3.107 \(\int \frac{x^4}{\sin ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=171 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}-\frac{5 \sqrt{\frac{5 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{6 a^5}-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}} \]

[Out]

(-2*x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2)) - (16*x^3)/(3*a^2*Sqrt[ArcSin[a*x]]) + (20*x^5)/(3*Sqrt[Arc
Sin[a*x]]) - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^5) + (3*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/P
i]*Sqrt[ArcSin[a*x]]])/(2*a^5) - (5*Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(6*a^5)

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Rubi [A]  time = 0.434386, antiderivative size = 235, normalized size of antiderivative = 1.37, number of steps used = 19, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4633, 4719, 4635, 4406, 3304, 3352} \[ \frac{4 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}-\frac{25 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}-\frac{4 \sqrt{\frac{2 \pi }{3}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}+\frac{25 \sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}-\frac{5 \sqrt{\frac{5 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{6 a^5}-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcSin[a*x]^(5/2),x]

[Out]

(-2*x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2)) - (16*x^3)/(3*a^2*Sqrt[ArcSin[a*x]]) + (20*x^5)/(3*Sqrt[Arc
Sin[a*x]]) - (25*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^5) + (4*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi
]*Sqrt[ArcSin[a*x]]])/a^5 + (25*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(2*a^5) - (4*Sqrt[(2*Pi)/3]
*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/a^5 - (5*Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(6*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 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

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^4}{\sin ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{8 \int \frac{x^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx}{3 a}-\frac{1}{3} (10 a) \int \frac{x^5}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{100}{3} \int \frac{x^4}{\sqrt{\sin ^{-1}(a x)}} \, dx+\frac{16 \int \frac{x^2}{\sqrt{\sin ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac{100 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^4(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 \sqrt{x}}-\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac{100 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{8 \sqrt{x}}-\frac{3 \cos (3 x)}{16 \sqrt{x}}+\frac{\cos (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}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{25 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{12 a^5}+\frac{4 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac{4 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac{25 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{6 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{25 \operatorname{Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{6 a^5}+\frac{8 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a^5}-\frac{8 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a^5}-\frac{25 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{25 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}+\frac{4 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}+\frac{25 \sqrt{\frac{\pi }{6}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}-\frac{4 \sqrt{\frac{2 \pi }{3}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}-\frac{5 \sqrt{\frac{5 \pi }{2}} C\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{6 a^5}\\ \end{align*}

Mathematica [C]  time = 0.328869, size = 418, normalized size = 2.44 \[ \frac{\frac{i e^{i \sin ^{-1}(a x)} \left (-2 \sin ^{-1}(a x)+i\right )-2 \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )}{24 \sin ^{-1}(a x)^{3/2}}-\frac{e^{-i \sin ^{-1}(a x)} \left (2 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )-2 i \sin ^{-1}(a x)+1\right )}{24 \sin ^{-1}(a x)^{3/2}}-\frac{i e^{3 i \sin ^{-1}(a x)} \left (-6 \sin ^{-1}(a x)+i\right )-6 \sqrt{3} \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-3 i \sin ^{-1}(a x)\right )}{16 \sin ^{-1}(a x)^{3/2}}+\frac{e^{-3 i \sin ^{-1}(a x)} \left (6 \sqrt{3} e^{3 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},3 i \sin ^{-1}(a x)\right )-6 i \sin ^{-1}(a x)+1\right )}{16 \sin ^{-1}(a x)^{3/2}}+\frac{i e^{5 i \sin ^{-1}(a x)} \left (-10 \sin ^{-1}(a x)+i\right )-10 \sqrt{5} \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-5 i \sin ^{-1}(a x)\right )}{48 \sin ^{-1}(a x)^{3/2}}-\frac{e^{-5 i \sin ^{-1}(a x)} \left (10 \sqrt{5} e^{5 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},5 i \sin ^{-1}(a x)\right )-10 i \sin ^{-1}(a x)+1\right )}{48 \sin ^{-1}(a x)^{3/2}}}{a^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/ArcSin[a*x]^(5/2),x]

[Out]

((I*E^(I*ArcSin[a*x])*(I - 2*ArcSin[a*x]) - 2*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-I)*ArcSin[a*x]])/(24*ArcSi
n[a*x]^(3/2)) - (1 - (2*I)*ArcSin[a*x] + 2*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, I*ArcSin[a*x]])/
(24*E^(I*ArcSin[a*x])*ArcSin[a*x]^(3/2)) - (I*E^((3*I)*ArcSin[a*x])*(I - 6*ArcSin[a*x]) - 6*Sqrt[3]*((-I)*ArcS
in[a*x])^(3/2)*Gamma[1/2, (-3*I)*ArcSin[a*x]])/(16*ArcSin[a*x]^(3/2)) + (1 - (6*I)*ArcSin[a*x] + 6*Sqrt[3]*E^(
(3*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcSin[a*x]])/(16*E^((3*I)*ArcSin[a*x])*ArcSin[a*x]^
(3/2)) + (I*E^((5*I)*ArcSin[a*x])*(I - 10*ArcSin[a*x]) - 10*Sqrt[5]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-5*I)
*ArcSin[a*x]])/(48*ArcSin[a*x]^(3/2)) - (1 - (10*I)*ArcSin[a*x] + 10*Sqrt[5]*E^((5*I)*ArcSin[a*x])*(I*ArcSin[a
*x])^(3/2)*Gamma[1/2, (5*I)*ArcSin[a*x]])/(48*E^((5*I)*ArcSin[a*x])*ArcSin[a*x]^(3/2)))/a^5

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Maple [A]  time = 0.07, size = 175, normalized size = 1. \begin{align*}{\frac{1}{24\,{a}^{5}} \left ( 18\,\sqrt{2}\sqrt{\pi }\sqrt{3}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}-10\,\sqrt{2}\sqrt{\pi }\sqrt{5}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{5}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}-4\,\sqrt{2}\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}+4\,ax\arcsin \left ( ax \right ) +10\,\arcsin \left ( ax \right ) \sin \left ( 5\,\arcsin \left ( ax \right ) \right ) -18\,\arcsin \left ( ax \right ) \sin \left ( 3\,\arcsin \left ( ax \right ) \right ) -2\,\sqrt{-{a}^{2}{x}^{2}+1}-\cos \left ( 5\,\arcsin \left ( ax \right ) \right ) +3\,\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arcsin(a*x)^(5/2),x)

[Out]

1/24/a^5*(18*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)-1
0*2^(1/2)*Pi^(1/2)*5^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)-4*2^(1/2)*Pi
^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)+4*a*x*arcsin(a*x)+10*arcsin(a*x)*sin(5*a
rcsin(a*x))-18*arcsin(a*x)*sin(3*arcsin(a*x))-2*(-a^2*x^2+1)^(1/2)-cos(5*arcsin(a*x))+3*cos(3*arcsin(a*x)))/ar
csin(a*x)^(3/2)

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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)^(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(x^4/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{x^{4}}{\operatorname{asin}^{\frac{5}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/asin(a*x)**(5/2),x)

[Out]

Integral(x**4/asin(a*x)**(5/2), x)

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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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsin(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(x^4/arcsin(a*x)^(5/2), x)