Integrand size = 28, antiderivative size = 82 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx=-\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c^3}+\frac {\log (a+b \arcsin (c x))}{2 b c^3}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c^3} \] Output:
-1/2*cos(2*a/b)*Ci(2*(a+b*arcsin(c*x))/b)/b/c^3+1/2*ln(a+b*arcsin(c*x))/b/ c^3-1/2*sin(2*a/b)*Si(2*(a+b*arcsin(c*x))/b)/b/c^3
Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx=-\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\log (a+b \arcsin (c x))+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{2 b c^3} \] Input:
Integrate[x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]
Output:
-1/2*(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])])/(b*c^3)
Time = 0.45 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5224, 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.
\(\displaystyle \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle \frac {\int \frac {\sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b c^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )^2}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b c^3}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {\int \left (\frac {1}{2 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\frac {1}{2} \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{2} \log (a+b \arcsin (c x))}{b c^3}\) |
Input:
Int[x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]
Output:
(-1/2*(Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b]) + Log[a + b*Ar cSin[c*x]]/2 - (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/2)/(b *c^3)
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_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x ^2)^p] Subst[Int[x^n*Sin[-a/b + x/b]^m*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 + 2, 0] && IGtQ[m, 0]
Time = 0.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {-\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )-\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )+\ln \left (a +b \arcsin \left (c x \right )\right )}{2 c^{3} b}\) | \(65\) |
Input:
int(x^2/(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
Output:
1/2/c^3*(-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
\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx=\int { \frac {x^{2}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \arcsin \left (c x\right ) + a\right )}} \,d x } \] Input:
integrate(x^2/(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x, algorithm="fricas")
Output:
integral(-sqrt(-c^2*x^2 + 1)*x^2/(a*c^2*x^2 + (b*c^2*x^2 - b)*arcsin(c*x) - a), x)
\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx=\int \frac {x^{2}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \] Input:
integrate(x**2/(-c**2*x**2+1)**(1/2)/(a+b*asin(c*x)),x)
Output:
Integral(x**2/(sqrt(-(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))), x)
\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx=\int { \frac {x^{2}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \arcsin \left (c x\right ) + a\right )}} \,d x } \] Input:
integrate(x^2/(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x)),x, algorithm="maxima")
Output:
integrate(x^2/(sqrt(-c^2*x^2 + 1)*(b*arcsin(c*x) + a)), x)
Time = 0.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.27 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx=-\frac {\cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c^{3}} - \frac {\cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {\operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{2 \, b c^{3}} + \frac {\log \left (b \arcsin \left (c x\right ) + a\right )}{2 \, b c^{3}} \] Input:
integrate(x^2/(-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^3) - cos(a/b)*sin(a/b )*sin_integral(2*a/b + 2*arcsin(c*x))/(b*c^3) + 1/2*cos_integral(2*a/b + 2 *arcsin(c*x))/(b*c^3) + 1/2*log(b*arcsin(c*x) + a)/(b*c^3)
Timed out. \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx=\int \frac {x^2}{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \] Input:
int(x^2/((a + b*asin(c*x))*(1 - c^2*x^2)^(1/2)),x)
Output:
int(x^2/((a + b*asin(c*x))*(1 - c^2*x^2)^(1/2)), x)
\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx=\int \frac {x^{2}}{\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b +\sqrt {-c^{2} x^{2}+1}\, a}d x \] Input:
int(x^2/(-c^2*x^2+1)^(1/2)/(a+b*asin(c*x)),x)
Output:
int(x**2/(sqrt( - c**2*x**2 + 1)*asin(c*x)*b + sqrt( - c**2*x**2 + 1)*a),x )