\(\int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx\) [115]

Optimal result

Integrand size = 12, antiderivative size = 191 \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {8 x}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^3}{5 \arcsin (a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\arcsin (a x)}}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\arcsin (a x)}}+\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{15 a^3}-\frac {6 \sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{5 a^3} \]

[Out]

-8/15*x/a^2/arcsin(a*x)^(3/2)+4/5*x^3/arcsin(a*x)^(3/2)+2/15*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1
/2)*Pi^(1/2)/a^3-6/5*FresnelS(6^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3-2/5*x^2*(-a^2*x^2+1)^(1
/2)/a/arcsin(a*x)^(5/2)-16/15*(-a^2*x^2+1)^(1/2)/a^3/arcsin(a*x)^(1/2)+24/5*x^2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*
x)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 191, 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, 4727, 3386, 3432, 4717, 4809} \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{15 a^3}-\frac {6 \sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{5 a^3}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\arcsin (a x)}}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {8 x}{15 a^2 \arcsin (a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\arcsin (a x)}}+\frac {4 x^3}{5 \arcsin (a x)^{3/2}} \]

[In]

Int[x^2/ArcSin[a*x]^(7/2),x]

[Out]

(-2*x^2*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) - (8*x)/(15*a^2*ArcSin[a*x]^(3/2)) + (4*x^3)/(5*ArcSin[a*x]
^(3/2)) - (16*Sqrt[1 - a^2*x^2])/(15*a^3*Sqrt[ArcSin[a*x]]) + (24*x^2*Sqrt[1 - a^2*x^2])/(5*a*Sqrt[ArcSin[a*x]
]) + (2*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a^3) - (6*Sqrt[6*Pi]*FresnelS[Sqrt[6/Pi]*Sqrt[A
rcSin[a*x]]])/(5*a^3)

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 4717

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

Rule 4727

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[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[
-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x]
&& IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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 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]

Rule 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c
^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x],
x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}+\frac {4 \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)^{5/2}} \, dx}{5 a}-\frac {1}{5} (6 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \arcsin (a x)^{5/2}} \, dx \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {8 x}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^3}{5 \arcsin (a x)^{3/2}}-\frac {12}{5} \int \frac {x^2}{\arcsin (a x)^{3/2}} \, dx+\frac {8 \int \frac {1}{\arcsin (a x)^{3/2}} \, dx}{15 a^2} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {8 x}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^3}{5 \arcsin (a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\arcsin (a x)}}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\arcsin (a x)}}-\frac {24 \text {Subst}\left (\int \left (-\frac {\sin (x)}{4 \sqrt {x}}+\frac {3 \sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{5 a^3}-\frac {16 \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \, dx}{15 a} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {8 x}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^3}{5 \arcsin (a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\arcsin (a x)}}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\arcsin (a x)}}-\frac {16 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{15 a^3}+\frac {6 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{5 a^3}-\frac {18 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{5 a^3} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {8 x}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^3}{5 \arcsin (a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\arcsin (a x)}}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\arcsin (a x)}}-\frac {32 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{15 a^3}+\frac {12 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{5 a^3}-\frac {36 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{5 a^3} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {8 x}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^3}{5 \arcsin (a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\arcsin (a x)}}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\arcsin (a x)}}+\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{15 a^3}-\frac {6 \sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{5 a^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.47 \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\frac {3 e^{3 i \arcsin (a x)} \left (1+2 i \arcsin (a x)-12 \arcsin (a x)^2\right )+e^{i \arcsin (a x)} \left (-3-2 i \arcsin (a x)+4 \arcsin (a x)^2\right )-4 \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )+e^{-i \arcsin (a x)} \left (-3+2 i \arcsin (a x)+4 \arcsin (a x)^2+4 e^{i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )+36 \sqrt {3} \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )-3 e^{-3 i \arcsin (a x)} \left (-1+2 i \arcsin (a x)+12 \arcsin (a x)^2+12 \sqrt {3} e^{3 i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )\right )}{60 a^3 \arcsin (a x)^{5/2}} \]

[In]

Integrate[x^2/ArcSin[a*x]^(7/2),x]

[Out]

(3*E^((3*I)*ArcSin[a*x])*(1 + (2*I)*ArcSin[a*x] - 12*ArcSin[a*x]^2) + E^(I*ArcSin[a*x])*(-3 - (2*I)*ArcSin[a*x
] + 4*ArcSin[a*x]^2) - 4*Sqrt[(-I)*ArcSin[a*x]]*ArcSin[a*x]^2*Gamma[1/2, (-I)*ArcSin[a*x]] + (-3 + (2*I)*ArcSi
n[a*x] + 4*ArcSin[a*x]^2 + 4*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(5/2)*Gamma[1/2, I*ArcSin[a*x]])/E^(I*ArcSin[a*
x]) + 36*Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]*ArcSin[a*x]^2*Gamma[1/2, (-3*I)*ArcSin[a*x]] - (3*(-1 + (2*I)*ArcSin[a
*x] + 12*ArcSin[a*x]^2 + 12*Sqrt[3]*E^((3*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(5/2)*Gamma[1/2, (3*I)*ArcSin[a*x]])
)/E^((3*I)*ArcSin[a*x]))/(60*a^3*ArcSin[a*x]^(5/2))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.81

method	result	size
default	\(-\frac {36 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-4 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}+36 \arcsin \left (a x \right )^{2} \cos \left (3 \arcsin \left (a x \right )\right )-4 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a x \arcsin \left (a x \right )+6 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )-3 \cos \left (3 \arcsin \left (a x \right )\right )+3 \sqrt {-a^{2} x^{2}+1}}{30 a^{3} \arcsin \left (a x \right )^{\frac {5}{2}}}\)	\(154\)

[In]

int(x^2/arcsin(a*x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/30/a^3*(36*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)-
4*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)+36*arcsin(a*x)^2*cos(3*arcsi
n(a*x))-4*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)-2*a*x*arcsin(a*x)+6*arcsin(a*x)*sin(3*arcsin(a*x))-3*cos(3*arcsin(a
*x))+3*(-a^2*x^2+1)^(1/2))/arcsin(a*x)^(5/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2/arcsin(a*x)^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x^{2}}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(x**2/asin(a*x)**(7/2),x)

[Out]

Integral(x**2/asin(a*x)**(7/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^2/arcsin(a*x)^(7/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\int { \frac {x^{2}}{\arcsin \left (a x\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate(x^2/arcsin(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^2/arcsin(a*x)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x^2}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \]

[In]

int(x^2/asin(a*x)^(7/2),x)

[Out]

int(x^2/asin(a*x)^(7/2), x)