Integrand size = 25, antiderivative size = 82 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c}+\frac {\log (a+b \arcsin (c x))}{2 b c}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c} \] Output:
1/2*cos(2*a/b)*Ci(2*(a+b*arcsin(c*x))/b)/b/c+1/2*ln(a+b*arcsin(c*x))/b/c+1 /2*sin(2*a/b)*Si(2*(a+b*arcsin(c*x))/b)/b/c
Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+\log (a+b \arcsin (c x))+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{2 b c} \] Input:
Integrate[Sqrt[1 - c^2*x^2]/(a + b*ArcSin[c*x]),x]
Output:
(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c*x])] + Log[a + b*ArcSin[c*x]] + Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])])/(2*b*c)
Time = 0.47 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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.
\(\displaystyle \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx\) |
\(\Big \downarrow \) 5168 |
\(\displaystyle \frac {\int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )^2}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b c}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {\int \left (\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 (a+b \arcsin (c x))}+\frac {1}{2 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b c}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{2} \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{2} \log (a+b \arcsin (c x))}{b c}\) |
Input:
Int[Sqrt[1 - c^2*x^2]/(a + b*ArcSin[c*x]),x]
Output:
((Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/2 + Log[a + b*ArcSi n[c*x]]/2 + (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/2)/(b*c)
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]))
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]
Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )+\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )+\ln \left (a +b \arcsin \left (c x \right )\right )}{2 c b}\) | \(63\) |
Input:
int((-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
Output:
1/2/c*(Si(2*arcsin(c*x)+2*a/b)*sin(2*a/b)+Ci(2*arcsin(c*x)+2*a/b)*cos(2*a/ b)+ln(a+b*arcsin(c*x)))/b
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{b \arcsin \left (c x\right ) + a} \,d x } \] Input:
integrate((-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x, algorithm="fricas")
Output:
integral(sqrt(-c^2*x^2 + 1)/(b*arcsin(c*x) + a), x)
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \] Input:
integrate((-c**2*x**2+1)**(1/2)/(a+b*asin(c*x)),x)
Output:
Integral(sqrt(-(c*x - 1)*(c*x + 1))/(a + b*asin(c*x)), x)
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{b \arcsin \left (c x\right ) + a} \,d x } \] Input:
integrate((-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x, algorithm="maxima")
Output:
integrate(sqrt(-c^2*x^2 + 1)/(b*arcsin(c*x) + a), x)
Time = 0.15 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\frac {\cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c} + \frac {\cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c} - \frac {\operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{2 \, b c} + \frac {\log \left (b \arcsin \left (c x\right ) + a\right )}{2 \, b c} \] Input:
integrate((-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x, algorithm="giac")
Output:
cos(a/b)^2*cos_integral(2*a/b + 2*arcsin(c*x))/(b*c) + cos(a/b)*sin(a/b)*s in_integral(2*a/b + 2*arcsin(c*x))/(b*c) - 1/2*cos_integral(2*a/b + 2*arcs in(c*x))/(b*c) + 1/2*log(b*arcsin(c*x) + a)/(b*c)
Timed out. \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \] Input:
int((1 - c^2*x^2)^(1/2)/(a + b*asin(c*x)),x)
Output:
int((1 - c^2*x^2)^(1/2)/(a + b*asin(c*x)), x)
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\int \frac {\sqrt {-c^{2} x^{2}+1}}{\mathit {asin} \left (c x \right ) b +a}d x \] Input:
int((-c^2*x^2+1)^(1/2)/(a+b*asin(c*x)),x)
Output:
int(sqrt( - c**2*x**2 + 1)/(asin(c*x)*b + a),x)