3.67 \(\int \frac{x^4}{\sin ^{-1}(a x)^4} \, dx\)

Optimal. Leaf size=158 \[ \frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{48 a^5}-\frac{27 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Si}\left (5 \sin ^{-1}(a x)\right )}{96 a^5}+\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \sin ^{-1}(a x)}-\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sin ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \sin ^{-1}(a x)}+\frac{5 x^5}{6 \sin ^{-1}(a x)^2} \]

[Out]

-(x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^3) - (2*x^3)/(3*a^2*ArcSin[a*x]^2) + (5*x^5)/(6*ArcSin[a*x]^2) - (2*
x^2*Sqrt[1 - a^2*x^2])/(a^3*ArcSin[a*x]) + (25*x^4*Sqrt[1 - a^2*x^2])/(6*a*ArcSin[a*x]) + SinIntegral[ArcSin[a
*x]]/(48*a^5) - (27*SinIntegral[3*ArcSin[a*x]])/(32*a^5) + (125*SinIntegral[5*ArcSin[a*x]])/(96*a^5)

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Rubi [A]  time = 0.314241, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4633, 4719, 4631, 3299} \[ \frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{48 a^5}-\frac{27 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Si}\left (5 \sin ^{-1}(a x)\right )}{96 a^5}+\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \sin ^{-1}(a x)}-\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sin ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \sin ^{-1}(a x)}+\frac{5 x^5}{6 \sin ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcSin[a*x]^4,x]

[Out]

-(x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^3) - (2*x^3)/(3*a^2*ArcSin[a*x]^2) + (5*x^5)/(6*ArcSin[a*x]^2) - (2*
x^2*Sqrt[1 - a^2*x^2])/(a^3*ArcSin[a*x]) + (25*x^4*Sqrt[1 - a^2*x^2])/(6*a*ArcSin[a*x]) + SinIntegral[ArcSin[a
*x]]/(48*a^5) - (27*SinIntegral[3*ArcSin[a*x]])/(32*a^5) + (125*SinIntegral[5*ArcSin[a*x]])/(96*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 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x^4}{\sin ^{-1}(a x)^4} \, dx &=-\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}+\frac{4 \int \frac{x^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx}{3 a}-\frac{1}{3} (5 a) \int \frac{x^5}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx\\ &=-\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sin ^{-1}(a x)^2}+\frac{5 x^5}{6 \sin ^{-1}(a x)^2}-\frac{25}{6} \int \frac{x^4}{\sin ^{-1}(a x)^2} \, dx+\frac{2 \int \frac{x^2}{\sin ^{-1}(a x)^2} \, dx}{a^2}\\ &=-\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sin ^{-1}(a x)^2}+\frac{5 x^5}{6 \sin ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \sin ^{-1}(a x)}+\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \sin ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{4 x}+\frac{3 \sin (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac{25 \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{8 x}+\frac{9 \sin (3 x)}{16 x}-\frac{5 \sin (5 x)}{16 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{6 a^5}\\ &=-\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sin ^{-1}(a x)^2}+\frac{5 x^5}{6 \sin ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \sin ^{-1}(a x)}+\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{2 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{48 a^5}+\frac{125 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{96 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{2 a^5}-\frac{75 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{32 a^5}\\ &=-\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{2 x^3}{3 a^2 \sin ^{-1}(a x)^2}+\frac{5 x^5}{6 \sin ^{-1}(a x)^2}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \sin ^{-1}(a x)}+\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \sin ^{-1}(a x)}+\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{48 a^5}-\frac{27 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Si}\left (5 \sin ^{-1}(a x)\right )}{96 a^5}\\ \end{align*}

Mathematica [A]  time = 0.338395, size = 159, normalized size = 1.01 \[ \frac{-32 a^4 x^4 \sqrt{1-a^2 x^2}+80 a^5 x^5 \sin ^{-1}(a x)+400 a^4 x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2-64 a^3 x^3 \sin ^{-1}(a x)-192 a^2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2+2 \sin ^{-1}(a x)^3 \text{Si}\left (\sin ^{-1}(a x)\right )-81 \sin ^{-1}(a x)^3 \text{Si}\left (3 \sin ^{-1}(a x)\right )+125 \sin ^{-1}(a x)^3 \text{Si}\left (5 \sin ^{-1}(a x)\right )}{96 a^5 \sin ^{-1}(a x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/ArcSin[a*x]^4,x]

[Out]

(-32*a^4*x^4*Sqrt[1 - a^2*x^2] - 64*a^3*x^3*ArcSin[a*x] + 80*a^5*x^5*ArcSin[a*x] - 192*a^2*x^2*Sqrt[1 - a^2*x^
2]*ArcSin[a*x]^2 + 400*a^4*x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2 + 2*ArcSin[a*x]^3*SinIntegral[ArcSin[a*x]] - 81
*ArcSin[a*x]^3*SinIntegral[3*ArcSin[a*x]] + 125*ArcSin[a*x]^3*SinIntegral[5*ArcSin[a*x]])/(96*a^5*ArcSin[a*x]^
3)

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Maple [A]  time = 0.04, size = 171, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{5}} \left ( -{\frac{1}{24\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{ax}{48\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}+{\frac{1}{48\,\arcsin \left ( ax \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{\it Si} \left ( \arcsin \left ( ax \right ) \right ) }{48}}+{\frac{\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) }{16\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}}}-{\frac{3\,\sin \left ( 3\,\arcsin \left ( ax \right ) \right ) }{32\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}-{\frac{9\,\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) }{32\,\arcsin \left ( ax \right ) }}-{\frac{27\,{\it Si} \left ( 3\,\arcsin \left ( ax \right ) \right ) }{32}}-{\frac{\cos \left ( 5\,\arcsin \left ( ax \right ) \right ) }{48\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}}}+{\frac{5\,\sin \left ( 5\,\arcsin \left ( ax \right ) \right ) }{96\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}+{\frac{25\,\cos \left ( 5\,\arcsin \left ( ax \right ) \right ) }{96\,\arcsin \left ( ax \right ) }}+{\frac{125\,{\it Si} \left ( 5\,\arcsin \left ( ax \right ) \right ) }{96}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(-1/24/arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)+1/48*a*x/arcsin(a*x)^2+1/48/arcsin(a*x)*(-a^2*x^2+1)^(1/2)+1/48*
Si(arcsin(a*x))+1/16/arcsin(a*x)^3*cos(3*arcsin(a*x))-3/32/arcsin(a*x)^2*sin(3*arcsin(a*x))-9/32/arcsin(a*x)*c
os(3*arcsin(a*x))-27/32*Si(3*arcsin(a*x))-1/48/arcsin(a*x)^3*cos(5*arcsin(a*x))+5/96/arcsin(a*x)^2*sin(5*arcsi
n(a*x))+25/96/arcsin(a*x)*cos(5*arcsin(a*x))+125/96*Si(5*arcsin(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{3} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3} \int \frac{{\left (125 \, a^{4} x^{5} - 136 \, a^{2} x^{3} + 24 \, x\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{{\left (a^{5} x^{2} - a^{3}\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )}\,{d x} +{\left (2 \, a^{2} x^{4} -{\left (25 \, a^{2} x^{4} - 12 \, x^{2}\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2}\right )} \sqrt{a x + 1} \sqrt{-a x + 1} -{\left (5 \, a^{3} x^{5} - 4 \, a x^{3}\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )}{6 \, a^{3} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsin(a*x)^4,x, algorithm="maxima")

[Out]

-1/6*(6*a^3*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3*integrate(1/6*(125*a^4*x^5 - 136*a^2*x^3 + 24*x)*sqrt
(a*x + 1)*sqrt(-a*x + 1)/((a^5*x^2 - a^3)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))), x) + (2*a^2*x^4 - (25*a
^2*x^4 - 12*x^2)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)*sqrt(a*x + 1)*sqrt(-a*x + 1) - (5*a^3*x^5 - 4*a
*x^3)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)))/(a^3*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4}}{\arcsin \left (a x\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsin(a*x)^4,x, algorithm="fricas")

[Out]

integral(x^4/arcsin(a*x)^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{asin}^{4}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**4/asin(a*x)**4, x)

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Giac [A]  time = 1.27056, size = 338, normalized size = 2.14 \begin{align*} \frac{5 \,{\left (a^{2} x^{2} - 1\right )}^{2} x}{6 \, a^{4} \arcsin \left (a x\right )^{2}} + \frac{25 \,{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{5} \arcsin \left (a x\right )} + \frac{{\left (a^{2} x^{2} - 1\right )} x}{a^{4} \arcsin \left (a x\right )^{2}} + \frac{125 \, \operatorname{Si}\left (5 \, \arcsin \left (a x\right )\right )}{96 \, a^{5}} - \frac{27 \, \operatorname{Si}\left (3 \, \arcsin \left (a x\right )\right )}{32 \, a^{5}} + \frac{\operatorname{Si}\left (\arcsin \left (a x\right )\right )}{48 \, a^{5}} - \frac{19 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{3 \, a^{5} \arcsin \left (a x\right )} + \frac{x}{6 \, a^{4} \arcsin \left (a x\right )^{2}} - \frac{{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1}}{3 \, a^{5} \arcsin \left (a x\right )^{3}} + \frac{13 \, \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{5} \arcsin \left (a x\right )} + \frac{2 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{3 \, a^{5} \arcsin \left (a x\right )^{3}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{3 \, a^{5} \arcsin \left (a x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/6*(a^2*x^2 - 1)^2*x/(a^4*arcsin(a*x)^2) + 25/6*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)) + (a^2*x
^2 - 1)*x/(a^4*arcsin(a*x)^2) + 125/96*sin_integral(5*arcsin(a*x))/a^5 - 27/32*sin_integral(3*arcsin(a*x))/a^5
+ 1/48*sin_integral(arcsin(a*x))/a^5 - 19/3*(-a^2*x^2 + 1)^(3/2)/(a^5*arcsin(a*x)) + 1/6*x/(a^4*arcsin(a*x)^2
) - 1/3*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)^3) + 13/6*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)) + 2
/3*(-a^2*x^2 + 1)^(3/2)/(a^5*arcsin(a*x)^3) - 1/3*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)^3)