Integrand size = 29, antiderivative size = 319 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x}+\frac {4 b c^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{3 \sqrt {d-c^2 d x^2}} \]
-1/3*b^2*c^2*(-c^2*x^2+1)/x/(-c^2*d*x^2+d)^(1/2)-1/3*b*c*(a+b*arcsin(c*x)) *(-c^2*x^2+1)^(1/2)/x^2/(-c^2*d*x^2+d)^(1/2)-2/3*I*c^3*(a+b*arcsin(c*x))^2 *(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+4/3*b*c^3*(a+b*arcsin(c*x))*ln(1- (I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-2/3* I*b^2*c^3*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/(-c^2 *d*x^2+d)^(1/2)-1/3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/d/x^3-2/3*c^2 *(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/d/x
Time = 0.76 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {1-c^2 x^2} \left (a b c x+a^2 \sqrt {1-c^2 x^2}+2 a^2 c^2 x^2 \sqrt {1-c^2 x^2}+b^2 c^2 x^2 \sqrt {1-c^2 x^2}+b^2 \left (2 i c^3 x^3+\sqrt {1-c^2 x^2}+2 c^2 x^2 \sqrt {1-c^2 x^2}\right ) \arcsin (c x)^2-b \arcsin (c x) \left (-b c x-2 a \sqrt {1-c^2 x^2} \left (1+2 c^2 x^2\right )+4 b c^3 x^3 \log \left (1-e^{2 i \arcsin (c x)}\right )\right )-4 a b c^3 x^3 \log (c x)+2 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{3 x^3 \sqrt {d-c^2 d x^2}} \]
-1/3*(Sqrt[1 - c^2*x^2]*(a*b*c*x + a^2*Sqrt[1 - c^2*x^2] + 2*a^2*c^2*x^2*S qrt[1 - c^2*x^2] + b^2*c^2*x^2*Sqrt[1 - c^2*x^2] + b^2*((2*I)*c^3*x^3 + Sq rt[1 - c^2*x^2] + 2*c^2*x^2*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 - b*ArcSin[c* x]*(-(b*c*x) - 2*a*Sqrt[1 - c^2*x^2]*(1 + 2*c^2*x^2) + 4*b*c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x])]) - 4*a*b*c^3*x^3*Log[c*x] + (2*I)*b^2*c^3*x^3*Poly Log[2, E^((2*I)*ArcSin[c*x])]))/(x^3*Sqrt[d - c^2*d*x^2])
Time = 1.20 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.80, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {5204, 5138, 242, 5186, 5136, 3042, 25, 4200, 25, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}}dx+\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x^3}dx}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}}dx+\frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {1}{2} b c \int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx-\frac {a+b \arcsin (c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}}dx+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {a+b \arcsin (c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 5186 |
| \(\displaystyle \frac {2}{3} c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x}dx}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {a+b \arcsin (c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle \frac {2}{3} c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {a+b \arcsin (c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} c^2 \left (\frac {2 b c \sqrt {1-c^2 x^2} \int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {a+b \arcsin (c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {2 b c \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {a+b \arcsin (c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {a+b \arcsin (c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {a+b \arcsin (c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {a+b \arcsin (c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {a+b \arcsin (c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\right )+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {a+b \arcsin (c x)}{2 x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 d x^3}\) |
-1/3*(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(d*x^3) + (2*b*c*Sqrt[1 - c^2*x^2]*(-1/2*(b*c*Sqrt[1 - c^2*x^2])/x - (a + b*ArcSin[c*x])/(2*x^2)))/ (3*Sqrt[d - c^2*d*x^2]) + (2*c^2*(-((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x ])^2)/(d*x)) + (2*b*c*Sqrt[1 - c^2*x^2]*(((-1/2*I)*(a + b*ArcSin[c*x])^2)/ b - (2*I)*((I/2)*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])] + (b*P olyLog[2, E^((2*I)*ArcSin[c*x])])/4)))/Sqrt[d - c^2*d*x^2]))/3
3.3.43.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
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[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] , x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[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_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 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*A rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* c*(n/(f*(m + 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] && ILtQ[m, -1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2318 vs. \(2 (301 ) = 602\).
Time = 0.26 (sec) , antiderivative size = 2319, normalized size of antiderivative = 7.27
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2319\) |
| parts | \(\text {Expression too large to display}\) | \(2319\) |
-4/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*(-c^2*x^2+ 1)*arcsin(c*x)*c^6-2*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/ d*x^2*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2*c^5-2/3*I*b^2*(-d*(c^2*x^2-1))^(1/2 )/(3*c^4*x^4-2*c^2*x^2-1)/d*x*(-c^2*x^2+1)*arcsin(c*x)*c^4+a*b*(-d*(c^2*x^ 2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*c^3*(-c^2*x^2+1)^(1/2)+2/3*a*b*(-d*( c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x^3*arcsin(c*x)+4/3*b^2*(-d*(c ^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x*arcsin(c*x)^2*c^2-1/3*I*b^2*( -d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*(-c^2*x^2+1)^(1/2)*c^3-1/3 *b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*c^6+1/3*b^2*(-d* (c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x^3*arcsin(c*x)^2-2/3*b^2*(-d *(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^5*c^8+a^2*(-1/3/d/x^3*(-c^ 2*d*x^2+d)^(1/2)-2/3*c^2/d/x*(-c^2*d*x^2+d)^(1/2))-4*I*a*b*(-d*(c^2*x^2-1) )^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^2*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^5-4 /3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*(-c^2*x^2+1) *c^6-2/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x*(-c^2*x^ 2+1)*c^4+8/3*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1) *arcsin(c*x)*c^3-2/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/ d*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2*c^3-4/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3 *c^4*x^4-2*c^2*x^2-1)/d*x^5*arcsin(c*x)*c^8+2/3*I*b^2*(-d*(c^2*x^2-1))^(1/ 2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*arcsin(c*x)*c^6-I*b^2*(-d*(c^2*x^2-1))...
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{4}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^ 2)/(c^2*d*x^6 - d*x^4), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{4}} \,d x } \]
1/3*(4*c^2*log(x)/sqrt(d) - 1/(sqrt(d)*x^2))*a*b*c - 2/3*a*b*(2*sqrt(-c^2* d*x^2 + d)*c^2/(d*x) + sqrt(-c^2*d*x^2 + d)/(d*x^3))*arcsin(c*x) - 1/3*a^2 *(2*sqrt(-c^2*d*x^2 + d)*c^2/(d*x) + sqrt(-c^2*d*x^2 + d)/(d*x^3)) + b^2*i ntegrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2/(sqrt(c*x + 1)*sqrt( -c*x + 1)*x^4), x)/sqrt(d)
Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,\sqrt {d-c^2\,d\,x^2}} \,d x \]