\(\int (a+b \arcsin (c+d x))^n \, dx\) [175]

Optimal result

Integrand size = 12, antiderivative size = 147 \[ \int (a+b \arcsin (c+d x))^n \, dx=-\frac {i e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c+d x))}{b}\right )}{2 d}+\frac {i e^{\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c+d x))}{b}\right )}{2 d} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4887, 4719, 3388, 2212} \[ \int (a+b \arcsin (c+d x))^n \, dx=\frac {i e^{\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {i (a+b \arcsin (c+d x))}{b}\right )}{2 d}-\frac {i e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i (a+b \arcsin (c+d x))}{b}\right )}{2 d} \]

[In]

Int[(a + b*ArcSin[c + d*x])^n,x]

[Out]

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

Rule 2212

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]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[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]

Rule 4719

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[-a/b + x/b], x], x,
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 4887

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSin[x])^n, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b \arcsin (x))^n \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int x^n \cos \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{b d} \\ & = \frac {\text {Subst}\left (\int e^{-i \left (\frac {a}{b}-\frac {x}{b}\right )} x^n \, dx,x,a+b \arcsin (c+d x)\right )}{2 b d}+\frac {\text {Subst}\left (\int e^{i \left (\frac {a}{b}-\frac {x}{b}\right )} x^n \, dx,x,a+b \arcsin (c+d x)\right )}{2 b 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}+\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.88 \[ \int (a+b \arcsin (c+d x))^n \, dx=-\frac {i e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c+d x))}{b}\right )-e^{\frac {2 i a}{b}} \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )}{2 d} \]

[In]

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

[Out]

((-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))

Maple [F]

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

[In]

int((a+b*arcsin(d*x+c))^n,x)

[Out]

int((a+b*arcsin(d*x+c))^n,x)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate((a+b*asin(d*x+c))**n,x)

[Out]

Integral((a + b*asin(c + d*x))**n, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*asin(c + d*x))^n,x)

[Out]

int((a + b*asin(c + d*x))^n, x)