Integrand size = 22, antiderivative size = 388 \[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {x \arcsin (a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {\arcsin (a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \arcsin (a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \arcsin (a x)^2 \log \left (1+e^{2 i \arcsin (a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}} \]
1/3*x*arcsin(a*x)^3/c/(-a^2*c*x^2+c)^(3/2)+x*arcsin(a*x)/c^2/(-a^2*c*x^2+c )^(1/2)+2/3*x*arcsin(a*x)^3/c^2/(-a^2*c*x^2+c)^(1/2)-1/2*arcsin(a*x)^2/a/c ^2/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(1/2)-2/3*I*arcsin(a*x)^3*(-a^2*x^2+1 )^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)+2*arcsin(a*x)^2*ln(1+(I*a*x+(-a^2*x^2+1 )^(1/2))^2)*(-a^2*x^2+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)+1/2*ln(-a^2*x^2+ 1)*(-a^2*x^2+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)-2*I*arcsin(a*x)*polylog(2 ,-(I*a*x+(-a^2*x^2+1)^(1/2))^2)*(-a^2*x^2+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1 /2)+polylog(3,-(I*a*x+(-a^2*x^2+1)^(1/2))^2)*(-a^2*x^2+1)^(1/2)/a/c^2/(-a^ 2*c*x^2+c)^(1/2)
Time = 0.49 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.54 \[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {6 a x \arcsin (a x)}{\sqrt {1-a^2 x^2}}+\frac {3 \arcsin (a x)^2}{-1+a^2 x^2}-4 i \arcsin (a x)^3+\frac {2 a x \arcsin (a x)^3}{\left (1-a^2 x^2\right )^{3/2}}+\frac {4 a x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}+12 \arcsin (a x)^2 \log \left (1+e^{2 i \arcsin (a x)}\right )+3 \log \left (1-a^2 x^2\right )-12 i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (a x)}\right )\right )}{6 a c \left (c-a^2 c x^2\right )^{3/2}} \]
((1 - a^2*x^2)^(3/2)*((6*a*x*ArcSin[a*x])/Sqrt[1 - a^2*x^2] + (3*ArcSin[a* x]^2)/(-1 + a^2*x^2) - (4*I)*ArcSin[a*x]^3 + (2*a*x*ArcSin[a*x]^3)/(1 - a^ 2*x^2)^(3/2) + (4*a*x*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2] + 12*ArcSin[a*x]^2* Log[1 + E^((2*I)*ArcSin[a*x])] + 3*Log[1 - a^2*x^2] - (12*I)*ArcSin[a*x]*P olyLog[2, -E^((2*I)*ArcSin[a*x])] + 6*PolyLog[3, -E^((2*I)*ArcSin[a*x])])) /(6*a*c*(c - a^2*c*x^2)^(3/2))
Time = 1.50 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.76, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5162, 5160, 5180, 3042, 4202, 2620, 3011, 2720, 5182, 5160, 240, 7143}
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.
| \(\displaystyle \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)^2}{1-a^2 x^2}dx}{c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5180 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \int \frac {a x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}d\arcsin (a x)}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \int \arcsin (a x)^2 \tan (\arcsin (a x))d\arcsin (a x)}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (\frac {1}{3} i \arcsin (a x)^3-2 i \int \frac {e^{2 i \arcsin (a x)} \arcsin (a x)^2}{1+e^{2 i \arcsin (a x)}}d\arcsin (a x)\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (\frac {1}{3} i \arcsin (a x)^3-2 i \left (i \int \arcsin (a x) \log \left (1+e^{2 i \arcsin (a x)}\right )d\arcsin (a x)-\frac {1}{2} i \arcsin (a x)^2 \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (\frac {1}{3} i \arcsin (a x)^3-2 i \left (i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \int \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )d\arcsin (a x)\right )-\frac {1}{2} i \arcsin (a x)^2 \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (\frac {1}{3} i \arcsin (a x)^3-2 i \left (i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )-\frac {1}{4} \int e^{-2 i \arcsin (a x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )de^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \arcsin (a x)^2 \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \left (\frac {\arcsin (a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\arcsin (a x)}{\left (1-a^2 x^2\right )^{3/2}}dx}{a}\right )}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (\frac {1}{3} i \arcsin (a x)^3-2 i \left (i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )-\frac {1}{4} \int e^{-2 i \arcsin (a x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )de^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \arcsin (a x)^2 \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \left (\frac {\arcsin (a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}-a \int \frac {x}{1-a^2 x^2}dx}{a}\right )}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (\frac {1}{3} i \arcsin (a x)^3-2 i \left (i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )-\frac {1}{4} \int e^{-2 i \arcsin (a x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )de^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \arcsin (a x)^2 \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (\frac {1}{3} i \arcsin (a x)^3-2 i \left (i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )-\frac {1}{4} \int e^{-2 i \arcsin (a x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )de^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \arcsin (a x)^2 \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}-\frac {a \sqrt {1-a^2 x^2} \left (\frac {\arcsin (a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}+\frac {\log \left (1-a^2 x^2\right )}{2 a}}{a}\right )}{c^2 \sqrt {c-a^2 c x^2}}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {a \sqrt {1-a^2 x^2} \left (\frac {\arcsin (a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}+\frac {\log \left (1-a^2 x^2\right )}{2 a}}{a}\right )}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \left (\frac {1}{3} i \arcsin (a x)^3-2 i \left (i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (a x)}\right )\right )-\frac {1}{2} i \arcsin (a x)^2 \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
(x*ArcSin[a*x]^3)/(3*c*(c - a^2*c*x^2)^(3/2)) - (a*Sqrt[1 - a^2*x^2]*(ArcS in[a*x]^2/(2*a^2*(1 - a^2*x^2)) - ((x*ArcSin[a*x])/Sqrt[1 - a^2*x^2] + Log [1 - a^2*x^2]/(2*a))/a))/(c^2*Sqrt[c - a^2*c*x^2]) + (2*((x*ArcSin[a*x]^3) /(c*Sqrt[c - a^2*c*x^2]) - (3*Sqrt[1 - a^2*x^2]*((I/3)*ArcSin[a*x]^3 - (2* I)*((-1/2*I)*ArcSin[a*x]^2*Log[1 + E^((2*I)*ArcSin[a*x])] + I*((I/2)*ArcSi n[a*x]*PolyLog[2, -E^((2*I)*ArcSin[a*x])] - PolyLog[3, -E^((2*I)*ArcSin[a* x])]/4))))/(a*c*Sqrt[c - a^2*c*x^2])))/(3*c)
3.3.100.3.1 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[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
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), x_Symbol] :> Simp[-e^(-1) Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.23 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}-2 i \sqrt {-a^{2} x^{2}+1}-3 a x \right ) \arcsin \left (a x \right ) \left (-6 i \arcsin \left (a x \right ) a^{4} x^{4}-6 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+6 i \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-6 a^{4} x^{4}+6 \arcsin \left (a x \right )^{2} a^{2} x^{2}+12 i \arcsin \left (a x \right ) a^{2} x^{2}+9 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x -6 i \sqrt {-a^{2} x^{2}+1}\, a x +18 a^{2} x^{2}-8 \arcsin \left (a x \right )^{2}-6 i \arcsin \left (a x \right )-12\right )}{6 c^{3} \left (3 a^{6} x^{6}-10 a^{4} x^{4}+11 a^{2} x^{2}-4\right ) a}+\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )}{a \,c^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{a \,c^{3} \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (4 i \arcsin \left (a x \right )^{3}-6 \arcsin \left (a x \right )^{2} \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+6 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-3 \operatorname {polylog}\left (3, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )\right )}{3 a \,c^{3} \left (a^{2} x^{2}-1\right )}\) | \(535\) |
-1/6*(-c*(a^2*x^2-1))^(1/2)*(2*I*(-a^2*x^2+1)^(1/2)*a^2*x^2+2*a^3*x^3-2*I* (-a^2*x^2+1)^(1/2)-3*a*x)*arcsin(a*x)*(-6*I*arcsin(a*x)*a^4*x^4-6*arcsin(a *x)*(-a^2*x^2+1)^(1/2)*a^3*x^3+6*I*(-a^2*x^2+1)^(1/2)*a^3*x^3-6*a^4*x^4+6* arcsin(a*x)^2*a^2*x^2+12*I*arcsin(a*x)*a^2*x^2+9*arcsin(a*x)*(-a^2*x^2+1)^ (1/2)*a*x-6*I*(-a^2*x^2+1)^(1/2)*a*x+18*a^2*x^2-8*arcsin(a*x)^2-6*I*arcsin (a*x)-12)/c^3/(3*a^6*x^6-10*a^4*x^4+11*a^2*x^2-4)/a+2*(-c*(a^2*x^2-1))^(1/ 2)*(-a^2*x^2+1)^(1/2)/a/c^3/(a^2*x^2-1)*ln(I*a*x+(-a^2*x^2+1)^(1/2))-(-c*( a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/c^3/(a^2*x^2-1)*ln(1+(I*a*x+(-a^2*x ^2+1)^(1/2))^2)+1/3*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)*(4*I*arcsin( a*x)^3-6*arcsin(a*x)^2*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2)+6*I*arcsin(a*x)* polylog(2,-(I*a*x+(-a^2*x^2+1)^(1/2))^2)-3*polylog(3,-(I*a*x+(-a^2*x^2+1)^ (1/2))^2))/a/c^3/(a^2*x^2-1)
\[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
integral(-sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^3/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)
\[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Time = 0.52 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.27 \[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {1}{2} \, a {\left (\frac {1}{a^{4} c^{\frac {5}{2}} x^{2} - a^{2} c^{\frac {5}{2}}} + \frac {2 \, \log \left (a x + 1\right )}{a^{2} c^{\frac {5}{2}}} + \frac {2 \, \log \left (a x - 1\right )}{a^{2} c^{\frac {5}{2}}}\right )} \arcsin \left (a x\right )^{2} + \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a^{2} c x^{2} + c} c^{2}} + \frac {x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c}\right )} \arcsin \left (a x\right )^{3} \]
1/2*a*(1/(a^4*c^(5/2)*x^2 - a^2*c^(5/2)) + 2*log(a*x + 1)/(a^2*c^(5/2)) + 2*log(a*x - 1)/(a^2*c^(5/2)))*arcsin(a*x)^2 + 1/3*(2*x/(sqrt(-a^2*c*x^2 + c)*c^2) + x/((-a^2*c*x^2 + c)^(3/2)*c))*arcsin(a*x)^3
\[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]