\(\int \frac {x^2}{\arcsin (a x)^{5/2}} \, dx\) [109]

Optimal result

Integrand size = 12, antiderivative size = 125 \[ \int \frac {x^2}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {4 x^3}{\sqrt {\arcsin (a x)}}-\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^3}+\frac {\sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^3} \]

[Out]

-1/3*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3+FresnelC(6^(1/2)/Pi^(1/2)*arcsin(a*x)^(
1/2))*6^(1/2)*Pi^(1/2)/a^3-2/3*x^2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(3/2)-8/3*x/a^2/arcsin(a*x)^(1/2)+4*x^3/ar
csin(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4729, 4807, 4731, 4491, 3385, 3433, 4719} \[ \int \frac {x^2}{\arcsin (a x)^{5/2}} \, dx=-\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^3}+\frac {\sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {4 x^3}{\sqrt {\arcsin (a x)}} \]

[In]

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

[Out]

(-2*x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2)) - (8*x)/(3*a^2*Sqrt[ArcSin[a*x]]) + (4*x^3)/Sqrt[ArcSin[a*x
]] - (Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^3) + (Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin
[a*x]]])/a^3

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[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 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[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 4719

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

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^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}+\frac {4 \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}} \, dx}{3 a}-(2 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}} \, dx \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {4 x^3}{\sqrt {\arcsin (a x)}}-12 \int \frac {x^2}{\sqrt {\arcsin (a x)}} \, dx+\frac {8 \int \frac {1}{\sqrt {\arcsin (a x)}} \, dx}{3 a^2} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {4 x^3}{\sqrt {\arcsin (a x)}}+\frac {8 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{3 a^3}-\frac {12 \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {4 x^3}{\sqrt {\arcsin (a x)}}+\frac {16 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{3 a^3}-\frac {12 \text {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {x}}-\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {4 x^3}{\sqrt {\arcsin (a x)}}+\frac {8 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^3}-\frac {3 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{a^3}+\frac {3 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {4 x^3}{\sqrt {\arcsin (a x)}}+\frac {8 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^3}-\frac {6 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{a^3}+\frac {6 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {4 x^3}{\sqrt {\arcsin (a x)}}-\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^3}+\frac {\sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.22 \[ \int \frac {x^2}{\arcsin (a x)^{5/2}} \, dx=\frac {\frac {i e^{i \arcsin (a x)} (i-2 \arcsin (a x))-2 (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )}{12 \arcsin (a x)^{3/2}}-\frac {e^{-i \arcsin (a x)} \left (1-2 i \arcsin (a x)+2 e^{i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )}{12 \arcsin (a x)^{3/2}}-\frac {i e^{3 i \arcsin (a x)} (i-6 \arcsin (a x))-6 \sqrt {3} (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )}{12 \arcsin (a x)^{3/2}}+\frac {e^{-3 i \arcsin (a x)} \left (1-6 i \arcsin (a x)+6 \sqrt {3} e^{3 i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )\right )}{12 \arcsin (a x)^{3/2}}}{a^3} \]

[In]

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

[Out]

((I*E^(I*ArcSin[a*x])*(I - 2*ArcSin[a*x]) - 2*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-I)*ArcSin[a*x]])/(12*ArcSi
n[a*x]^(3/2)) - (1 - (2*I)*ArcSin[a*x] + 2*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, I*ArcSin[a*x]])/
(12*E^(I*ArcSin[a*x])*ArcSin[a*x]^(3/2)) - (I*E^((3*I)*ArcSin[a*x])*(I - 6*ArcSin[a*x]) - 6*Sqrt[3]*((-I)*ArcS
in[a*x])^(3/2)*Gamma[1/2, (-3*I)*ArcSin[a*x]])/(12*ArcSin[a*x]^(3/2)) + (1 - (6*I)*ArcSin[a*x] + 6*Sqrt[3]*E^(
(3*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcSin[a*x]])/(12*E^((3*I)*ArcSin[a*x])*ArcSin[a*x]^
(3/2)))/a^3

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {-6 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+2 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}-2 a x \arcsin \left (a x \right )+6 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )+\sqrt {-a^{2} x^{2}+1}-\cos \left (3 \arcsin \left (a x \right )\right )}{6 a^{3} \arcsin \left (a x \right )^{\frac {3}{2}}}\)	\(117\)

[In]

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

[Out]

-1/6/a^3*(-6*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)+2
*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)-2*a*x*arcsin(a*x)+6*arcsin(a*
x)*sin(3*arcsin(a*x))+(-a^2*x^2+1)^(1/2)-cos(3*arcsin(a*x)))/arcsin(a*x)^(3/2)

Fricas [F(-2)]

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

[In]

integrate(x^2/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]

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

[In]

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

[Out]

Integral(x**2/asin(a*x)**(5/2), x)

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(x^2/arcsin(a*x)^(5/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2/asin(a*x)^(5/2), x)