\(\int x \arcsin (a x)^4 \, dx\) [33]

Optimal result

Integrand size = 8, antiderivative size = 111 \[ \int x \arcsin (a x)^4 \, dx=\frac {3 x^2}{4}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a}+\frac {3 \arcsin (a x)^2}{4 a^2}-\frac {3}{2} x^2 \arcsin (a x)^2+\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{a}-\frac {\arcsin (a x)^4}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^4 \] Output:

3/4*x^2-3/2*x*(-a^2*x^2+1)^(1/2)*arcsin(a*x)/a+3/4*arcsin(a*x)^2/a^2-3/2*x 
^2*arcsin(a*x)^2+x*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^3/a-1/4*arcsin(a*x)^4/a^ 
2+1/2*x^2*arcsin(a*x)^4
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.86 \[ \int x \arcsin (a x)^4 \, dx=\frac {3 a^2 x^2-6 a x \sqrt {1-a^2 x^2} \arcsin (a x)+\left (3-6 a^2 x^2\right ) \arcsin (a x)^2+4 a x \sqrt {1-a^2 x^2} \arcsin (a x)^3+\left (-1+2 a^2 x^2\right ) \arcsin (a x)^4}{4 a^2} \] Input:

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

Output:

(3*a^2*x^2 - 6*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + (3 - 6*a^2*x^2)*ArcSin[ 
a*x]^2 + 4*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3 + (-1 + 2*a^2*x^2)*ArcSin[a 
*x]^4)/(4*a^2)
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5138, 5210, 5138, 5152, 5210, 15, 5152}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \arcsin (a x)^4 \, dx\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^4-2 a \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^4-2 a \left (\frac {\int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {3 \int x \arcsin (a x)^2dx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^4-2 a \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 a}+\frac {\int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^4-2 a \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^4-2 a \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^4-2 a \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^4-2 a \left (\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}+\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 a}\right )\)

Input:

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

Output:

(x^2*ArcSin[a*x]^4)/2 - 2*a*(-1/2*(x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2 
+ ArcSin[a*x]^4/(8*a^3) + (3*((x^2*ArcSin[a*x]^2)/2 - a*(x^2/(4*a) - (x*Sq 
rt[1 - a^2*x^2]*ArcSin[a*x])/(2*a^2) + ArcSin[a*x]^2/(4*a^3))))/(2*a))
 

Defintions of rubi rules used

rule 15

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

rule 5138

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[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 5152

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

rule 5210

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

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(\frac {-\frac {\cos \left (2 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )^{4}}{4}+\frac {\sin \left (2 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )^{3}}{2}+\frac {3 \arcsin \left (a x \right )^{2} \cos \left (2 \arcsin \left (a x \right )\right )}{4}-\frac {3 \cos \left (2 \arcsin \left (a x \right )\right )}{8}-\frac {3 \arcsin \left (a x \right ) \sin \left (2 \arcsin \left (a x \right )\right )}{4}}{a^{2}}\)	\(73\)
default	\(\frac {-\frac {\cos \left (2 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )^{4}}{4}+\frac {\sin \left (2 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )^{3}}{2}+\frac {3 \arcsin \left (a x \right )^{2} \cos \left (2 \arcsin \left (a x \right )\right )}{4}-\frac {3 \cos \left (2 \arcsin \left (a x \right )\right )}{8}-\frac {3 \arcsin \left (a x \right ) \sin \left (2 \arcsin \left (a x \right )\right )}{4}}{a^{2}}\)	\(73\)
orering	\(\frac {\left (31 a^{4} x^{4}-60 a^{2} x^{2}+40\right ) \arcsin \left (a x \right )^{4}}{32 a^{4} x^{2}}-\frac {\left (15 a^{4} x^{4}-52 a^{2} x^{2}+40\right ) \left (\arcsin \left (a x \right )^{4}+\frac {4 x \arcsin \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}\right )}{32 x^{2} a^{4}}+\frac {\left (5 a^{4} x^{4}-22 a^{2} x^{2}+20\right ) \left (\frac {8 \arcsin \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {12 x \arcsin \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}+\frac {4 x^{2} \arcsin \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{32 x \,a^{4}}+\frac {3 \left (a x -1\right ) \left (a x +1\right ) \left (\frac {36 \arcsin \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}+\frac {16 \arcsin \left (a x \right )^{3} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {24 x \arcsin \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {36 x^{2} \arcsin \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {12 x^{3} \arcsin \left (a x \right )^{3} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{16 a^{4}}+\frac {x \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (\frac {96 \arcsin \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {192 \arcsin \left (a x \right )^{2} a^{4} x}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {84 \arcsin \left (a x \right )^{3} a^{5} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {16 \arcsin \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {24 x \,a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {144 x^{2} \arcsin \left (a x \right ) a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {180 x^{3} \arcsin \left (a x \right )^{2} a^{6}}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {60 x^{4} \arcsin \left (a x \right )^{3} a^{7}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{32 a^{4}}\)	\(533\)

Input:

int(x*arcsin(a*x)^4,x,method=_RETURNVERBOSE)
 

Output:

1/a^2*(-1/4*cos(2*arcsin(a*x))*arcsin(a*x)^4+1/2*sin(2*arcsin(a*x))*arcsin 
(a*x)^3+3/4*arcsin(a*x)^2*cos(2*arcsin(a*x))-3/8*cos(2*arcsin(a*x))-3/4*ar 
csin(a*x)*sin(2*arcsin(a*x)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.74 \[ \int x \arcsin (a x)^4 \, dx=\frac {{\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{4} + 3 \, a^{2} x^{2} - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2} + 2 \, {\left (2 \, a x \arcsin \left (a x\right )^{3} - 3 \, a x \arcsin \left (a x\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{4 \, a^{2}} \] Input:

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

Output:

1/4*((2*a^2*x^2 - 1)*arcsin(a*x)^4 + 3*a^2*x^2 - 3*(2*a^2*x^2 - 1)*arcsin( 
a*x)^2 + 2*(2*a*x*arcsin(a*x)^3 - 3*a*x*arcsin(a*x))*sqrt(-a^2*x^2 + 1))/a 
^2
 

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94 \[ \int x \arcsin (a x)^4 \, dx=\begin {cases} \frac {x^{2} \operatorname {asin}^{4}{\left (a x \right )}}{2} - \frac {3 x^{2} \operatorname {asin}^{2}{\left (a x \right )}}{2} + \frac {3 x^{2}}{4} + \frac {x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{a} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{2 a} - \frac {\operatorname {asin}^{4}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {asin}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((x**2*asin(a*x)**4/2 - 3*x**2*asin(a*x)**2/2 + 3*x**2/4 + x*sqrt 
(-a**2*x**2 + 1)*asin(a*x)**3/a - 3*x*sqrt(-a**2*x**2 + 1)*asin(a*x)/(2*a) 
 - asin(a*x)**4/(4*a**2) + 3*asin(a*x)**2/(4*a**2), Ne(a, 0)), (0, True))
 

Maxima [F]

\[ \int x \arcsin (a x)^4 \, dx=\int { x \arcsin \left (a x\right )^{4} \,d x } \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.14 \[ \int x \arcsin (a x)^4 \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{a} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{4}}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )^{4}}{4 \, a^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{2 \, a} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{2 \, a^{2}} - \frac {3 \, \arcsin \left (a x\right )^{2}}{4 \, a^{2}} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}}{4 \, a^{2}} + \frac {3}{8 \, a^{2}} \] Input:

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

Output:

sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a + 1/2*(a^2*x^2 - 1)*arcsin(a*x)^4/a^2 
 + 1/4*arcsin(a*x)^4/a^2 - 3/2*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)/a - 3/2*(a 
^2*x^2 - 1)*arcsin(a*x)^2/a^2 - 3/4*arcsin(a*x)^2/a^2 + 3/4*(a^2*x^2 - 1)/ 
a^2 + 3/8/a^2
 

Mupad [F(-1)]

Timed out. \[ \int x \arcsin (a x)^4 \, dx=\int x\,{\mathrm {asin}\left (a\,x\right )}^4 \,d x \] Input:

int(x*asin(a*x)^4,x)
 

Output:

int(x*asin(a*x)^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88 \[ \int x \arcsin (a x)^4 \, dx=\frac {2 \mathit {asin} \left (a x \right )^{4} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{4}+4 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )^{3} a x -6 \mathit {asin} \left (a x \right )^{2} a^{2} x^{2}+3 \mathit {asin} \left (a x \right )^{2}-6 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x +3 a^{2} x^{2}}{4 a^{2}} \] Input:

int(x*asin(a*x)^4,x)
 

Output:

(2*asin(a*x)**4*a**2*x**2 - asin(a*x)**4 + 4*sqrt( - a**2*x**2 + 1)*asin(a 
*x)**3*a*x - 6*asin(a*x)**2*a**2*x**2 + 3*asin(a*x)**2 - 6*sqrt( - a**2*x* 
*2 + 1)*asin(a*x)*a*x + 3*a**2*x**2)/(4*a**2)