Integrand size = 25, antiderivative size = 150 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b x (a+b \arcsin (c x))}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b x (a+b \arcsin (c x))}{3 c d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{6 c^2 d^3} \]
1/12*b^2/c^2/d^3/(-c^2*x^2+1)-1/6*b*x*(a+b*arcsin(c*x))/c/d^3/(-c^2*x^2+1) ^(3/2)+1/4*(a+b*arcsin(c*x))^2/c^2/d^3/(-c^2*x^2+1)^2-1/6*b^2*ln(-c^2*x^2+ 1)/c^2/d^3-1/3*b*x*(a+b*arcsin(c*x))/c/d^3/(-c^2*x^2+1)^(1/2)
Time = 0.94 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.08 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {3 a^2+b^2-b^2 c^2 x^2-6 a b c x \sqrt {1-c^2 x^2}+4 a b c^3 x^3 \sqrt {1-c^2 x^2}+2 b \left (3 a+b c x \sqrt {1-c^2 x^2} \left (-3+2 c^2 x^2\right )\right ) \arcsin (c x)+3 b^2 \arcsin (c x)^2-2 b^2 \left (-1+c^2 x^2\right )^2 \log \left (1-c^2 x^2\right )}{12 c^2 d^3 \left (-1+c^2 x^2\right )^2} \]
(3*a^2 + b^2 - b^2*c^2*x^2 - 6*a*b*c*x*Sqrt[1 - c^2*x^2] + 4*a*b*c^3*x^3*S qrt[1 - c^2*x^2] + 2*b*(3*a + b*c*x*Sqrt[1 - c^2*x^2]*(-3 + 2*c^2*x^2))*Ar cSin[c*x] + 3*b^2*ArcSin[c*x]^2 - 2*b^2*(-1 + c^2*x^2)^2*Log[1 - c^2*x^2]) /(12*c^2*d^3*(-1 + c^2*x^2)^2)
Time = 0.45 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5182, 5162, 241, 5160, 240}
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.
| \(\displaystyle \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {2}{3} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {x}{\left (1-c^2 x^2\right )^2}dx+\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 c d^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {2}{3} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 c d^3}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-b c \int \frac {x}{1-c^2 x^2}dx\right )+\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 c d^3}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 c d^3}\) |
(a + b*ArcSin[c*x])^2/(4*c^2*d^3*(1 - c^2*x^2)^2) - (b*(-1/6*b/(c*(1 - c^2 *x^2)) + (x*(a + b*ArcSin[c*x]))/(3*(1 - c^2*x^2)^(3/2)) + (2*((x*(a + b*A rcSin[c*x]))/Sqrt[1 - c^2*x^2] + (b*Log[1 - c^2*x^2])/(2*c)))/3))/(2*c*d^3 )
3.3.4.3.1 Defintions of rubi rules used
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp[b *c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSin[c*x ])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1 ))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b*Ar cSin[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2 *x^2)^p] Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x ]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
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 = 0.13 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.80
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2}}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{4 \left (c^{2} x^{2}-1\right )^{2}}+\frac {\arcsin \left (c x \right ) c x \sqrt {-c^{2} x^{2}+1}}{6 \left (c^{2} x^{2}-1\right )^{2}}+\frac {1}{12 c^{2} x^{2}-12}-\frac {\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c x}{3 \left (c^{2} x^{2}-1\right )}+\frac {\ln \left (-c^{2} x^{2}+1\right )}{6}\right )}{d^{3}}-\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{12 \left (c x +1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{12 \left (c x -1\right )}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}\right )}{d^{3}}}{c^{2}}\) | \(270\) |
| default | \(\frac {\frac {a^{2}}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{4 \left (c^{2} x^{2}-1\right )^{2}}+\frac {\arcsin \left (c x \right ) c x \sqrt {-c^{2} x^{2}+1}}{6 \left (c^{2} x^{2}-1\right )^{2}}+\frac {1}{12 c^{2} x^{2}-12}-\frac {\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c x}{3 \left (c^{2} x^{2}-1\right )}+\frac {\ln \left (-c^{2} x^{2}+1\right )}{6}\right )}{d^{3}}-\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{12 \left (c x +1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{12 \left (c x -1\right )}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}\right )}{d^{3}}}{c^{2}}\) | \(270\) |
| parts | \(\frac {a^{2}}{4 d^{3} c^{2} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{4 \left (c^{2} x^{2}-1\right )^{2}}+\frac {\arcsin \left (c x \right ) c x \sqrt {-c^{2} x^{2}+1}}{6 \left (c^{2} x^{2}-1\right )^{2}}+\frac {1}{12 c^{2} x^{2}-12}-\frac {\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c x}{3 \left (c^{2} x^{2}-1\right )}+\frac {\ln \left (-c^{2} x^{2}+1\right )}{6}\right )}{d^{3} c^{2}}-\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{12 \left (c x +1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{12 \left (c x -1\right )}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}\right )}{d^{3} c^{2}}\) | \(275\) |
1/c^2*(1/4*a^2/d^3/(c^2*x^2-1)^2-b^2/d^3*(-1/4/(c^2*x^2-1)^2*arcsin(c*x)^2 +1/6*arcsin(c*x)*c*x*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)^2+1/12/(c^2*x^2-1)-1/3 *(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)*c*x+1/6*ln(-c^2*x^2+1))-2*a*b/ d^3*(-1/4/(c^2*x^2-1)^2*arcsin(c*x)-1/48/(c*x+1)^2*(-(c*x+1)^2+2*c*x+2)^(1 /2)-1/12/(c*x+1)*(-(c*x+1)^2+2*c*x+2)^(1/2)-1/12/(c*x-1)*(-(c*x-1)^2-2*c*x +2)^(1/2)+1/48/(c*x-1)^2*(-(c*x-1)^2-2*c*x+2)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.10 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {b^{2} c^{2} x^{2} - 3 \, b^{2} \arcsin \left (c x\right )^{2} - 6 \, a b \arcsin \left (c x\right ) - 3 \, a^{2} - b^{2} + 2 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} - 1\right ) - 2 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x + {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{12 \, {\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \]
-1/12*(b^2*c^2*x^2 - 3*b^2*arcsin(c*x)^2 - 6*a*b*arcsin(c*x) - 3*a^2 - b^2 + 2*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*log(c^2*x^2 - 1) - 2*(2*a*b*c^3*x ^3 - 3*a*b*c*x + (2*b^2*c^3*x^3 - 3*b^2*c*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3)
\[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2} x}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x \operatorname {asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
-(Integral(a**2*x/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integr al(b**2*x*asin(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + I ntegral(2*a*b*x*asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x)) /d**3
\[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
1/4*a^2/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) + 1/4*(b^2*arctan2(c*x, sq rt(c*x + 1)*sqrt(-c*x + 1))^2 + 4*(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3)* integrate(-1/2*(4*a*b*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - sqr t(c*x + 1)*sqrt(-c*x + 1)*b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/ (c^7*d^3*x^6 - 3*c^5*d^3*x^4 + 3*c^3*d^3*x^2 - c*d^3), x))/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (134) = 268\).
Time = 0.36 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.63 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b^{2} c^{2} x^{4} \arcsin \left (c x\right )^{2}}{4 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {a b c^{2} x^{4} \arcsin \left (c x\right )}{2 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {a^{2} c^{2} x^{4}}{4 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {b^{2} c x^{3} \arcsin \left (c x\right )}{6 \, {\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} d^{3}} - \frac {b^{2} x^{2} \arcsin \left (c x\right )^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{3}} + \frac {a b c x^{3}}{6 \, {\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} d^{3}} - \frac {a b x^{2} \arcsin \left (c x\right )}{{\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac {a^{2} x^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac {b^{2} x^{2}}{12 \, {\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac {b^{2} x \arcsin \left (c x\right )}{2 \, \sqrt {-c^{2} x^{2} + 1} c d^{3}} + \frac {b^{2} \arcsin \left (c x\right )^{2}}{4 \, c^{2} d^{3}} - \frac {a b x}{2 \, \sqrt {-c^{2} x^{2} + 1} c d^{3}} + \frac {a b \arcsin \left (c x\right )}{2 \, c^{2} d^{3}} - \frac {b^{2} \log \left (2\right )}{3 \, c^{2} d^{3}} - \frac {b^{2} \log \left ({\left | -c^{2} x^{2} + 1 \right |}\right )}{6 \, c^{2} d^{3}} + \frac {a^{2}}{4 \, c^{2} d^{3}} + \frac {b^{2}}{12 \, c^{2} d^{3}} \]
1/4*b^2*c^2*x^4*arcsin(c*x)^2/((c^2*x^2 - 1)^2*d^3) + 1/2*a*b*c^2*x^4*arcs in(c*x)/((c^2*x^2 - 1)^2*d^3) + 1/4*a^2*c^2*x^4/((c^2*x^2 - 1)^2*d^3) + 1/ 6*b^2*c*x^3*arcsin(c*x)/((c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*d^3) - 1/2*b^2*x ^2*arcsin(c*x)^2/((c^2*x^2 - 1)*d^3) + 1/6*a*b*c*x^3/((c^2*x^2 - 1)*sqrt(- c^2*x^2 + 1)*d^3) - a*b*x^2*arcsin(c*x)/((c^2*x^2 - 1)*d^3) - 1/2*a^2*x^2/ ((c^2*x^2 - 1)*d^3) - 1/12*b^2*x^2/((c^2*x^2 - 1)*d^3) - 1/2*b^2*x*arcsin( c*x)/(sqrt(-c^2*x^2 + 1)*c*d^3) + 1/4*b^2*arcsin(c*x)^2/(c^2*d^3) - 1/2*a* b*x/(sqrt(-c^2*x^2 + 1)*c*d^3) + 1/2*a*b*arcsin(c*x)/(c^2*d^3) - 1/3*b^2*l og(2)/(c^2*d^3) - 1/6*b^2*log(abs(-c^2*x^2 + 1))/(c^2*d^3) + 1/4*a^2/(c^2* d^3) + 1/12*b^2/(c^2*d^3)
Timed out. \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]