\(\int \frac {x^3}{\arcsin (a x)^{5/2}} \, dx\) [108]

Optimal result

Integrand size = 12, antiderivative size = 126 \[ \int \frac {x^3}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\arcsin (a x)}}+\frac {16 x^4}{3 \sqrt {\arcsin (a x)}}+\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^4}-\frac {4 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{3 a^4} \]

[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)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4729, 4807, 4731, 4491, 3386, 3432, 12} \[ \int \frac {x^3}{\arcsin (a x)^{5/2}} \, dx=\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^4}-\frac {4 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{3 a^4}-\frac {4 x^2}{a^2 \sqrt {\arcsin (a x)}}-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}+\frac {16 x^4}{3 \sqrt {\arcsin (a x)}} \]

[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 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 4729

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 4731

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

Rule 4807

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.59 \[ \int \frac {x^3}{\arcsin (a x)^{5/2}} \, dx=\frac {-4 \arcsin (a x) \left (e^{-2 i \arcsin (a x)}+e^{2 i \arcsin (a x)}-\sqrt {2} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-2 i \arcsin (a x)\right )-\sqrt {2} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},2 i \arcsin (a x)\right )\right )+4 \arcsin (a x) \left (e^{-4 i \arcsin (a x)}+e^{4 i \arcsin (a x)}-2 \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-4 i \arcsin (a x)\right )-2 \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},4 i \arcsin (a x)\right )\right )-2 \sin (2 \arcsin (a x))+\sin (4 \arcsin (a x))}{12 a^4 \arcsin (a x)^{3/2}} \]

[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))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.87

method	result	size
default	\(-\frac {-16 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+16 \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+8 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )-8 \arcsin \left (a x \right ) \cos \left (4 \arcsin \left (a x \right )\right )+2 \sin \left (2 \arcsin \left (a x \right )\right )-\sin \left (4 \arcsin \left (a x \right )\right )}{12 a^{4} \arcsin \left (a x \right )^{\frac {3}{2}}}\)	\(109\)

[In]

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

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F]

\[ \int \frac {x^3}{\arcsin (a x)^{5/2}} \, dx=\int \frac {x^{3}}{\operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral(x**3/asin(a*x)**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^3/arcsin(a*x)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\arcsin (a x)^{5/2}} \, dx=\int \frac {x^3}{{\mathrm {asin}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

int(x^3/asin(a*x)^(5/2), x)