Integrand size = 10, antiderivative size = 250 \[ \int x^4 \arcsin (a x)^4 \, dx=\frac {16576 x}{5625 a^4}+\frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}-\frac {16576 \sqrt {1-a^2 x^2} \arcsin (a x)}{5625 a^5}-\frac {1088 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{5625 a^3}-\frac {24 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{625 a}-\frac {32 x \arcsin (a x)^2}{25 a^4}-\frac {16 x^3 \arcsin (a x)^2}{75 a^2}-\frac {12}{125} x^5 \arcsin (a x)^2+\frac {32 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{75 a^5}+\frac {16 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{75 a^3}+\frac {4 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{25 a}+\frac {1}{5} x^5 \arcsin (a x)^4 \]
16576/5625*x/a^4+1088/16875*x^3/a^2+24/3125*x^5-32/25*x*arcsin(a*x)^2/a^4- 16/75*x^3*arcsin(a*x)^2/a^2-12/125*x^5*arcsin(a*x)^2+1/5*x^5*arcsin(a*x)^4 -16576/5625*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^5-1088/5625*x^2*arcsin(a*x)*( -a^2*x^2+1)^(1/2)/a^3-24/625*x^4*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a+32/75*ar csin(a*x)^3*(-a^2*x^2+1)^(1/2)/a^5+16/75*x^2*arcsin(a*x)^3*(-a^2*x^2+1)^(1 /2)/a^3+4/25*x^4*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/a
Time = 0.06 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.60 \[ \int x^4 \arcsin (a x)^4 \, dx=\frac {8 a x \left (31080+680 a^2 x^2+81 a^4 x^4\right )-120 \sqrt {1-a^2 x^2} \left (2072+136 a^2 x^2+27 a^4 x^4\right ) \arcsin (a x)-900 a x \left (120+20 a^2 x^2+9 a^4 x^4\right ) \arcsin (a x)^2+4500 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \arcsin (a x)^3+16875 a^5 x^5 \arcsin (a x)^4}{84375 a^5} \]
(8*a*x*(31080 + 680*a^2*x^2 + 81*a^4*x^4) - 120*Sqrt[1 - a^2*x^2]*(2072 + 136*a^2*x^2 + 27*a^4*x^4)*ArcSin[a*x] - 900*a*x*(120 + 20*a^2*x^2 + 9*a^4* x^4)*ArcSin[a*x]^2 + 4500*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*Ar cSin[a*x]^3 + 16875*a^5*x^5*ArcSin[a*x]^4)/(84375*a^5)
Time = 2.14 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.65, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {5138, 5210, 5138, 5210, 15, 5138, 5182, 5130, 5182, 24, 5210, 15, 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.
| \(\displaystyle \int x^4 \arcsin (a x)^4 \, dx\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \int \frac {x^5 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {3 \int x^4 \arcsin (a x)^2dx}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \int \frac {x^5 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{5 a}+\frac {4 \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int x^2 \arcsin (a x)^2dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {\int x^4dx}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}\right )\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int x^2 \arcsin (a x)^2dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}+\frac {x^5}{25 a}\right )\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}+\frac {x^5}{25 a}\right )\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \left (\frac {2 \left (\frac {3 \int \arcsin (a x)^2dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}+\frac {x^5}{25 a}\right )\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \left (\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}+\frac {x^5}{25 a}\right )\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \left (\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}+\frac {x^5}{25 a}\right )\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \left (\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}+\frac {x^5}{25 a}\right )\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \left (\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (\frac {2 \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int x^2dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}\right )}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int x^2dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}+\frac {x^5}{25 a}\right )\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \left (\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (\frac {2 \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {x^3}{9 a}\right )}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {x^3}{9 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}+\frac {x^5}{25 a}\right )\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (\frac {4 \left (\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (\frac {2 \left (\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {x^3}{9 a}\right )}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (\frac {4 \left (\frac {2 \left (\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {x^3}{9 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}+\frac {x^5}{25 a}\right )\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^4-\frac {4}{5} a \left (-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{5 a^2}+\frac {4 \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {2 \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )}{3 a^2}+\frac {x^3}{9 a}\right )}{a}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \arcsin (a x)^2-\frac {2}{5} a \left (-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)}{5 a^2}+\frac {4 \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {2 \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )}{3 a^2}+\frac {x^3}{9 a}\right )}{5 a^2}+\frac {x^5}{25 a}\right )\right )}{5 a}\right )\) |
(x^5*ArcSin[a*x]^4)/5 - (4*a*(-1/5*(x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a ^2 + (3*((x^5*ArcSin[a*x]^2)/5 - (2*a*(x^5/(25*a) - (x^4*Sqrt[1 - a^2*x^2] *ArcSin[a*x])/(5*a^2) + (4*(x^3/(9*a) - (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x] )/(3*a^2) + (2*(x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2))/(3*a^2)))/(5*a ^2)))/5))/(5*a) + (4*(-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2 + ((x ^3*ArcSin[a*x]^2)/3 - (2*a*(x^3/(9*a) - (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x] )/(3*a^2) + (2*(x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2))/(3*a^2)))/3)/a + (2*(-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2) + (3*(x*ArcSin[a*x]^2 - 2* a*(x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2)))/a))/(3*a^2)))/(5*a^2)))/5
3.1.33.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_.)*((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]
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]
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]
Time = 0.05 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{5} x^{5} \arcsin \left (a x \right )^{4}}{5}+\frac {4 \arcsin \left (a x \right )^{3} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {12 a^{5} x^{5} \arcsin \left (a x \right )^{2}}{125}-\frac {8 \arcsin \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}+\frac {24 a^{5} x^{5}}{3125}+\frac {1088 a^{3} x^{3}}{16875}+\frac {16576 a x}{5625}-\frac {16 a^{3} x^{3} \arcsin \left (a x \right )^{2}}{75}-\frac {32 \arcsin \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {32 a x \arcsin \left (a x \right )^{2}}{25}-\frac {64 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{25}}{a^{5}}\) | \(197\) |
| default | \(\frac {\frac {a^{5} x^{5} \arcsin \left (a x \right )^{4}}{5}+\frac {4 \arcsin \left (a x \right )^{3} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {12 a^{5} x^{5} \arcsin \left (a x \right )^{2}}{125}-\frac {8 \arcsin \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}+\frac {24 a^{5} x^{5}}{3125}+\frac {1088 a^{3} x^{3}}{16875}+\frac {16576 a x}{5625}-\frac {16 a^{3} x^{3} \arcsin \left (a x \right )^{2}}{75}-\frac {32 \arcsin \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {32 a x \arcsin \left (a x \right )^{2}}{25}-\frac {64 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{25}}{a^{5}}\) | \(197\) |
1/a^5*(1/5*a^5*x^5*arcsin(a*x)^4+4/75*arcsin(a*x)^3*(3*a^4*x^4+4*a^2*x^2+8 )*(-a^2*x^2+1)^(1/2)-12/125*a^5*x^5*arcsin(a*x)^2-8/625*arcsin(a*x)*(3*a^4 *x^4+4*a^2*x^2+8)*(-a^2*x^2+1)^(1/2)+24/3125*a^5*x^5+1088/16875*a^3*x^3+16 576/5625*a*x-16/75*a^3*x^3*arcsin(a*x)^2-32/225*arcsin(a*x)*(a^2*x^2+2)*(- a^2*x^2+1)^(1/2)-32/25*a*x*arcsin(a*x)^2-64/25*arcsin(a*x)*(-a^2*x^2+1)^(1 /2))
Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.54 \[ \int x^4 \arcsin (a x)^4 \, dx=\frac {16875 \, a^{5} x^{5} \arcsin \left (a x\right )^{4} + 648 \, a^{5} x^{5} + 5440 \, a^{3} x^{3} - 900 \, {\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \arcsin \left (a x\right )^{2} + 248640 \, a x + 60 \, \sqrt {-a^{2} x^{2} + 1} {\left (75 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \arcsin \left (a x\right )^{3} - 2 \, {\left (27 \, a^{4} x^{4} + 136 \, a^{2} x^{2} + 2072\right )} \arcsin \left (a x\right )\right )}}{84375 \, a^{5}} \]
1/84375*(16875*a^5*x^5*arcsin(a*x)^4 + 648*a^5*x^5 + 5440*a^3*x^3 - 900*(9 *a^5*x^5 + 20*a^3*x^3 + 120*a*x)*arcsin(a*x)^2 + 248640*a*x + 60*sqrt(-a^2 *x^2 + 1)*(75*(3*a^4*x^4 + 4*a^2*x^2 + 8)*arcsin(a*x)^3 - 2*(27*a^4*x^4 + 136*a^2*x^2 + 2072)*arcsin(a*x)))/a^5
Time = 0.76 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.96 \[ \int x^4 \arcsin (a x)^4 \, dx=\begin {cases} \frac {x^{5} \operatorname {asin}^{4}{\left (a x \right )}}{5} - \frac {12 x^{5} \operatorname {asin}^{2}{\left (a x \right )}}{125} + \frac {24 x^{5}}{3125} + \frac {4 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{25 a} - \frac {24 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{625 a} - \frac {16 x^{3} \operatorname {asin}^{2}{\left (a x \right )}}{75 a^{2}} + \frac {1088 x^{3}}{16875 a^{2}} + \frac {16 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{75 a^{3}} - \frac {1088 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{5625 a^{3}} - \frac {32 x \operatorname {asin}^{2}{\left (a x \right )}}{25 a^{4}} + \frac {16576 x}{5625 a^{4}} + \frac {32 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{75 a^{5}} - \frac {16576 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{5625 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**5*asin(a*x)**4/5 - 12*x**5*asin(a*x)**2/125 + 24*x**5/3125 + 4*x**4*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(25*a) - 24*x**4*sqrt(-a**2*x**2 + 1)*asin(a*x)/(625*a) - 16*x**3*asin(a*x)**2/(75*a**2) + 1088*x**3/(1687 5*a**2) + 16*x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(75*a**3) - 1088*x**2* sqrt(-a**2*x**2 + 1)*asin(a*x)/(5625*a**3) - 32*x*asin(a*x)**2/(25*a**4) + 16576*x/(5625*a**4) + 32*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(75*a**5) - 16 576*sqrt(-a**2*x**2 + 1)*asin(a*x)/(5625*a**5), Ne(a, 0)), (0, True))
Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.83 \[ \int x^4 \arcsin (a x)^4 \, dx=\frac {1}{5} \, x^{5} \arcsin \left (a x\right )^{4} + \frac {4}{75} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arcsin \left (a x\right )^{3} - \frac {4}{84375} \, {\left (2 \, a {\left (\frac {15 \, {\left (27 \, \sqrt {-a^{2} x^{2} + 1} a^{2} x^{4} + 136 \, \sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arcsin \left (a x\right )}{a^{5}} - \frac {81 \, a^{4} x^{5} + 680 \, a^{2} x^{3} + 31080 \, x}{a^{6}}\right )} + \frac {225 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \arcsin \left (a x\right )^{2}}{a^{5}}\right )} a \]
1/5*x^5*arcsin(a*x)^4 + 4/75*(3*sqrt(-a^2*x^2 + 1)*x^4/a^2 + 4*sqrt(-a^2*x ^2 + 1)*x^2/a^4 + 8*sqrt(-a^2*x^2 + 1)/a^6)*a*arcsin(a*x)^3 - 4/84375*(2*a *(15*(27*sqrt(-a^2*x^2 + 1)*a^2*x^4 + 136*sqrt(-a^2*x^2 + 1)*x^2 + 2072*sq rt(-a^2*x^2 + 1)/a^2)*arcsin(a*x)/a^5 - (81*a^4*x^5 + 680*a^2*x^3 + 31080* x)/a^6) + 225*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)*arcsin(a*x)^2/a^5)*a
Time = 0.28 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.22 \[ \int x^4 \arcsin (a x)^4 \, dx=\frac {{\left (a^{2} x^{2} - 1\right )}^{2} x \arcsin \left (a x\right )^{4}}{5 \, a^{4}} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{4}}{5 \, a^{4}} - \frac {12 \, {\left (a^{2} x^{2} - 1\right )}^{2} x \arcsin \left (a x\right )^{2}}{125 \, a^{4}} + \frac {x \arcsin \left (a x\right )^{4}}{5 \, a^{4}} + \frac {4 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{25 \, a^{5}} - \frac {152 \, {\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{2}}{375 \, a^{4}} - \frac {8 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (a x\right )^{3}}{15 \, a^{5}} + \frac {24 \, {\left (a^{2} x^{2} - 1\right )}^{2} x}{3125 \, a^{4}} - \frac {596 \, x \arcsin \left (a x\right )^{2}}{375 \, a^{4}} - \frac {24 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{625 \, a^{5}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{5 \, a^{5}} + \frac {6736 \, {\left (a^{2} x^{2} - 1\right )} x}{84375 \, a^{4}} + \frac {304 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (a x\right )}{1125 \, a^{5}} + \frac {254728 \, x}{84375 \, a^{4}} - \frac {1192 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{375 \, a^{5}} \]
1/5*(a^2*x^2 - 1)^2*x*arcsin(a*x)^4/a^4 + 2/5*(a^2*x^2 - 1)*x*arcsin(a*x)^ 4/a^4 - 12/125*(a^2*x^2 - 1)^2*x*arcsin(a*x)^2/a^4 + 1/5*x*arcsin(a*x)^4/a ^4 + 4/25*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/a^5 - 152/375*( a^2*x^2 - 1)*x*arcsin(a*x)^2/a^4 - 8/15*(-a^2*x^2 + 1)^(3/2)*arcsin(a*x)^3 /a^5 + 24/3125*(a^2*x^2 - 1)^2*x/a^4 - 596/375*x*arcsin(a*x)^2/a^4 - 24/62 5*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*arcsin(a*x)/a^5 + 4/5*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/a^5 + 6736/84375*(a^2*x^2 - 1)*x/a^4 + 304/1125*(-a^2*x^ 2 + 1)^(3/2)*arcsin(a*x)/a^5 + 254728/84375*x/a^4 - 1192/375*sqrt(-a^2*x^2 + 1)*arcsin(a*x)/a^5
Timed out. \[ \int x^4 \arcsin (a x)^4 \, dx=\int x^4\,{\mathrm {asin}\left (a\,x\right )}^4 \,d x \]