Integrand size = 6, antiderivative size = 69 \[ \int \arcsin (a x)^4 \, dx=24 x-\frac {24 \sqrt {1-a^2 x^2} \arcsin (a x)}{a}-12 x \arcsin (a x)^2+\frac {4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{a}+x \arcsin (a x)^4 \] Output:
24*x-24*(-a^2*x^2+1)^(1/2)*arcsin(a*x)/a-12*x*arcsin(a*x)^2+4*(-a^2*x^2+1) ^(1/2)*arcsin(a*x)^3/a+x*arcsin(a*x)^4
Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \arcsin (a x)^4 \, dx=24 x-\frac {24 \sqrt {1-a^2 x^2} \arcsin (a x)}{a}-12 x \arcsin (a x)^2+\frac {4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{a}+x \arcsin (a x)^4 \] Input:
Integrate[ArcSin[a*x]^4,x]
Output:
24*x - (24*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a - 12*x*ArcSin[a*x]^2 + (4*Sqrt [1 - a^2*x^2]*ArcSin[a*x]^3)/a + x*ArcSin[a*x]^4
Time = 0.47 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5130, 5182, 5130, 5182, 24}
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.
\(\displaystyle \int \arcsin (a x)^4 \, dx\) |
\(\Big \downarrow \) 5130 |
\(\displaystyle x \arcsin (a x)^4-4 a \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle x \arcsin (a x)^4-4 a \left (\frac {3 \int \arcsin (a x)^2dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )\) |
\(\Big \downarrow \) 5130 |
\(\displaystyle x \arcsin (a x)^4-4 a \left (\frac {3 \left (x \arcsin (a x)^2-2 a \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle x \arcsin (a x)^4-4 a \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle x \arcsin (a x)^4-4 a \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )\) |
Input:
Int[ArcSin[a*x]^4,x]
Output:
x*ArcSin[a*x]^4 - 4*a*(-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2) + (3*(x*Ar cSin[a*x]^2 - 2*a*(x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2)))/a)
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {a x \arcsin \left (a x \right )^{4}+4 \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}-12 \arcsin \left (a x \right )^{2} a x +24 a x -24 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\) | \(67\) |
default | \(\frac {a x \arcsin \left (a x \right )^{4}+4 \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}-12 \arcsin \left (a x \right )^{2} a x +24 a x -24 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\) | \(67\) |
orering | \(x \arcsin \left (a x \right )^{4}+\frac {8 \arcsin \left (a x \right )^{3}}{a \sqrt {-a^{2} x^{2}+1}}+\frac {\left (5 a^{2} x^{2}-2\right ) x \left (\frac {12 \arcsin \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}+\frac {4 \arcsin \left (a x \right )^{3} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{2}}+\frac {\left (a x -1\right ) \left (a x +1\right ) \left (5 a^{2} x^{2}+1\right ) \left (\frac {24 \arcsin \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {36 \arcsin \left (a x \right )^{2} a^{4} x}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {12 \arcsin \left (a x \right )^{3} a^{5} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {4 \arcsin \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{4}}+\frac {x \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (\frac {24 a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {144 \arcsin \left (a x \right ) a^{5} x}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {180 \arcsin \left (a x \right )^{2} a^{6} x^{2}}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {48 \arcsin \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {60 \arcsin \left (a x \right )^{3} a^{7} x^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {36 \arcsin \left (a x \right )^{3} a^{5} x}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{a^{4}}\) | \(373\) |
Input:
int(arcsin(a*x)^4,x,method=_RETURNVERBOSE)
Output:
1/a*(a*x*arcsin(a*x)^4+4*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)-12*arcsin(a*x)^2 *a*x+24*a*x-24*arcsin(a*x)*(-a^2*x^2+1)^(1/2))
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int \arcsin (a x)^4 \, dx=\frac {a x \arcsin \left (a x\right )^{4} - 12 \, a x \arcsin \left (a x\right )^{2} + 24 \, a x + 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arcsin \left (a x\right )^{3} - 6 \, \arcsin \left (a x\right )\right )}}{a} \] Input:
integrate(arcsin(a*x)^4,x, algorithm="fricas")
Output:
(a*x*arcsin(a*x)^4 - 12*a*x*arcsin(a*x)^2 + 24*a*x + 4*sqrt(-a^2*x^2 + 1)* (arcsin(a*x)^3 - 6*arcsin(a*x)))/a
Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int \arcsin (a x)^4 \, dx=\begin {cases} x \operatorname {asin}^{4}{\left (a x \right )} - 12 x \operatorname {asin}^{2}{\left (a x \right )} + 24 x + \frac {4 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{a} - \frac {24 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(asin(a*x)**4,x)
Output:
Piecewise((x*asin(a*x)**4 - 12*x*asin(a*x)**2 + 24*x + 4*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/a - 24*sqrt(-a**2*x**2 + 1)*asin(a*x)/a, Ne(a, 0)), (0, T rue))
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09 \[ \int \arcsin (a x)^4 \, dx=x \arcsin \left (a x\right )^{4} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a} - 12 \, {\left (\frac {x \arcsin \left (a x\right )^{2}}{a} - \frac {2 \, {\left (x - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a}\right )} a \] Input:
integrate(arcsin(a*x)^4,x, algorithm="maxima")
Output:
x*arcsin(a*x)^4 + 4*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/a - 12*(x*arcsin(a*x) ^2/a - 2*(x - sqrt(-a^2*x^2 + 1)*arcsin(a*x)/a)/a)*a
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int \arcsin (a x)^4 \, dx=x \arcsin \left (a x\right )^{4} - 12 \, x \arcsin \left (a x\right )^{2} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a} + 24 \, x - \frac {24 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \] Input:
integrate(arcsin(a*x)^4,x, algorithm="giac")
Output:
x*arcsin(a*x)^4 - 12*x*arcsin(a*x)^2 + 4*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/ a + 24*x - 24*sqrt(-a^2*x^2 + 1)*arcsin(a*x)/a
Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.70 \[ \int \arcsin (a x)^4 \, dx=x\,\left ({\mathrm {asin}\left (a\,x\right )}^4-12\,{\mathrm {asin}\left (a\,x\right )}^2+24\right )+\frac {4\,\mathrm {asin}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}\,\left ({\mathrm {asin}\left (a\,x\right )}^2-6\right )}{a} \] Input:
int(asin(a*x)^4,x)
Output:
x*(asin(a*x)^4 - 12*asin(a*x)^2 + 24) + (4*asin(a*x)*(1 - a^2*x^2)^(1/2)*( asin(a*x)^2 - 6))/a
Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \arcsin (a x)^4 \, dx=\frac {\mathit {asin} \left (a x \right )^{4} a x +4 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )^{3}-12 \mathit {asin} \left (a x \right )^{2} a x -24 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+24 a x}{a} \] Input:
int(asin(a*x)^4,x)
Output:
(asin(a*x)**4*a*x + 4*sqrt( - a**2*x**2 + 1)*asin(a*x)**3 - 12*asin(a*x)** 2*a*x - 24*sqrt( - a**2*x**2 + 1)*asin(a*x) + 24*a*x)/a