3.117 \(\int \frac{1}{\sin ^{-1}(a x)^{7/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{8 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}-\frac{2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{8 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a}+\frac{4 x}{15 \sin ^{-1}(a x)^{3/2}} \]

[Out]

(-2*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) + (4*x)/(15*ArcSin[a*x]^(3/2)) + (8*Sqrt[1 - a^2*x^2])/(15*a*Sq
rt[ArcSin[a*x]]) + (8*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a)

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Rubi [A]  time = 0.165702, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4621, 4719, 4723, 3305, 3351} \[ \frac{8 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}-\frac{2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{8 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a}+\frac{4 x}{15 \sin ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) + (4*x)/(15*ArcSin[a*x]^(3/2)) + (8*Sqrt[1 - a^2*x^2])/(15*a*Sq
rt[ArcSin[a*x]]) + (8*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*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 4723

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

Rule 3305

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 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[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)^{7/2}} \, dx &=-\frac{2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{1}{5} (2 a) \int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{4 x}{15 \sin ^{-1}(a x)^{3/2}}-\frac{4}{15} \int \frac{1}{\sin ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{4 x}{15 \sin ^{-1}(a x)^{3/2}}+\frac{8 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}+\frac{1}{15} (8 a) \int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{4 x}{15 \sin ^{-1}(a x)^{3/2}}+\frac{8 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{4 x}{15 \sin ^{-1}(a x)^{3/2}}+\frac{8 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{15 a}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{4 x}{15 \sin ^{-1}(a x)^{3/2}}+\frac{8 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}+\frac{8 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a}\\ \end{align*}

Mathematica [C]  time = 0.259962, size = 143, normalized size = 1.36 \[ \frac{-8 \sqrt{-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )+e^{-i \sin ^{-1}(a x)} \left (8 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )+8 \sin ^{-1}(a x)^2+4 i \sin ^{-1}(a x)-6\right )+2 e^{i \sin ^{-1}(a x)} \left (4 \sin ^{-1}(a x)^2-2 i \sin ^{-1}(a x)-3\right )}{30 a \sin ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*E^(I*ArcSin[a*x])*(-3 - (2*I)*ArcSin[a*x] + 4*ArcSin[a*x]^2) - 8*Sqrt[(-I)*ArcSin[a*x]]*ArcSin[a*x]^2*Gamma
[1/2, (-I)*ArcSin[a*x]] + (-6 + (4*I)*ArcSin[a*x] + 8*ArcSin[a*x]^2 + 8*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(5/2
)*Gamma[1/2, I*ArcSin[a*x]])/E^(I*ArcSin[a*x]))/(30*a*ArcSin[a*x]^(5/2))

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Maple [A]  time = 0.04, size = 110, normalized size = 1.1 \begin{align*}{\frac{\sqrt{2}}{15\,a\sqrt{\pi } \left ( \arcsin \left ( ax \right ) \right ) ^{3}} \left ( 8\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}\pi \,{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +4\, \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }\sqrt{-{a}^{2}{x}^{2}+1}+2\, \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }xa-3\,\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)^(7/2),x)

[Out]

1/15/a*2^(1/2)/Pi^(1/2)/arcsin(a*x)^3*(8*arcsin(a*x)^3*Pi*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))+4*arcsi
n(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*(-a^2*x^2+1)^(1/2)+2*arcsin(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*x*a-3*2^(1/2)*arcsin(a*x
)^(1/2)*Pi^(1/2)*(-a^2*x^2+1)^(1/2))

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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)^(7/2),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)^(7/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/asin(a*x)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\arcsin \left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsin(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^(-7/2), x)