3.100 \(\int \frac{x^5}{\sin ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=127 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^6}+\frac{\sqrt{3 \pi } \text{FresnelC}\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^6}+\frac{5 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a^6}-\frac{2 x^5 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]

[Out]

(-2*x^5*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^6 +
(Sqrt[3*Pi]*FresnelC[2*Sqrt[3/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^6) + (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sq
rt[Pi]])/(8*a^6)

________________________________________________________________________________________

Rubi [A]  time = 0.106607, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4631, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^6}+\frac{\sqrt{3 \pi } \text{FresnelC}\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^6}+\frac{5 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a^6}-\frac{2 x^5 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/ArcSin[a*x]^(3/2),x]

[Out]

(-2*x^5*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^6 +
(Sqrt[3*Pi]*FresnelC[2*Sqrt[3/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^6) + (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sq
rt[Pi]])/(8*a^6)

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 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{x^5}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x^5 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \left (\frac{5 \cos (2 x)}{16 \sqrt{x}}-\frac{\cos (4 x)}{2 \sqrt{x}}+\frac{3 \cos (6 x)}{16 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^6}\\ &=-\frac{2 x^5 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (6 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^6}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^6}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^6}\\ &=-\frac{2 x^5 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}+\frac{3 \operatorname{Subst}\left (\int \cos \left (6 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{4 a^6}+\frac{5 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{4 a^6}-\frac{2 \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a^6}\\ &=-\frac{2 x^5 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^6}+\frac{\sqrt{3 \pi } C\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^6}+\frac{5 \sqrt{\pi } C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a^6}\\ \end{align*}

Mathematica [C]  time = 0.135244, size = 231, normalized size = 1.82 \[ -\frac{5 i \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )-5 i \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )-8 i \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \sin ^{-1}(a x)\right )+8 i \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \sin ^{-1}(a x)\right )+i \sqrt{6} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-6 i \sin ^{-1}(a x)\right )-i \sqrt{6} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},6 i \sin ^{-1}(a x)\right )+10 \sin \left (2 \sin ^{-1}(a x)\right )-8 \sin \left (4 \sin ^{-1}(a x)\right )+2 \sin \left (6 \sin ^{-1}(a x)\right )}{32 a^6 \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^5/ArcSin[a*x]^(3/2),x]

[Out]

-((5*I)*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] - (5*I)*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamm
a[1/2, (2*I)*ArcSin[a*x]] - (8*I)*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-4*I)*ArcSin[a*x]] + (8*I)*Sqrt[I*ArcSin[
a*x]]*Gamma[1/2, (4*I)*ArcSin[a*x]] + I*Sqrt[6]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-6*I)*ArcSin[a*x]] - I*Sqrt
[6]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (6*I)*ArcSin[a*x]] + 10*Sin[2*ArcSin[a*x]] - 8*Sin[4*ArcSin[a*x]] + 2*Sin[6
*ArcSin[a*x]])/(32*a^6*Sqrt[ArcSin[a*x]])

________________________________________________________________________________________

Maple [A]  time = 0.067, size = 121, normalized size = 1. \begin{align*} -{\frac{1}{16\,{a}^{6}} \left ( -2\,\sqrt{\pi }\sqrt{3}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{6}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \sqrt{\arcsin \left ( ax \right ) }+8\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -10\,\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +5\,\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) -4\,\sin \left ( 4\,\arcsin \left ( ax \right ) \right ) +\sin \left ( 6\,\arcsin \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/arcsin(a*x)^(3/2),x)

[Out]

-1/16/a^6/arcsin(a*x)^(1/2)*(-2*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*6^(1/2)*arcsin(a*x)^(1/2))*arcsin(a
*x)^(1/2)+8*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))-10*arcsin(a*x)^(
1/2)*Pi^(1/2)*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))+5*sin(2*arcsin(a*x))-4*sin(4*arcsin(a*x))+sin(6*arcsin(a*
x)))

________________________________________________________________________________________

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^5/arcsin(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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^5/arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\operatorname{asin}^{\frac{3}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/asin(a*x)**(3/2),x)

[Out]

Integral(x**5/asin(a*x)**(3/2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\arcsin \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(x^5/arcsin(a*x)^(3/2), x)