3.32 \(\int x^5 \sin ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=282 \[ \frac{65 x^4}{3456 a^2}+\frac{245 x^2}{1152 a^4}+\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{9 a}-\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{54 a}-\frac{5 x^4 \sin ^{-1}(a x)^2}{48 a^2}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{36 a^3}-\frac{65 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{864 a^3}-\frac{5 x^2 \sin ^{-1}(a x)^2}{16 a^4}+\frac{5 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{24 a^5}-\frac{245 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{576 a^5}-\frac{5 \sin ^{-1}(a x)^4}{96 a^6}+\frac{245 \sin ^{-1}(a x)^2}{1152 a^6}+\frac{1}{6} x^6 \sin ^{-1}(a x)^4-\frac{1}{18} x^6 \sin ^{-1}(a x)^2+\frac{x^6}{324} \]

[Out]

(245*x^2)/(1152*a^4) + (65*x^4)/(3456*a^2) + x^6/324 - (245*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(576*a^5) - (65*x
^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(864*a^3) - (x^5*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(54*a) + (245*ArcSin[a*x]^2)
/(1152*a^6) - (5*x^2*ArcSin[a*x]^2)/(16*a^4) - (5*x^4*ArcSin[a*x]^2)/(48*a^2) - (x^6*ArcSin[a*x]^2)/18 + (5*x*
Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(24*a^5) + (5*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(36*a^3) + (x^5*Sqrt[1 - a
^2*x^2]*ArcSin[a*x]^3)/(9*a) - (5*ArcSin[a*x]^4)/(96*a^6) + (x^6*ArcSin[a*x]^4)/6

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Rubi [A]  time = 0.869246, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4627, 4707, 4641, 30} \[ \frac{65 x^4}{3456 a^2}+\frac{245 x^2}{1152 a^4}+\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{9 a}-\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{54 a}-\frac{5 x^4 \sin ^{-1}(a x)^2}{48 a^2}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{36 a^3}-\frac{65 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{864 a^3}-\frac{5 x^2 \sin ^{-1}(a x)^2}{16 a^4}+\frac{5 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{24 a^5}-\frac{245 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{576 a^5}-\frac{5 \sin ^{-1}(a x)^4}{96 a^6}+\frac{245 \sin ^{-1}(a x)^2}{1152 a^6}+\frac{1}{6} x^6 \sin ^{-1}(a x)^4-\frac{1}{18} x^6 \sin ^{-1}(a x)^2+\frac{x^6}{324} \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcSin[a*x]^4,x]

[Out]

(245*x^2)/(1152*a^4) + (65*x^4)/(3456*a^2) + x^6/324 - (245*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(576*a^5) - (65*x
^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(864*a^3) - (x^5*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(54*a) + (245*ArcSin[a*x]^2)
/(1152*a^6) - (5*x^2*ArcSin[a*x]^2)/(16*a^4) - (5*x^4*ArcSin[a*x]^2)/(48*a^2) - (x^6*ArcSin[a*x]^2)/18 + (5*x*
Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(24*a^5) + (5*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(36*a^3) + (x^5*Sqrt[1 - a
^2*x^2]*ArcSin[a*x]^3)/(9*a) - (5*ArcSin[a*x]^4)/(96*a^6) + (x^6*ArcSin[a*x]^4)/6

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^5 \sin ^{-1}(a x)^4 \, dx &=\frac{1}{6} x^6 \sin ^{-1}(a x)^4-\frac{1}{3} (2 a) \int \frac{x^6 \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \sin ^{-1}(a x)^4-\frac{1}{3} \int x^5 \sin ^{-1}(a x)^2 \, dx-\frac{5 \int \frac{x^4 \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac{1}{18} x^6 \sin ^{-1}(a x)^2+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{36 a^3}+\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \sin ^{-1}(a x)^4-\frac{5 \int \frac{x^2 \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{12 a^3}-\frac{5 \int x^3 \sin ^{-1}(a x)^2 \, dx}{12 a^2}+\frac{1}{9} a \int \frac{x^6 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{54 a}-\frac{5 x^4 \sin ^{-1}(a x)^2}{48 a^2}-\frac{1}{18} x^6 \sin ^{-1}(a x)^2+\frac{5 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{24 a^5}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{36 a^3}+\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \sin ^{-1}(a x)^4+\frac{\int x^5 \, dx}{54}-\frac{5 \int \frac{\sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{24 a^5}-\frac{5 \int x \sin ^{-1}(a x)^2 \, dx}{8 a^4}+\frac{5 \int \frac{x^4 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{54 a}+\frac{5 \int \frac{x^4 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{24 a}\\ &=\frac{x^6}{324}-\frac{65 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{864 a^3}-\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{54 a}-\frac{5 x^2 \sin ^{-1}(a x)^2}{16 a^4}-\frac{5 x^4 \sin ^{-1}(a x)^2}{48 a^2}-\frac{1}{18} x^6 \sin ^{-1}(a x)^2+\frac{5 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{24 a^5}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{36 a^3}+\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{9 a}-\frac{5 \sin ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \sin ^{-1}(a x)^4+\frac{5 \int \frac{x^2 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{72 a^3}+\frac{5 \int \frac{x^2 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{32 a^3}+\frac{5 \int \frac{x^2 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{8 a^3}+\frac{5 \int x^3 \, dx}{216 a^2}+\frac{5 \int x^3 \, dx}{96 a^2}\\ &=\frac{65 x^4}{3456 a^2}+\frac{x^6}{324}-\frac{245 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{576 a^5}-\frac{65 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{864 a^3}-\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{54 a}-\frac{5 x^2 \sin ^{-1}(a x)^2}{16 a^4}-\frac{5 x^4 \sin ^{-1}(a x)^2}{48 a^2}-\frac{1}{18} x^6 \sin ^{-1}(a x)^2+\frac{5 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{24 a^5}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{36 a^3}+\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{9 a}-\frac{5 \sin ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \sin ^{-1}(a x)^4+\frac{5 \int \frac{\sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{144 a^5}+\frac{5 \int \frac{\sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{64 a^5}+\frac{5 \int \frac{\sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{16 a^5}+\frac{5 \int x \, dx}{144 a^4}+\frac{5 \int x \, dx}{64 a^4}+\frac{5 \int x \, dx}{16 a^4}\\ &=\frac{245 x^2}{1152 a^4}+\frac{65 x^4}{3456 a^2}+\frac{x^6}{324}-\frac{245 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{576 a^5}-\frac{65 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{864 a^3}-\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{54 a}+\frac{245 \sin ^{-1}(a x)^2}{1152 a^6}-\frac{5 x^2 \sin ^{-1}(a x)^2}{16 a^4}-\frac{5 x^4 \sin ^{-1}(a x)^2}{48 a^2}-\frac{1}{18} x^6 \sin ^{-1}(a x)^2+\frac{5 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{24 a^5}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{36 a^3}+\frac{x^5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{9 a}-\frac{5 \sin ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \sin ^{-1}(a x)^4\\ \end{align*}

Mathematica [A]  time = 0.0977085, size = 167, normalized size = 0.59 \[ \frac{a^2 x^2 \left (32 a^4 x^4+195 a^2 x^2+2205\right )+108 \left (16 a^6 x^6-5\right ) \sin ^{-1}(a x)^4+144 a x \sqrt{1-a^2 x^2} \left (8 a^4 x^4+10 a^2 x^2+15\right ) \sin ^{-1}(a x)^3-9 \left (64 a^6 x^6+120 a^4 x^4+360 a^2 x^2-245\right ) \sin ^{-1}(a x)^2-6 a x \sqrt{1-a^2 x^2} \left (32 a^4 x^4+130 a^2 x^2+735\right ) \sin ^{-1}(a x)}{10368 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*ArcSin[a*x]^4,x]

[Out]

(a^2*x^2*(2205 + 195*a^2*x^2 + 32*a^4*x^4) - 6*a*x*Sqrt[1 - a^2*x^2]*(735 + 130*a^2*x^2 + 32*a^4*x^4)*ArcSin[a
*x] - 9*(-245 + 360*a^2*x^2 + 120*a^4*x^4 + 64*a^6*x^6)*ArcSin[a*x]^2 + 144*a*x*Sqrt[1 - a^2*x^2]*(15 + 10*a^2
*x^2 + 8*a^4*x^4)*ArcSin[a*x]^3 + 108*(-5 + 16*a^6*x^6)*ArcSin[a*x]^4)/(10368*a^6)

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Maple [A]  time = 0.1, size = 320, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{6}} \left ({\frac{{a}^{6}{x}^{6} \left ( \arcsin \left ( ax \right ) \right ) ^{4}}{6}}-{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{72} \left ( -8\,\sqrt{-{a}^{2}{x}^{2}+1}{a}^{5}{x}^{5}-10\,{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}-15\,ax\sqrt{-{a}^{2}{x}^{2}+1}+15\,\arcsin \left ( ax \right ) \right ) }-{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{2}{a}^{6}{x}^{6}}{18}}+{\frac{\arcsin \left ( ax \right ) }{432} \left ( -8\,\sqrt{-{a}^{2}{x}^{2}+1}{a}^{5}{x}^{5}-10\,{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}-15\,ax\sqrt{-{a}^{2}{x}^{2}+1}+15\,\arcsin \left ( ax \right ) \right ) }+{\frac{115\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{1152}}+{\frac{{a}^{6}{x}^{6}}{324}}+{\frac{65\,{a}^{4}{x}^{4}}{3456}}+{\frac{245\,{a}^{2}{x}^{2}}{1152}}-{\frac{5\,{a}^{4}{x}^{4} \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{48}}+{\frac{5\,\arcsin \left ( ax \right ) }{192} \left ( -2\,{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}-3\,ax\sqrt{-{a}^{2}{x}^{2}+1}+3\,\arcsin \left ( ax \right ) \right ) }-{\frac{ \left ( 5\,{a}^{2}{x}^{2}-5 \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{16}}-{\frac{5\,\arcsin \left ( ax \right ) }{16} \left ( ax\sqrt{-{a}^{2}{x}^{2}+1}+\arcsin \left ( ax \right ) \right ) }+{\frac{5\, \left ( \arcsin \left ( ax \right ) \right ) ^{4}}{32}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^6*(1/6*a^6*x^6*arcsin(a*x)^4-1/72*arcsin(a*x)^3*(-8*(-a^2*x^2+1)^(1/2)*a^5*x^5-10*a^3*x^3*(-a^2*x^2+1)^(1/
2)-15*a*x*(-a^2*x^2+1)^(1/2)+15*arcsin(a*x))-1/18*arcsin(a*x)^2*a^6*x^6+1/432*arcsin(a*x)*(-8*(-a^2*x^2+1)^(1/
2)*a^5*x^5-10*a^3*x^3*(-a^2*x^2+1)^(1/2)-15*a*x*(-a^2*x^2+1)^(1/2)+15*arcsin(a*x))+115/1152*arcsin(a*x)^2+1/32
4*a^6*x^6+65/3456*a^4*x^4+245/1152*a^2*x^2-5/48*a^4*x^4*arcsin(a*x)^2+5/192*arcsin(a*x)*(-2*a^3*x^3*(-a^2*x^2+
1)^(1/2)-3*a*x*(-a^2*x^2+1)^(1/2)+3*arcsin(a*x))-5/16*(a^2*x^2-1)*arcsin(a*x)^2-5/16*arcsin(a*x)*(a*x*(-a^2*x^
2+1)^(1/2)+arcsin(a*x))+5/32*arcsin(a*x)^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arcsin(a*x)^4,x, algorithm="maxima")

[Out]

1/6*x^6*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^4 + 2*a*integrate(1/3*sqrt(a*x + 1)*sqrt(-a*x + 1)*x^6*arct
an2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3/(a^2*x^2 - 1), x)

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Fricas [A]  time = 2.21951, size = 383, normalized size = 1.36 \begin{align*} \frac{32 \, a^{6} x^{6} + 195 \, a^{4} x^{4} + 108 \,{\left (16 \, a^{6} x^{6} - 5\right )} \arcsin \left (a x\right )^{4} + 2205 \, a^{2} x^{2} - 9 \,{\left (64 \, a^{6} x^{6} + 120 \, a^{4} x^{4} + 360 \, a^{2} x^{2} - 245\right )} \arcsin \left (a x\right )^{2} + 6 \, \sqrt{-a^{2} x^{2} + 1}{\left (24 \,{\left (8 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 15 \, a x\right )} \arcsin \left (a x\right )^{3} -{\left (32 \, a^{5} x^{5} + 130 \, a^{3} x^{3} + 735 \, a x\right )} \arcsin \left (a x\right )\right )}}{10368 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arcsin(a*x)^4,x, algorithm="fricas")

[Out]

1/10368*(32*a^6*x^6 + 195*a^4*x^4 + 108*(16*a^6*x^6 - 5)*arcsin(a*x)^4 + 2205*a^2*x^2 - 9*(64*a^6*x^6 + 120*a^
4*x^4 + 360*a^2*x^2 - 245)*arcsin(a*x)^2 + 6*sqrt(-a^2*x^2 + 1)*(24*(8*a^5*x^5 + 10*a^3*x^3 + 15*a*x)*arcsin(a
*x)^3 - (32*a^5*x^5 + 130*a^3*x^3 + 735*a*x)*arcsin(a*x)))/a^6

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Sympy [A]  time = 33.0018, size = 269, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{asin}^{4}{\left (a x \right )}}{6} - \frac{x^{6} \operatorname{asin}^{2}{\left (a x \right )}}{18} + \frac{x^{6}}{324} + \frac{x^{5} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{9 a} - \frac{x^{5} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{54 a} - \frac{5 x^{4} \operatorname{asin}^{2}{\left (a x \right )}}{48 a^{2}} + \frac{65 x^{4}}{3456 a^{2}} + \frac{5 x^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{36 a^{3}} - \frac{65 x^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{864 a^{3}} - \frac{5 x^{2} \operatorname{asin}^{2}{\left (a x \right )}}{16 a^{4}} + \frac{245 x^{2}}{1152 a^{4}} + \frac{5 x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{24 a^{5}} - \frac{245 x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{576 a^{5}} - \frac{5 \operatorname{asin}^{4}{\left (a x \right )}}{96 a^{6}} + \frac{245 \operatorname{asin}^{2}{\left (a x \right )}}{1152 a^{6}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**6*asin(a*x)**4/6 - x**6*asin(a*x)**2/18 + x**6/324 + x**5*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(9*a
) - x**5*sqrt(-a**2*x**2 + 1)*asin(a*x)/(54*a) - 5*x**4*asin(a*x)**2/(48*a**2) + 65*x**4/(3456*a**2) + 5*x**3*
sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(36*a**3) - 65*x**3*sqrt(-a**2*x**2 + 1)*asin(a*x)/(864*a**3) - 5*x**2*asin(
a*x)**2/(16*a**4) + 245*x**2/(1152*a**4) + 5*x*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(24*a**5) - 245*x*sqrt(-a**2*
x**2 + 1)*asin(a*x)/(576*a**5) - 5*asin(a*x)**4/(96*a**6) + 245*asin(a*x)**2/(1152*a**6), Ne(a, 0)), (0, True)
)

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Giac [A]  time = 1.37513, size = 489, normalized size = 1.73 \begin{align*} \frac{{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{9 \, a^{5}} + \frac{{\left (a^{2} x^{2} - 1\right )}^{3} \arcsin \left (a x\right )^{4}}{6 \, a^{6}} - \frac{13 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x \arcsin \left (a x\right )^{3}}{36 \, a^{5}} + \frac{{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{4}}{2 \, a^{6}} - \frac{{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{54 \, a^{5}} + \frac{11 \, \sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{24 \, a^{5}} - \frac{{\left (a^{2} x^{2} - 1\right )}^{3} \arcsin \left (a x\right )^{2}}{18 \, a^{6}} + \frac{{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{4}}{2 \, a^{6}} + \frac{97 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x \arcsin \left (a x\right )}{864 \, a^{5}} - \frac{13 \,{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{2}}{48 \, a^{6}} + \frac{11 \, \arcsin \left (a x\right )^{4}}{96 \, a^{6}} - \frac{299 \, \sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{576 \, a^{5}} + \frac{{\left (a^{2} x^{2} - 1\right )}^{3}}{324 \, a^{6}} - \frac{11 \,{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{16 \, a^{6}} + \frac{97 \,{\left (a^{2} x^{2} - 1\right )}^{2}}{3456 \, a^{6}} - \frac{299 \, \arcsin \left (a x\right )^{2}}{1152 \, a^{6}} + \frac{299 \,{\left (a^{2} x^{2} - 1\right )}}{1152 \, a^{6}} + \frac{9971}{82944 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a^5 + 1/6*(a^2*x^2 - 1)^3*arcsin(a*x)^4/a^6 - 13/36*(-a
^2*x^2 + 1)^(3/2)*x*arcsin(a*x)^3/a^5 + 1/2*(a^2*x^2 - 1)^2*arcsin(a*x)^4/a^6 - 1/54*(a^2*x^2 - 1)^2*sqrt(-a^2
*x^2 + 1)*x*arcsin(a*x)/a^5 + 11/24*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a^5 - 1/18*(a^2*x^2 - 1)^3*arcsin(a*x)^
2/a^6 + 1/2*(a^2*x^2 - 1)*arcsin(a*x)^4/a^6 + 97/864*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)/a^5 - 13/48*(a^2*x^2 -
1)^2*arcsin(a*x)^2/a^6 + 11/96*arcsin(a*x)^4/a^6 - 299/576*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)/a^5 + 1/324*(a^2*
x^2 - 1)^3/a^6 - 11/16*(a^2*x^2 - 1)*arcsin(a*x)^2/a^6 + 97/3456*(a^2*x^2 - 1)^2/a^6 - 299/1152*arcsin(a*x)^2/
a^6 + 299/1152*(a^2*x^2 - 1)/a^6 + 9971/82944/a^6