3.108 \(\int \frac {x^3}{\sin ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac {4 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^4}-\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^4}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}} \]

[Out]

-4/3*FresnelS(2*arcsin(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^4+4/3*FresnelS(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^
(1/2)*Pi^(1/2)/a^4-2/3*x^3*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(3/2)-4*x^2/a^2/arcsin(a*x)^(1/2)+16/3*x^4/arcsin(
a*x)^(1/2)

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Rubi [A]  time = 0.34, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4633, 4719, 4635, 4406, 3305, 3351, 12} \[ \frac {4 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^4}-\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^4}-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/ArcSin[a*x]^(5/2),x]

[Out]

(-2*x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2)) - (4*x^2)/(a^2*Sqrt[ArcSin[a*x]]) + (16*x^4)/(3*Sqrt[ArcSin
[a*x]]) + (4*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^4) - (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSi
n[a*x]])/Sqrt[Pi]])/(3*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 4633

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

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 4719

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

Rubi steps

\begin {align*} \int \frac {x^3}{\sin ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac {2 \int \frac {x^2}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx}{a}-\frac {1}{3} (8 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}-\frac {64}{3} \int \frac {x^3}{\sqrt {\sin ^{-1}(a x)}} \, dx+\frac {8 \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {8 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}-\frac {64 \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^3(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {8 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}-\frac {64 \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 )}{3 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {8 \operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^4}+\frac {4 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}-\frac {16 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {16 \operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{3 a^4}+\frac {8 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{a^4}-\frac {32 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{3 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {4 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^4}-\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^4}\\ \end {align*}

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Mathematica [C]  time = 0.38, size = 200, normalized size = 1.59 \[ \frac {-2 \sin \left (2 \sin ^{-1}(a x)\right )+\sin \left (4 \sin ^{-1}(a x)\right )-4 \sin ^{-1}(a x) \left (e^{-2 i \sin ^{-1}(a x)}+e^{2 i \sin ^{-1}(a x)}-\sqrt {2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \sin ^{-1}(a x)\right )-\sqrt {2} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \sin ^{-1}(a x)\right )\right )+4 \sin ^{-1}(a x) \left (e^{-4 i \sin ^{-1}(a x)}+e^{4 i \sin ^{-1}(a x)}-2 \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \sin ^{-1}(a x)\right )-2 \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \sin ^{-1}(a x)\right )\right )}{12 a^4 \sin ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/ArcSin[a*x]^(5/2),x]

[Out]

(-4*ArcSin[a*x]*(E^((-2*I)*ArcSin[a*x]) + E^((2*I)*ArcSin[a*x]) - Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-
2*I)*ArcSin[a*x]] - Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (2*I)*ArcSin[a*x]]) + 4*ArcSin[a*x]*(E^((-4*I)*ArcS
in[a*x]) + E^((4*I)*ArcSin[a*x]) - 2*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-4*I)*ArcSin[a*x]] - 2*Sqrt[I*ArcSin[a
*x]]*Gamma[1/2, (4*I)*ArcSin[a*x]]) - 2*Sin[2*ArcSin[a*x]] + Sin[4*ArcSin[a*x]])/(12*a^4*ArcSin[a*x]^(3/2))

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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(x^3/arcsin(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 [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)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.08, size = 109, normalized size = 0.87 \[ -\frac {-16 \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+16 \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+8 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )-8 \arcsin \left (a x \right ) \cos \left (4 \arcsin \left (a x \right )\right )+2 \sin \left (2 \arcsin \left (a x \right )\right )-\sin \left (4 \arcsin \left (a x \right )\right )}{12 a^{4} \arcsin \left (a x \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/arcsin(a*x)^(5/2),x)

[Out]

-1/12/a^4*(-16*2^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)+16*Pi^(1/2)*F
resnelS(2*arcsin(a*x)^(1/2)/Pi^(1/2))*arcsin(a*x)^(3/2)+8*arcsin(a*x)*cos(2*arcsin(a*x))-8*arcsin(a*x)*cos(4*a
rcsin(a*x))+2*sin(2*arcsin(a*x))-sin(4*arcsin(a*x)))/arcsin(a*x)^(3/2)

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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)^(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 \[ \int \frac {x^3}{{\mathrm {asin}\left (a\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/asin(a*x)^(5/2),x)

[Out]

int(x^3/asin(a*x)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/asin(a*x)**(5/2),x)

[Out]

Integral(x**3/asin(a*x)**(5/2), x)

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