\(\int \frac {1}{\arcsin (a x)^{5/2}} \, dx\) [111]

Optimal result

Integrand size = 8, antiderivative size = 76 \[ \int \frac {1}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}+\frac {4 x}{3 \sqrt {\arcsin (a x)}}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4717, 4807, 4719, 3385, 3433} \[ \int \frac {1}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a}+\frac {4 x}{3 \sqrt {\arcsin (a x)}} \]

[In]

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

[Out]

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.82 \[ \int \frac {1}{\arcsin (a x)^{5/2}} \, dx=\frac {-2 i e^{i \arcsin (a x)} (-i+2 \arcsin (a x))-4 (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )}{6 a \arcsin (a x)^{3/2}}+\frac {e^{-i \arcsin (a x)} \left (-2+4 i \arcsin (a x)-4 e^{i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )}{6 a \arcsin (a x)^{3/2}} \]

[In]

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

[Out]

((-2*I)*E^(I*ArcSin[a*x])*(-I + 2*ArcSin[a*x]) - 4*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-I)*ArcSin[a*x]])/(6*a
*ArcSin[a*x]^(3/2)) + (-2 + (4*I)*ArcSin[a*x] - 4*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, I*ArcSin[
a*x]])/(6*a*E^(I*ArcSin[a*x])*ArcSin[a*x]^(3/2))

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.09

method	result	size
default	\(-\frac {\sqrt {2}\, \left (4 \arcsin \left (a x \right )^{2} \pi \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-2 \arcsin \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a x +\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{3 a \sqrt {\pi }\, \arcsin \left (a x \right )^{2}}\)	\(83\)

[In]

int(1/arcsin(a*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/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]

\[ \int \frac {1}{\arcsin (a x)^{5/2}} \, dx=\int \frac {1}{\operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(1/asin(a*x)**(5/2),x)

[Out]

Integral(asin(a*x)**(-5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/arcsin(a*x)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{\arcsin (a x)^{5/2}} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/arcsin(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^(-5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\arcsin (a x)^{5/2}} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

int(1/asin(a*x)^(5/2),x)

[Out]

int(1/asin(a*x)^(5/2), x)