Integrand size = 25, antiderivative size = 89 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b x (a+b \arcsin (c x))}{c d^2 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^2 d^2} \]
1/2*(a+b*arcsin(c*x))^2/c^2/d^2/(-c^2*x^2+1)-1/2*b^2*ln(-c^2*x^2+1)/c^2/d^ 2-b*x*(a+b*arcsin(c*x))/c/d^2/(-c^2*x^2+1)^(1/2)
Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.84 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {\frac {2 b c x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{-1+c^2 x^2}+b^2 \log \left (1-c^2 x^2\right )}{2 c^2 d^2} \]
-1/2*((2*b*c*x*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^2] + (a + b*ArcSin[c*x] )^2/(-1 + c^2*x^2) + b^2*Log[1 - c^2*x^2])/(c^2*d^2)
Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5182, 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.
\(\displaystyle \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle \frac {(a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}\) |
\(\Big \downarrow \) 5160 |
\(\displaystyle \frac {(a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-b c \int \frac {x}{1-c^2 x^2}dx\right )}{c d^2}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {(a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )}{c d^2}\) |
(a + b*ArcSin[c*x])^2/(2*c^2*d^2*(1 - c^2*x^2)) - (b*((x*(a + b*ArcSin[c*x ]))/Sqrt[1 - c^2*x^2] + (b*Log[1 - c^2*x^2])/(2*c)))/(c*d^2)
3.2.95.3.1 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[((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_.)*(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]
Leaf count of result is larger than twice the leaf count of optimal. \(171\) vs. \(2(83)=166\).
Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.93
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{2 \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c x}{c^{2} x^{2}-1}-\frac {\ln \left (-c^{2} x^{2}+1\right )}{2}\right )}{d^{2}}+\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 c x -4}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 c x +4}\right )}{d^{2}}}{c^{2}}\) | \(172\) |
default | \(\frac {-\frac {a^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{2 \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c x}{c^{2} x^{2}-1}-\frac {\ln \left (-c^{2} x^{2}+1\right )}{2}\right )}{d^{2}}+\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 c x -4}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 c x +4}\right )}{d^{2}}}{c^{2}}\) | \(172\) |
parts | \(-\frac {a^{2}}{2 d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{2 \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c x}{c^{2} x^{2}-1}-\frac {\ln \left (-c^{2} x^{2}+1\right )}{2}\right )}{d^{2} c^{2}}+\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 c x -4}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 c x +4}\right )}{d^{2} c^{2}}\) | \(177\) |
1/c^2*(-1/2*a^2/d^2/(c^2*x^2-1)+b^2/d^2*(-1/2/(c^2*x^2-1)*arcsin(c*x)^2+(- c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)*c*x-1/2*ln(-c^2*x^2+1))+2*a*b/d^2 *(-1/2/(c^2*x^2-1)*arcsin(c*x)+1/4/(c*x-1)*(-(c*x-1)^2-2*c*x+2)^(1/2)+1/4/ (c*x+1)*(-(c*x+1)^2+2*c*x+2)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \log \left (c^{2} x^{2} - 1\right ) - 2 \, {\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1}}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \]
-1/2*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2 + (b^2*c^2*x^2 - b^2)*lo g(c^2*x^2 - 1) - 2*(b^2*c*x*arcsin(c*x) + a*b*c*x)*sqrt(-c^2*x^2 + 1))/(c^ 4*d^2*x^2 - c^2*d^2)
\[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2} x}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x \operatorname {asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
(Integral(a**2*x/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b**2*x*asin( c*x)**2/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(2*a*b*x*asin(c*x)/(c* *4*x**4 - 2*c**2*x**2 + 1), x))/d**2
Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (82) = 164\).
Time = 0.30 (sec) , antiderivative size = 293, normalized size of antiderivative = 3.29 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {1}{2} \, {\left ({\left (\frac {\sqrt {-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{7} d^{4} x + c^{6} d^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{7} d^{4} x - c^{6} d^{4}}\right )} c^{2} - \frac {2 \, \arcsin \left (c x\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} a b - \frac {1}{2} \, {\left (c^{3} {\left (\frac {\log \left (c x + 1\right )}{c^{5} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{5} d^{2}}\right )} - {\left (\frac {\sqrt {-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{7} d^{4} x + c^{6} d^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{7} d^{4} x - c^{6} d^{4}}\right )} c^{2} \arcsin \left (c x\right )\right )} b^{2} - \frac {b^{2} \arcsin \left (c x\right )^{2}}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} - \frac {a^{2}}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \]
1/2*((sqrt(-c^2*x^2 + 1)*c^2*d^2/(c^7*d^4*x + c^6*d^4) + sqrt(-c^2*x^2 + 1 )*c^2*d^2/(c^7*d^4*x - c^6*d^4))*c^2 - 2*arcsin(c*x)/(c^4*d^2*x^2 - c^2*d^ 2))*a*b - 1/2*(c^3*(log(c*x + 1)/(c^5*d^2) + log(c*x - 1)/(c^5*d^2)) - (sq rt(-c^2*x^2 + 1)*c^2*d^2/(c^7*d^4*x + c^6*d^4) + sqrt(-c^2*x^2 + 1)*c^2*d^ 2/(c^7*d^4*x - c^6*d^4))*c^2*arcsin(c*x))*b^2 - 1/2*b^2*arcsin(c*x)^2/(c^4 *d^2*x^2 - c^2*d^2) - 1/2*a^2/(c^4*d^2*x^2 - c^2*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (82) = 164\).
Time = 0.32 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.29 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b^{2} x^{2} \arcsin \left (c x\right )^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac {a b x^{2} \arcsin \left (c x\right )}{{\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac {a^{2} x^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac {b^{2} x \arcsin \left (c x\right )}{\sqrt {-c^{2} x^{2} + 1} c d^{2}} + \frac {b^{2} \arcsin \left (c x\right )^{2}}{2 \, c^{2} d^{2}} - \frac {a b x}{\sqrt {-c^{2} x^{2} + 1} c d^{2}} + \frac {a b \arcsin \left (c x\right )}{c^{2} d^{2}} - \frac {b^{2} \log \left (2\right )}{c^{2} d^{2}} - \frac {b^{2} \log \left ({\left | -c^{2} x^{2} + 1 \right |}\right )}{2 \, c^{2} d^{2}} + \frac {a^{2}}{2 \, c^{2} d^{2}} \]
-1/2*b^2*x^2*arcsin(c*x)^2/((c^2*x^2 - 1)*d^2) - a*b*x^2*arcsin(c*x)/((c^2 *x^2 - 1)*d^2) - 1/2*a^2*x^2/((c^2*x^2 - 1)*d^2) - b^2*x*arcsin(c*x)/(sqrt (-c^2*x^2 + 1)*c*d^2) + 1/2*b^2*arcsin(c*x)^2/(c^2*d^2) - a*b*x/(sqrt(-c^2 *x^2 + 1)*c*d^2) + a*b*arcsin(c*x)/(c^2*d^2) - b^2*log(2)/(c^2*d^2) - 1/2* b^2*log(abs(-c^2*x^2 + 1))/(c^2*d^2) + 1/2*a^2/(c^2*d^2)
Timed out. \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]