\(\int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx\) [403]

Optimal result

Integrand size = 26, antiderivative size = 259 \[ \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^{1+n}}{2 b c (1+n) \sqrt {1-c^2 x^2}}-\frac {i 2^{-3-n} e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c x))}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i 2^{-3-n} e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c x))}{b}\right )}{c \sqrt {1-c^2 x^2}} \] Output:

1/2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^(1+n)/b/c/(1+n)/(-c^2*x^2+1)^(1 
/2)-I*2^(-3-n)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n*GAMMA(1+n,-2*I*(a+ 
b*arcsin(c*x))/b)/c/exp(2*I*a/b)/(-c^2*x^2+1)^(1/2)/((-I*(a+b*arcsin(c*x)) 
/b)^n)+I*2^(-3-n)*exp(2*I*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n*GA 
MMA(1+n,2*I*(a+b*arcsin(c*x))/b)/c/(-c^2*x^2+1)^(1/2)/((I*(a+b*arcsin(c*x) 
)/b)^n)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.70 \[ \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\frac {d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (\frac {4 a+4 b \arcsin (c x)}{b+b n}-i 2^{-n} e^{-\frac {2 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c x))}{b}\right )+i 2^{-n} e^{\frac {2 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c x))}{b}\right )\right )}{8 c \sqrt {d \left (1-c^2 x^2\right )}} \] Input:

Integrate[Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n,x]
 

Output:

(d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*((4*a + 4*b*ArcSin[c*x])/(b + b 
*n) - (I*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*E^(((2*I)*a)/b 
)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (I*E^(((2*I)*a)/b)*Gamma[1 + n, ((2* 
I)*(a + b*ArcSin[c*x]))/b])/(2^n*((I*(a + b*ArcSin[c*x]))/b)^n)))/(8*c*Sqr 
t[d*(1 - c^2*x^2)])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5168, 3042, 3793, 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.

\(\displaystyle \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx\)

\(\Big \downarrow \) 5168

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))^n \cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )d(a+b \arcsin (c x))}{b c \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))^n \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )^2d(a+b \arcsin (c x))}{b c \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \left (\frac {1}{2} \cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n+\frac {1}{2} (a+b \arcsin (c x))^n\right )d(a+b \arcsin (c x))}{b c \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {(a+b \arcsin (c x))^{n+1}}{2 (n+1)}-i b 2^{-n-3} e^{-\frac {2 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i (a+b \arcsin (c x))}{b}\right )+i b 2^{-n-3} e^{\frac {2 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i (a+b \arcsin (c x))}{b}\right )\right )}{b c \sqrt {1-c^2 x^2}}\)

Input:

Int[Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n,x]
 

Output:

(Sqrt[d - c^2*d*x^2]*((a + b*ArcSin[c*x])^(1 + n)/(2*(1 + n)) - (I*2^(-3 - 
 n)*b*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/ 
(E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (I*2^(-3 - n)*b*E^((( 
2*I)*a)/b)*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/ 
b])/((I*(a + b*ArcSin[c*x]))/b)^n))/(b*c*Sqrt[1 - c^2*x^2])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5168

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Subst[Int[ 
x^n*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, 0]
 

Maple [F]

\[\int \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{n}d x\]

Input:

int((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n,x)
 

Output:

int((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n,x)
 

Fricas [F]

\[ \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n,x, algorithm="fricas")
 

Output:

integral(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)^n, x)
 

Sympy [F]

\[ \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{n}\, dx \] Input:

integrate((-c**2*d*x**2+d)**(1/2)*(a+b*asin(c*x))**n,x)
 

Output:

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))**n, x)
 

Maxima [F]

\[ \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n,x, algorithm="maxima")
 

Output:

integrate(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)^n, x)
 

Giac [F(-2)]

Exception generated. \[ \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n,x, algorithm="giac")
 

Output:

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
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,\sqrt {d-c^2\,d\,x^2} \,d x \] Input:

int((a + b*asin(c*x))^n*(d - c^2*d*x^2)^(1/2),x)
 

Output:

int((a + b*asin(c*x))^n*(d - c^2*d*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\sqrt {d}\, \left (\int \left (\mathit {asin} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}d x \right ) \] Input:

int((-c^2*d*x^2+d)^(1/2)*(a+b*asin(c*x))^n,x)
 

Output:

sqrt(d)*int((asin(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1),x)