3.113 \(\int \frac{x^4}{\sin ^{-1}(a x)^{7/2}} \, dx\)

Optimal. Leaf size=264 \[ \frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^5}+\frac{8 \sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^5}-\frac{5 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}+\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\sin ^{-1}(a x)}}-\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{4 x^5}{3 \sin ^{-1}(a x)^{3/2}} \]

[Out]

(-2*x^4*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) - (16*x^3)/(15*a^2*ArcSin[a*x]^(3/2)) + (4*x^5)/(3*ArcSin[a
*x]^(3/2)) - (32*x^2*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcSin[a*x]]) + (40*x^4*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[ArcSi
n[a*x]]) + (Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a^5) - (5*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi
]*Sqrt[ArcSin[a*x]]])/a^5 + (8*Sqrt[6*Pi]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(5*a^5) + (5*Sqrt[(5*Pi)/2]*
FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^5)

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Rubi [A]  time = 0.39587, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4633, 4719, 4631, 3305, 3351} \[ \frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^5}+\frac{8 \sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^5}-\frac{5 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}+\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\sin ^{-1}(a x)}}-\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{4 x^5}{3 \sin ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcSin[a*x]^(7/2),x]

[Out]

(-2*x^4*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) - (16*x^3)/(15*a^2*ArcSin[a*x]^(3/2)) + (4*x^5)/(3*ArcSin[a
*x]^(3/2)) - (32*x^2*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcSin[a*x]]) + (40*x^4*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[ArcSi
n[a*x]]) + (Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a^5) - (5*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi
]*Sqrt[ArcSin[a*x]]])/a^5 + (8*Sqrt[6*Pi]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(5*a^5) + (5*Sqrt[(5*Pi)/2]*
FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^5)

Rule 4633

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 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 4631

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

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{x^4}{\sin ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{8 \int \frac{x^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx}{5 a}-(2 a) \int \frac{x^5}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^5}{3 \sin ^{-1}(a x)^{3/2}}-\frac{20}{3} \int \frac{x^4}{\sin ^{-1}(a x)^{3/2}} \, dx+\frac{16 \int \frac{x^2}{\sin ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^5}{3 \sin ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\sin ^{-1}(a x)}}+\frac{32 \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{4 \sqrt{x}}+\frac{3 \sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{5 a^5}-\frac{40 \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{8 \sqrt{x}}+\frac{9 \sin (3 x)}{16 \sqrt{x}}-\frac{5 \sin (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^5}{3 \sin ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\sin ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{6 a^5}+\frac{24 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^5}-\frac{15 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^5}{3 \sin ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 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 )}{5 a^5}+\frac{10 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}+\frac{48 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{5 a^5}-\frac{15 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^5}{3 \sin ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\sin ^{-1}(a x)}}+\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^5}-\frac{5 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}+\frac{8 \sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}\\ \end{align*}

Mathematica [C]  time = 0.772494, size = 417, normalized size = 1.58 \[ \frac{-8 \sqrt{-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )+108 \sqrt{3} \sqrt{-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \text{Gamma}\left (\frac{1}{2},-3 i \sin ^{-1}(a x)\right )-100 \sqrt{5} \sqrt{-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \text{Gamma}\left (\frac{1}{2},-5 i \sin ^{-1}(a x)\right )+e^{-i \sin ^{-1}(a x)} \left (8 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )+8 \sin ^{-1}(a x)^2+4 i \sin ^{-1}(a x)-6\right )-9 e^{-3 i \sin ^{-1}(a x)} \left (12 \sqrt{3} e^{3 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},3 i \sin ^{-1}(a x)\right )+12 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)-1\right )+e^{-5 i \sin ^{-1}(a x)} \left (100 \sqrt{5} e^{5 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},5 i \sin ^{-1}(a x)\right )+100 \sin ^{-1}(a x)^2+10 i \sin ^{-1}(a x)-3\right )+9 e^{3 i \sin ^{-1}(a x)} \left (-12 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)+1\right )+2 e^{i \sin ^{-1}(a x)} \left (4 \sin ^{-1}(a x)^2-2 i \sin ^{-1}(a x)-3\right )+e^{5 i \sin ^{-1}(a x)} \left (100 \sin ^{-1}(a x)^2-10 i \sin ^{-1}(a x)-3\right )}{240 a^5 \sin ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/ArcSin[a*x]^(7/2),x]

[Out]

(9*E^((3*I)*ArcSin[a*x])*(1 + (2*I)*ArcSin[a*x] - 12*ArcSin[a*x]^2) + 2*E^(I*ArcSin[a*x])*(-3 - (2*I)*ArcSin[a
*x] + 4*ArcSin[a*x]^2) + E^((5*I)*ArcSin[a*x])*(-3 - (10*I)*ArcSin[a*x] + 100*ArcSin[a*x]^2) - 8*Sqrt[(-I)*Arc
Sin[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*ArcS
in[a*x])*(I*ArcSin[a*x])^(5/2)*Gamma[1/2, I*ArcSin[a*x]])/E^(I*ArcSin[a*x]) + 108*Sqrt[3]*Sqrt[(-I)*ArcSin[a*x
]]*ArcSin[a*x]^2*Gamma[1/2, (-3*I)*ArcSin[a*x]] - (9*(-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]) - 100*Sqrt[5]
*Sqrt[(-I)*ArcSin[a*x]]*ArcSin[a*x]^2*Gamma[1/2, (-5*I)*ArcSin[a*x]] + (-3 + (10*I)*ArcSin[a*x] + 100*ArcSin[a
*x]^2 + 100*Sqrt[5]*E^((5*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(5/2)*Gamma[1/2, (5*I)*ArcSin[a*x]])/E^((5*I)*ArcSin
[a*x]))/(240*a^5*ArcSin[a*x]^(5/2))

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Maple [A]  time = 0.083, size = 225, normalized size = 0.9 \begin{align*} -{\frac{1}{120\,{a}^{5}} \left ( -100\,\sqrt{2}\sqrt{\pi }\sqrt{5}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{5}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}+108\,\sqrt{2}\sqrt{\pi }\sqrt{3}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}-8\,\sqrt{2}\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}+108\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) -100\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\cos \left ( 5\,\arcsin \left ( ax \right ) \right ) -8\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}-4\,ax\arcsin \left ( ax \right ) +18\,\arcsin \left ( ax \right ) \sin \left ( 3\,\arcsin \left ( ax \right ) \right ) -10\,\arcsin \left ( ax \right ) \sin \left ( 5\,\arcsin \left ( ax \right ) \right ) -9\,\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) +3\,\cos \left ( 5\,\arcsin \left ( ax \right ) \right ) +6\,\sqrt{-{a}^{2}{x}^{2}+1} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arcsin(a*x)^(7/2),x)

[Out]

-1/120/a^5*(-100*2^(1/2)*Pi^(1/2)*5^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/
2)+108*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)-8*2^(1/
2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)+108*arcsin(a*x)^2*cos(3*arcsin(a*x)
)-100*arcsin(a*x)^2*cos(5*arcsin(a*x))-8*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)-4*a*x*arcsin(a*x)+18*arcsin(a*x)*sin
(3*arcsin(a*x))-10*arcsin(a*x)*sin(5*arcsin(a*x))-9*cos(3*arcsin(a*x))+3*cos(5*arcsin(a*x))+6*(-a^2*x^2+1)^(1/
2))/arcsin(a*x)^(5/2)

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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(x^4/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(x^4/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(x**4/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{x^{4}}{\arcsin \left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsin(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^4/arcsin(a*x)^(7/2), x)