Integrand size = 12, antiderivative size = 147 \[ \int (a+b \arcsin (c+d x))^n \, dx=-\frac {i e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c+d x))}{b}\right )}{2 d}+\frac {i e^{\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c+d x))}{b}\right )}{2 d} \]
-1/2*I*(a+b*arcsin(d*x+c))^n*GAMMA(1+n,-I*(a+b*arcsin(d*x+c))/b)/d/exp(I*a /b)/((-I*(a+b*arcsin(d*x+c))/b)^n)+1/2*I*exp(I*a/b)*(a+b*arcsin(d*x+c))^n* GAMMA(1+n,I*(a+b*arcsin(d*x+c))/b)/d/((I*(a+b*arcsin(d*x+c))/b)^n)
Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.88 \[ \int (a+b \arcsin (c+d x))^n \, dx=-\frac {i e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c+d x))}{b}\right )-e^{\frac {2 i a}{b}} \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )}{2 d} \]
((-1/2*I)*(a + b*ArcSin[c + d*x])^n*(Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/b]/(((-I)*(a + b*ArcSin[c + d*x]))/b)^n - (E^(((2*I)*a)/b)*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/((I*(a + b*ArcSin[c + d*x]))/b)^n))/( d*E^((I*a)/b))
Time = 0.38 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5302, 5134, 3042, 3788, 26, 2612}
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.
\(\displaystyle \int (a+b \arcsin (c+d x))^n \, dx\) |
\(\Big \downarrow \) 5302 |
\(\displaystyle \frac {\int (a+b \arcsin (c+d x))^nd(c+d x)}{d}\) |
\(\Big \downarrow \) 5134 |
\(\displaystyle \frac {\int (a+b \arcsin (c+d x))^n \cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )d(a+b \arcsin (c+d x))}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (a+b \arcsin (c+d x))^n \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}+\frac {\pi }{2}\right )d(a+b \arcsin (c+d x))}{b d}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle \frac {\frac {1}{2} i \int -i e^{-\frac {i (a-c-d x)}{b}} (a+b \arcsin (c+d x))^nd(a+b \arcsin (c+d x))-\frac {1}{2} i \int i e^{\frac {i (a-c-d x)}{b}} (a+b \arcsin (c+d x))^nd(a+b \arcsin (c+d x))}{b d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {1}{2} \int e^{-\frac {i (a-c-d x)}{b}} (a+b \arcsin (c+d x))^nd(a+b \arcsin (c+d x))+\frac {1}{2} \int e^{\frac {i (a-c-d x)}{b}} (a+b \arcsin (c+d x))^nd(a+b \arcsin (c+d x))}{b d}\) |
\(\Big \downarrow \) 2612 |
\(\displaystyle \frac {\frac {1}{2} i b e^{\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {i (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} i b e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i (a+b \arcsin (c+d x))}{b}\right )}{b d}\) |
(((-1/2*I)*b*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/b])/(E^((I*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n) + ((I/2)*b *E^((I*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d *x]))/b])/((I*(a + b*ArcSin[c + d*x]))/b)^n)/(b*d)
3.2.75.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) )^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && !IntegerQ[m]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [I/2 Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e , f, m}, x] && IntegerQ[2*k]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) Su bst[Int[x^n*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
\[\int \left (a +b \arcsin \left (d x +c \right )\right )^{n}d x\]
\[ \int (a+b \arcsin (c+d x))^n \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} \,d x } \]
\[ \int (a+b \arcsin (c+d x))^n \, dx=\int \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{n}\, dx \]
\[ \int (a+b \arcsin (c+d x))^n \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} \,d x } \]
\[ \int (a+b \arcsin (c+d x))^n \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} \,d x } \]
Timed out. \[ \int (a+b \arcsin (c+d x))^n \, dx=\int {\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^n \,d x \]