Integrand size = 27, antiderivative size = 391 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=-\frac {e^{-\frac {i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c x))}{b}\right )}{8 c^2 \sqrt {1-c^2 x^2}}-\frac {e^{\frac {i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c x))}{b}\right )}{8 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{-1-n} e^{-\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 i (a+b \arcsin (c x))}{b}\right )}{8 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{-1-n} e^{\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 i (a+b \arcsin (c x))}{b}\right )}{8 c^2 \sqrt {1-c^2 x^2}} \]
-1/8*(a+b*arcsin(c*x))^n*GAMMA(1+n,-I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^ (1/2)/c^2/exp(I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)-1/8*e xp(I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2 +d)^(1/2)/c^2/((I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)-1/8*3^(-1-n)* (a+b*arcsin(c*x))^n*GAMMA(1+n,-3*I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(1/ 2)/c^2/exp(3*I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)-1/8*3^ (-1-n)*exp(3*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,3*I*(a+b*arcsin(c*x))/b) *(-c^2*d*x^2+d)^(1/2)/c^2/((I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)
Time = 0.67 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.70 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\frac {d e^{-\frac {3 i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (3 e^{\frac {2 i a}{b}} \left (-\left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c x))}{b}\right )-e^{\frac {2 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c x))}{b}\right )\right )-3^{-n} \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \left (\left (\frac {i (a+b \arcsin (c x))}{b}\right )^n \Gamma \left (1+n,-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^n \Gamma \left (1+n,\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{24 c^2 \sqrt {d \left (1-c^2 x^2\right )}} \]
(d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*(3*E^(((2*I)*a)/b)*(-(Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b]/(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (E^( ((2*I)*a)/b)*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/((I*(a + b*ArcSin[c* x]))/b)^n) - (((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-3*I)*(a + b*Ar cSin[c*x]))/b] + E^(((6*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b])/(3^n*((a + b*ArcSin[c*x])^2/b^2)^n))) /(24*c^2*E^(((3*I)*a)/b)*Sqrt[d*(1 - c^2*x^2)])
Time = 0.59 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5224, 25, 4906, 2009}
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.
\(\displaystyle \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int -(a+b \arcsin (c x))^n \cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )d(a+b \arcsin (c x))}{b c^2 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))^n \cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )d(a+b \arcsin (c x))}{b c^2 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \left (\frac {1}{4} \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n+\frac {1}{4} \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) (a+b \arcsin (c x))^n\right )d(a+b \arcsin (c x))}{b c^2 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {1}{8} b e^{-\frac {i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} b 3^{-n-1} e^{-\frac {3 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 i (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} b e^{\frac {i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {i (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} b 3^{-n-1} e^{\frac {3 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )}{b c^2 \sqrt {1-c^2 x^2}}\) |
(Sqrt[d - c^2*d*x^2]*(-1/8*(b*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b])/(E^((I*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (b* E^((I*a)/b)*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b]) /(8*((I*(a + b*ArcSin[c*x]))/b)^n) - (3^(-1 - n)*b*(a + b*ArcSin[c*x])^n*G amma[1 + n, ((-3*I)*(a + b*ArcSin[c*x]))/b])/(8*E^(((3*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (3^(-1 - n)*b*E^(((3*I)*a)/b)*(a + b*ArcSin[c*x] )^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b])/(8*((I*(a + b*ArcSin[c*x] ))/b)^n)))/(b*c^2*Sqrt[1 - c^2*x^2])
3.5.83.3.1 Defintions of rubi rules used
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x ^2)^p] Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int x \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{n}d x\]
\[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} x \,d x } \]
\[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\int x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{n}\, dx \]
\[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} x \,d x } \]
Exception generated. \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,\sqrt {d-c^2\,d\,x^2} \,d x \]