right )-3 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+12 \arcsin \left (a x \right ) \sin \left (4 \arcsin \left (a x \right )\right )-96 \arcsin \left (a x \right ) \sin \left (2 \arcsin \left (a x \right )\right )+48 \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )}{1024 a^{4} \sqrt {\arcsin \left (a x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1024/a^4/arcsin(a*x)^(1/2)*(128*arcsin(a*x)^2*cos(2*arcsin(a*x))-32*arcsin(a*x)^2*cos(4*arcsin(a*x))-3*2^(1
/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))+12*arcsin(a*x)*sin(4*arcsin(a*x)
)-96*arcsin(a*x)*sin(2*arcsin(a*x))+48*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*arcsin(a*x)^(1/2)/Pi^(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(x^3*arcsin(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 \[ \int x^3\,{\mathrm {asin}\left (a\,x\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*asin(a*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*asin(a*x)**(3/2), x)

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