Integrand size = 20, antiderivative size = 210 \[ \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {1}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {2}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \arcsin (a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \arcsin (a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \arcsin (a x)}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \]
1/5*x*arcsin(a*x)/c/(-a^2*c*x^2+c)^(5/2)+4/15*x*arcsin(a*x)/c^2/(-a^2*c*x^ 2+c)^(3/2)-1/20/a/c^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(1/2)+8/15*x*arcsi n(a*x)/c^3/(-a^2*c*x^2+c)^(1/2)-2/15/a/c^3/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+ c)^(1/2)+4/15*ln(-a^2*x^2+1)*(-a^2*x^2+1)^(1/2)/a/c^3/(-a^2*c*x^2+c)^(1/2)
Time = 0.16 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.53 \[ \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {\sqrt {c-a^2 c x^2} \left (4 a x \left (15-20 a^2 x^2+8 a^4 x^4\right ) \arcsin (a x)+\sqrt {1-a^2 x^2} \left (-11+8 a^2 x^2+16 \left (-1+a^2 x^2\right )^2 \log \left (-1+a^2 x^2\right )\right )\right )}{60 a c^4 \left (-1+a^2 x^2\right )^3} \]
-1/60*(Sqrt[c - a^2*c*x^2]*(4*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*ArcSin[a*x ] + Sqrt[1 - a^2*x^2]*(-11 + 8*a^2*x^2 + 16*(-1 + a^2*x^2)^2*Log[-1 + a^2* x^2])))/(a*c^4*(-1 + a^2*x^2)^3)
Time = 0.49 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5162, 241, 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.
| \(\displaystyle \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {4 \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}-\frac {a \sqrt {1-a^2 x^2} \int \frac {x}{\left (1-a^2 x^2\right )^3}dx}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \arcsin (a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {4 \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}+\frac {x \arcsin (a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {1}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}-\frac {a \sqrt {1-a^2 x^2} \int \frac {x}{\left (1-a^2 x^2\right )^2}dx}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \arcsin (a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \arcsin (a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {1}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {x \arcsin (a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}-\frac {1}{6 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}\right )}{5 c}+\frac {x \arcsin (a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {1}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (\frac {x \arcsin (a x)}{c \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2} \int \frac {x}{1-a^2 x^2}dx}{c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}-\frac {1}{6 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}\right )}{5 c}+\frac {x \arcsin (a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {1}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {4 \left (\frac {x \arcsin (a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \left (\frac {x \arcsin (a x)}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c \sqrt {c-a^2 c x^2}}\right )}{3 c}-\frac {1}{6 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}\right )}{5 c}+\frac {x \arcsin (a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {1}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}\) |
-1/20*1/(a*c^3*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x])/ (5*c*(c - a^2*c*x^2)^(5/2)) + (4*(-1/6*1/(a*c^2*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*((x*ArcSin [a*x])/(c*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[1 - a^2*x^2])/(2*a *c*Sqrt[c - a^2*c*x^2])))/(3*c)))/(5*c)
3.2.40.3.1 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]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.95
| method | result | size |
| default | \(\frac {16 i \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )}{15 a \,c^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-20 a^{3} x^{3}+8 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+15 a x -16 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+8 i \sqrt {-a^{2} x^{2}+1}\right ) \left (64 i a^{8} x^{8}+64 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}-280 i a^{6} x^{6}-248 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+160 a^{4} x^{4} \arcsin \left (a x \right )+456 i a^{4} x^{4}+340 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-380 a^{2} x^{2} \arcsin \left (a x \right )-328 i a^{2} x^{2}-165 a x \sqrt {-a^{2} x^{2}+1}+256 \arcsin \left (a x \right )+88 i\right )}{60 c^{4} \left (40 a^{10} x^{10}-215 a^{8} x^{8}+469 a^{6} x^{6}-517 a^{4} x^{4}+287 a^{2} x^{2}-64\right ) a}-\frac {8 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{15 a \,c^{4} \left (a^{2} x^{2}-1\right )}\) | \(409\) |
16/15*I*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/c^4/(a^2*x^2-1)*arcsin (a*x)-1/60*(-c*(a^2*x^2-1))^(1/2)*(8*a^5*x^5-20*a^3*x^3+8*I*(-a^2*x^2+1)^( 1/2)*a^4*x^4+15*a*x-16*I*(-a^2*x^2+1)^(1/2)*a^2*x^2+8*I*(-a^2*x^2+1)^(1/2) )*(64*I*a^8*x^8+64*(-a^2*x^2+1)^(1/2)*a^7*x^7-280*I*a^6*x^6-248*(-a^2*x^2+ 1)^(1/2)*a^5*x^5+160*a^4*x^4*arcsin(a*x)+456*I*a^4*x^4+340*a^3*x^3*(-a^2*x ^2+1)^(1/2)-380*a^2*x^2*arcsin(a*x)-328*I*a^2*x^2-165*a*x*(-a^2*x^2+1)^(1/ 2)+256*arcsin(a*x)+88*I)/c^4/(40*a^10*x^10-215*a^8*x^8+469*a^6*x^6-517*a^4 *x^4+287*a^2*x^2-64)/a-8/15*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/c^ 4/(a^2*x^2-1)*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2)
\[ \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\arcsin \left (a x\right )}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
integral(sqrt(-a^2*c*x^2 + c)*arcsin(a*x)/(a^8*c^4*x^8 - 4*a^6*c^4*x^6 + 6 *a^4*c^4*x^4 - 4*a^2*c^4*x^2 + c^4), x)
\[ \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {asin}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.71 \[ \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {1}{60} \, a {\left (\frac {3}{{\left (a^{6} c^{\frac {5}{2}} x^{4} - 2 \, a^{4} c^{\frac {5}{2}} x^{2} + a^{2} c^{\frac {5}{2}}\right )} c} - \frac {8}{{\left (a^{4} c^{\frac {3}{2}} x^{2} - a^{2} c^{\frac {3}{2}}\right )} c^{2}} + \frac {16 \, \log \left (x^{2} - \frac {1}{a^{2}}\right )}{a^{2} c^{\frac {7}{2}}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {-a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \arcsin \left (a x\right ) \]
-1/60*a*(3/((a^6*c^(5/2)*x^4 - 2*a^4*c^(5/2)*x^2 + a^2*c^(5/2))*c) - 8/((a ^4*c^(3/2)*x^2 - a^2*c^(3/2))*c^2) + 16*log(x^2 - 1/a^2)/(a^2*c^(7/2))) + 1/15*(8*x/(sqrt(-a^2*c*x^2 + c)*c^3) + 4*x/((-a^2*c*x^2 + c)^(3/2)*c^2) + 3*x/((-a^2*c*x^2 + c)^(5/2)*c))*arcsin(a*x)
Time = 0.35 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.61 \[ \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {1}{60} \, \sqrt {c} {\left (\frac {16 \, \log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{a c^{4}} - \frac {24 \, a^{4} x^{4} - 56 \, a^{2} x^{2} + 35}{{\left (a^{2} x^{2} - 1\right )}^{2} a c^{4}}\right )} - \frac {\sqrt {-a^{2} c x^{2} + c} {\left (4 \, {\left (\frac {2 \, a^{4} x^{2}}{c} - \frac {5 \, a^{2}}{c}\right )} x^{2} + \frac {15}{c}\right )} x \arcsin \left (a x\right )}{15 \, {\left (a^{2} c x^{2} - c\right )}^{3}} \]
-1/60*sqrt(c)*(16*log(abs(a^2*x^2 - 1))/(a*c^4) - (24*a^4*x^4 - 56*a^2*x^2 + 35)/((a^2*x^2 - 1)^2*a*c^4)) - 1/15*sqrt(-a^2*c*x^2 + c)*(4*(2*a^4*x^2/ c - 5*a^2/c)*x^2 + 15/c)*x*arcsin(a*x)/(a^2*c*x^2 - c)^3
Timed out. \[ \int \frac {\arcsin (a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {asin}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]