Integrand size = 12, antiderivative size = 59 \[ \int (a+b \arcsin (c+d x))^2 \, dx=-2 b^2 x+\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^2}{d} \] Output:
-2*b^2*x+2*b*(1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x+c))/d+(d*x+c)*(a+b*arcsin (d*x+c))^2/d
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03 \[ \int (a+b \arcsin (c+d x))^2 \, dx=\frac {-2 b^2 (c+d x)+2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))+(c+d x) (a+b \arcsin (c+d x))^2}{d} \] Input:
Integrate[(a + b*ArcSin[c + d*x])^2,x]
Output:
(-2*b^2*(c + d*x) + 2*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (c + d*x)*(a + b*ArcSin[c + d*x])^2)/d
Time = 0.31 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5302, 5130, 5182, 24}
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.
\(\displaystyle \int (a+b \arcsin (c+d x))^2 \, dx\) |
\(\Big \downarrow \) 5302 |
\(\displaystyle \frac {\int (a+b \arcsin (c+d x))^2d(c+d x)}{d}\) |
\(\Big \downarrow \) 5130 |
\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^2-2 b \int \frac {(c+d x) (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^2-2 b \left (b \int 1d(c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )}{d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^2-2 b \left (b (c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )}{d}\) |
Input:
Int[(a + b*ArcSin[c + d*x])^2,x]
Output:
((c + d*x)*(a + b*ArcSin[c + d*x])^2 - 2*b*(b*(c + d*x) - Sqrt[1 - (c + d* x)^2]*(a + b*ArcSin[c + d*x])))/d
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
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]
Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.53
method | result | size |
parts | \(x \,a^{2}+\frac {b^{2} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{2}-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )}{d}+\frac {2 a b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(90\) |
derivativedivides | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{2}-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+2 a b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(92\) |
default | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{2}-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+2 a b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(92\) |
orering | \(\frac {\left (d x +c \right ) \left (a +b \arcsin \left (d x +c \right )\right )^{2}}{d}-\frac {2 \left (c d x +c^{2}-1\right ) \left (a +b \arcsin \left (d x +c \right )\right ) b}{d \sqrt {1-\left (d x +c \right )^{2}}}+\frac {x \left (d x +c +1\right ) \left (d x +c -1\right ) \left (\frac {2 b^{2} d^{2}}{1-\left (d x +c \right )^{2}}+\frac {2 \left (a +b \arcsin \left (d x +c \right )\right ) b \,d^{2} \left (d x +c \right )}{\left (1-\left (d x +c \right )^{2}\right )^{\frac {3}{2}}}\right )}{d^{2}}\) | \(134\) |
Input:
int((a+b*arcsin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
x*a^2+b^2/d*((d*x+c)*arcsin(d*x+c)^2-2*d*x-2*c+2*arcsin(d*x+c)*(1-(d*x+c)^ 2)^(1/2))+2*a*b/d*((d*x+c)*arcsin(d*x+c)+(1-(d*x+c)^2)^(1/2))
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.59 \[ \int (a+b \arcsin (c+d x))^2 \, dx=\frac {{\left (a^{2} - 2 \, b^{2}\right )} d x + {\left (b^{2} d x + b^{2} c\right )} \arcsin \left (d x + c\right )^{2} + 2 \, {\left (a b d x + a b c\right )} \arcsin \left (d x + c\right ) + 2 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (b^{2} \arcsin \left (d x + c\right ) + a b\right )}}{d} \] Input:
integrate((a+b*arcsin(d*x+c))^2,x, algorithm="fricas")
Output:
((a^2 - 2*b^2)*d*x + (b^2*d*x + b^2*c)*arcsin(d*x + c)^2 + 2*(a*b*d*x + a* b*c)*arcsin(d*x + c) + 2*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(b^2*arcsin(d* x + c) + a*b))/d
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (51) = 102\).
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.42 \[ \int (a+b \arcsin (c+d x))^2 \, dx=\begin {cases} a^{2} x + \frac {2 a b c \operatorname {asin}{\left (c + d x \right )}}{d} + 2 a b x \operatorname {asin}{\left (c + d x \right )} + \frac {2 a b \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + b^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - 2 b^{2} x + \frac {2 b^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asin}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*asin(d*x+c))**2,x)
Output:
Piecewise((a**2*x + 2*a*b*c*asin(c + d*x)/d + 2*a*b*x*asin(c + d*x) + 2*a* b*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/d + b**2*c*asin(c + d*x)**2/d + b* *2*x*asin(c + d*x)**2 - 2*b**2*x + 2*b**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/d, Ne(d, 0)), (x*(a + b*asin(c))**2, True))
\[ \int (a+b \arcsin (c+d x))^2 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((a+b*arcsin(d*x+c))^2,x, algorithm="maxima")
Output:
(x*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 2*d*integrat e(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*x*arctan2(d*x + c, sqrt(d*x + c + 1 )*sqrt(-d*x - c + 1))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x))*b^2 + a^2*x + 2*( (d*x + c)*arcsin(d*x + c) + sqrt(-(d*x + c)^2 + 1))*a*b/d
Time = 0.13 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.88 \[ \int (a+b \arcsin (c+d x))^2 \, dx=\frac {{\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right )^{2}}{d} + \frac {2 \, {\left (d x + c\right )} a b \arcsin \left (d x + c\right )}{d} + \frac {2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{2} \arcsin \left (d x + c\right )}{d} + \frac {{\left (d x + c\right )} a^{2}}{d} - \frac {2 \, {\left (d x + c\right )} b^{2}}{d} + \frac {2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b}{d} \] Input:
integrate((a+b*arcsin(d*x+c))^2,x, algorithm="giac")
Output:
(d*x + c)*b^2*arcsin(d*x + c)^2/d + 2*(d*x + c)*a*b*arcsin(d*x + c)/d + 2* sqrt(-(d*x + c)^2 + 1)*b^2*arcsin(d*x + c)/d + (d*x + c)*a^2/d - 2*(d*x + c)*b^2/d + 2*sqrt(-(d*x + c)^2 + 1)*a*b/d
Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.49 \[ \int (a+b \arcsin (c+d x))^2 \, dx=a^2\,x+\frac {b^2\,\left ({\mathrm {asin}\left (c+d\,x\right )}^2-2\right )\,\left (c+d\,x\right )}{d}+\frac {2\,a\,b\,\left (\sqrt {1-{\left (c+d\,x\right )}^2}+\mathrm {asin}\left (c+d\,x\right )\,\left (c+d\,x\right )\right )}{d}+\frac {2\,b^2\,\mathrm {asin}\left (c+d\,x\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d} \] Input:
int((a + b*asin(c + d*x))^2,x)
Output:
a^2*x + (b^2*(asin(c + d*x)^2 - 2)*(c + d*x))/d + (2*a*b*((1 - (c + d*x)^2 )^(1/2) + asin(c + d*x)*(c + d*x)))/d + (2*b^2*asin(c + d*x)*(1 - (c + d*x )^2)^(1/2))/d
Time = 0.15 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.22 \[ \int (a+b \arcsin (c+d x))^2 \, dx=\frac {\mathit {asin} \left (d x +c \right )^{2} b^{2} c +\mathit {asin} \left (d x +c \right )^{2} b^{2} d x +2 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \mathit {asin} \left (d x +c \right ) b^{2}+2 \mathit {asin} \left (d x +c \right ) a b c +2 \mathit {asin} \left (d x +c \right ) a b d x +2 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, a b +a^{2} d x -2 b^{2} c -2 b^{2} d x}{d} \] Input:
int((a+b*asin(d*x+c))^2,x)
Output:
(asin(c + d*x)**2*b**2*c + asin(c + d*x)**2*b**2*d*x + 2*sqrt( - c**2 - 2* c*d*x - d**2*x**2 + 1)*asin(c + d*x)*b**2 + 2*asin(c + d*x)*a*b*c + 2*asin (c + d*x)*a*b*d*x + 2*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*a*b + a**2*d *x - 2*b**2*c - 2*b**2*d*x)/d