Integrand size = 29, antiderivative size = 183 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {i c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {d-c^2 d x^2}} \]
-I*c*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+2*b*c*(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)-I*b^2*c*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)-(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/d/x
Time = 0.69 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {1-c^2 x^2} \left (b^2 \left (i c x+\sqrt {1-c^2 x^2}\right ) \arcsin (c x)^2+2 b \arcsin (c x) \left (a \sqrt {1-c^2 x^2}-b c x \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+a \left (a \sqrt {1-c^2 x^2}-2 b c x \log (c x)\right )+i b^2 c x \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{x \sqrt {d-c^2 d x^2}} \]
-((Sqrt[1 - c^2*x^2]*(b^2*(I*c*x + Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 2*b* ArcSin[c*x]*(a*Sqrt[1 - c^2*x^2] - b*c*x*Log[1 - E^((2*I)*ArcSin[c*x])]) + a*(a*Sqrt[1 - c^2*x^2] - 2*b*c*x*Log[c*x]) + I*b^2*c*x*PolyLog[2, E^((2*I )*ArcSin[c*x])]))/(x*Sqrt[d - c^2*d*x^2]))
Time = 0.61 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.75, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 5186 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{x}dx}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b c \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\) |
-((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*PolyLog[2, E^((2*I)*ArcSin[c*x])])/ 4)))/Sqrt[d - c^2*d*x^2]
3.3.41.3.1 Defintions of rubi rules used
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_.)*((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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (187 ) = 374\).
Time = 0.21 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.33
| method | result | size |
| default | \(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{d x}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{\left (c^{2} x^{2}-1\right ) x d}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{d x \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )\) | \(427\) |
| parts | \(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{d x}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{\left (c^{2} x^{2}-1\right ) x d}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{d x \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )\) | \(427\) |
-a^2/d/x*(-c^2*d*x^2+d)^(1/2)+b^2*(-(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1) ^(1/2)*x*c+c^2*x^2-1)*arcsin(c*x)^2/(c^2*x^2-1)/x/d+2*I*(-c^2*x^2+1)^(1/2) *(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*(I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+ 1)^(1/2))+I*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+arcsin(c*x)^2+polyl og(2,-I*c*x-(-c^2*x^2+1)^(1/2))+polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)))*c)+2* a*b*(2*I*(-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-(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*arcsin(c* x)/(c^2*x^2-1)/x/d-(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*x^2-1) *ln((I*c*x+(-c^2*x^2+1)^(1/2))^2-1)*c)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{2}} \,d x } \]
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^4 - d*x^2), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{2} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{2}} \,d x } \]
-((-1)^(-2*c^2*d*x^2 + 2*d)*sqrt(d)*log(-2*c^2*d + 2*d/x^2) + sqrt(d)*log( x^2 - 1/c^2))*a*b*c/d + b^2*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2/(sqrt(c*x + 1)*sqrt(-c*x + 1)*x^2), x)/sqrt(d) - 2*sqrt(-c^2*d*x^ 2 + d)*a*b*arcsin(c*x)/(d*x) - sqrt(-c^2*d*x^2 + d)*a^2/(d*x)
Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^2\,\sqrt {d-c^2\,d\,x^2}} \,d x \]