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

Optimal. Leaf size=76 \[ -\frac{2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{4 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a}+\frac{4 x}{3 \sqrt{\sin ^{-1}(a x)}} \]

[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)

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Rubi [A]  time = 0.0989886, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4621, 4719, 4623, 3304, 3352} \[ -\frac{2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{4 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a}+\frac{4 x}{3 \sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[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 4621

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_)*((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]

Rule 4623

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 3304

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 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

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

Mathematica [C]  time = 0.127191, size = 138, normalized size = 1.82 \[ \frac{-4 \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )-2 i e^{i \sin ^{-1}(a x)} \left (2 \sin ^{-1}(a x)-i\right )}{6 a \sin ^{-1}(a x)^{3/2}}+\frac{e^{-i \sin ^{-1}(a x)} \left (-4 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )+4 i \sin ^{-1}(a x)-2\right )}{6 a \sin ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[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))

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Maple [A]  time = 0.042, size = 83, normalized size = 1.1 \begin{align*} -{\frac{\sqrt{2}}{3\,a\sqrt{\pi } \left ( \arcsin \left ( ax \right ) \right ) ^{2}} \left ( 4\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\pi \,{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -2\, \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }xa+\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)*x*a+2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*(-a^2*x^2+1)^(1/2))/arcsin(a*x)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{asin}^{\frac{5}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asin(a*x)**(-5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\arcsin \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsin(a*x)^(-5/2), x)