3.114 \(\int \frac{x^3}{\sin ^{-1}(a x)^{7/2}} \, dx\)

Optimal. Leaf size=190 \[ \frac{32 \sqrt{2 \pi } \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^4}-\frac{16 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{15 a^4}+\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}-\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}-\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{16 x^4}{15 \sin ^{-1}(a x)^{3/2}} \]

[Out]

(-2*x^3*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) - (4*x^2)/(5*a^2*ArcSin[a*x]^(3/2)) + (16*x^4)/(15*ArcSin[a
*x]^(3/2)) - (16*x*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcSin[a*x]]) + (128*x^3*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcSi
n[a*x]]) + (32*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a^4) - (16*Sqrt[Pi]*FresnelC[(2*Sqrt[A
rcSin[a*x]])/Sqrt[Pi]])/(15*a^4)

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Rubi [A]  time = 0.318782, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4633, 4719, 4631, 3304, 3352} \[ \frac{32 \sqrt{2 \pi } \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^4}-\frac{16 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{15 a^4}+\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}-\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}-\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{16 x^4}{15 \sin ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/ArcSin[a*x]^(7/2),x]

[Out]

(-2*x^3*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) - (4*x^2)/(5*a^2*ArcSin[a*x]^(3/2)) + (16*x^4)/(15*ArcSin[a
*x]^(3/2)) - (16*x*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcSin[a*x]]) + (128*x^3*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcSi
n[a*x]]) + (32*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a^4) - (16*Sqrt[Pi]*FresnelC[(2*Sqrt[A
rcSin[a*x]])/Sqrt[Pi]])/(15*a^4)

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 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^3}{\sin ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{6 \int \frac{x^2}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx}{5 a}-\frac{1}{5} (8 a) \int \frac{x^4}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{16 x^4}{15 \sin ^{-1}(a x)^{3/2}}-\frac{64}{15} \int \frac{x^3}{\sin ^{-1}(a x)^{3/2}} \, dx+\frac{8 \int \frac{x}{\sin ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{16 x^4}{15 \sin ^{-1}(a x)^{3/2}}-\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^4}-\frac{128 \operatorname{Subst}\left (\int \left (\frac{\cos (2 x)}{2 \sqrt{x}}-\frac{\cos (4 x)}{2 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{15 a^4}\\ &=-\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{16 x^4}{15 \sin ^{-1}(a x)^{3/2}}-\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}-\frac{64 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a^4}+\frac{64 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a^4}+\frac{32 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{5 a^4}\\ &=-\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{16 x^4}{15 \sin ^{-1}(a x)^{3/2}}-\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}+\frac{16 \sqrt{\pi } C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{5 a^4}-\frac{128 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{15 a^4}+\frac{128 \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{15 a^4}\\ &=-\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{16 x^4}{15 \sin ^{-1}(a x)^{3/2}}-\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\sin ^{-1}(a x)}}+\frac{32 \sqrt{2 \pi } C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^4}-\frac{16 \sqrt{\pi } C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{15 a^4}\\ \end{align*}

Mathematica [C]  time = 1.18683, size = 272, normalized size = 1.43 \[ \frac{4 \sin ^{-1}(a x) \left (-4 \sqrt{2} \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )+e^{-2 i \sin ^{-1}(a x)} \left (-4 \sqrt{2} e^{2 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )+4 i \sin ^{-1}(a x)-1\right )+i e^{2 i \sin ^{-1}(a x)} \left (-4 \sin ^{-1}(a x)+i\right )\right )-4 \sin ^{-1}(a x) \left (-16 \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-4 i \sin ^{-1}(a x)\right )+e^{-4 i \sin ^{-1}(a x)} \left (-16 e^{4 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},4 i \sin ^{-1}(a x)\right )+8 i \sin ^{-1}(a x)-1\right )+i e^{4 i \sin ^{-1}(a x)} \left (-8 \sin ^{-1}(a x)+i\right )\right )-6 \sin \left (2 \sin ^{-1}(a x)\right )+3 \sin \left (4 \sin ^{-1}(a x)\right )}{60 a^4 \sin ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/ArcSin[a*x]^(7/2),x]

[Out]

(4*ArcSin[a*x]*(I*E^((2*I)*ArcSin[a*x])*(I - 4*ArcSin[a*x]) - 4*Sqrt[2]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-
2*I)*ArcSin[a*x]] + (-1 + (4*I)*ArcSin[a*x] - 4*Sqrt[2]*E^((2*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2,
(2*I)*ArcSin[a*x]])/E^((2*I)*ArcSin[a*x])) - 4*ArcSin[a*x]*(I*E^((4*I)*ArcSin[a*x])*(I - 8*ArcSin[a*x]) - 16*
((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcSin[a*x]] + (-1 + (8*I)*ArcSin[a*x] - 16*E^((4*I)*ArcSin[a*x])*(
I*ArcSin[a*x])^(3/2)*Gamma[1/2, (4*I)*ArcSin[a*x]])/E^((4*I)*ArcSin[a*x])) - 6*Sin[2*ArcSin[a*x]] + 3*Sin[4*Ar
cSin[a*x]])/(60*a^4*ArcSin[a*x]^(5/2))

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Maple [A]  time = 0.055, size = 139, normalized size = 0.7 \begin{align*}{\frac{1}{60\,{a}^{4}} \left ( 128\,\sqrt{2}\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}-64\,\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}+32\,\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{2}-64\,\sin \left ( 4\,\arcsin \left ( ax \right ) \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{2}-8\,\arcsin \left ( ax \right ) \cos \left ( 2\,\arcsin \left ( ax \right ) \right ) +8\,\arcsin \left ( ax \right ) \cos \left ( 4\,\arcsin \left ( ax \right ) \right ) -6\,\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) +3\,\sin \left ( 4\,\arcsin \left ( ax \right ) \right ) \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^3/arcsin(a*x)^(7/2),x)

[Out]

1/60/a^4*(128*2^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)-64*Pi^(1/2)*Fr
esnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*arcsin(a*x)^(5/2)+32*sin(2*arcsin(a*x))*arcsin(a*x)^2-64*sin(4*arcsin(a*x
))*arcsin(a*x)^2-8*arcsin(a*x)*cos(2*arcsin(a*x))+8*arcsin(a*x)*cos(4*arcsin(a*x))-6*sin(2*arcsin(a*x))+3*sin(
4*arcsin(a*x)))/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^3/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^3/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**3/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^{3}}{\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^3/arcsin(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^3/arcsin(a*x)^(7/2), x)