Integrand size = 18, antiderivative size = 249 \[ \int \frac {d+e x^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {d \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {d \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}+\frac {e \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 c^3}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3} \]
-d*cos(a/b)*Si((a+b*arcsin(c*x))/b)/b^2/c-1/4*e*cos(a/b)*Si((a+b*arcsin(c* x))/b)/b^2/c^3+3/4*e*cos(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b^2/c^3+d*Ci((a+ b*arcsin(c*x))/b)*sin(a/b)/b^2/c+1/4*e*Ci((a+b*arcsin(c*x))/b)*sin(a/b)/b^ 2/c^3-3/4*e*Ci(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2/c^3-d*(-c^2*x^2+1)^(1 /2)/b/c/(a+b*arcsin(c*x))-e*x^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))
Time = 0.90 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.77 \[ \int \frac {d+e x^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {\frac {4 b c^2 d \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {4 b c^2 e x^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}-\left (4 c^2 d+e\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )+3 e \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+4 c^2 d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{4 b^2 c^3} \]
-1/4*((4*b*c^2*d*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (4*b*c^2*e*x^2*S qrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) - (4*c^2*d + e)*CosIntegral[a/b + Ar cSin[c*x]]*Sin[a/b] + 3*e*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + 4*c^2*d*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + e*Cos[a/b]*SinIntegral [a/b + ArcSin[c*x]] - 3*e*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])]) /(b^2*c^3)
Time = 0.59 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5172, 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.
| \(\displaystyle \int \frac {d+e x^2}{(a+b \arcsin (c x))^2} \, dx\) |
| \(\Big \downarrow \) 5172 |
| \(\displaystyle \int \left (\frac {d}{(a+b \arcsin (c x))^2}+\frac {e x^2}{(a+b \arcsin (c x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3}-\frac {e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3}+\frac {d \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}\) |
-((d*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) - (e*x^2*Sqrt[1 - c^2*x ^2])/(b*c*(a + b*ArcSin[c*x])) + (d*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin [a/b])/(b^2*c) + (e*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(4*b^2*c^ 3) - (3*e*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(4*b^2*c^3) - (d*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c) - (e*Cos[a/b]*S inIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2*c^3) + (3*e*Cos[(3*a)/b]*SinInte gral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2*c^3)
3.7.78.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G tQ[p, 0] || IGtQ[n, 0])
Time = 0.17 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.51
| method | result | size |
| derivativedivides | \(\frac {-\frac {d \left (\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\sqrt {-c^{2} x^{2}+1}\, b \right )}{\left (a +b \arcsin \left (c x \right )\right ) b^{2}}-\frac {e \left (\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\sqrt {-c^{2} x^{2}+1}\, b \right )}{4 c^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}+\frac {e \left (3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b -3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +\cos \left (3 \arcsin \left (c x \right )\right ) b \right )}{4 c^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}}{c}\) | \(376\) |
| default | \(\frac {-\frac {d \left (\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\sqrt {-c^{2} x^{2}+1}\, b \right )}{\left (a +b \arcsin \left (c x \right )\right ) b^{2}}-\frac {e \left (\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\sqrt {-c^{2} x^{2}+1}\, b \right )}{4 c^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}+\frac {e \left (3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b -3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +\cos \left (3 \arcsin \left (c x \right )\right ) b \right )}{4 c^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}}{c}\) | \(376\) |
1/c*(-d*(arcsin(c*x)*Si(arcsin(c*x)+a/b)*cos(a/b)*b-arcsin(c*x)*Ci(arcsin( c*x)+a/b)*sin(a/b)*b+Si(arcsin(c*x)+a/b)*cos(a/b)*a-Ci(arcsin(c*x)+a/b)*si n(a/b)*a+(-c^2*x^2+1)^(1/2)*b)/(a+b*arcsin(c*x))/b^2-1/4*e/c^2*(arcsin(c*x )*Si(arcsin(c*x)+a/b)*cos(a/b)*b-arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)* b+Si(arcsin(c*x)+a/b)*cos(a/b)*a-Ci(arcsin(c*x)+a/b)*sin(a/b)*a+(-c^2*x^2+ 1)^(1/2)*b)/(a+b*arcsin(c*x))/b^2+1/4*e/c^2*(3*arcsin(c*x)*Si(3*arcsin(c*x )+3*a/b)*cos(3*a/b)*b-3*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b+3 *Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a-3*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b) *a+cos(3*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^2)
\[ \int \frac {d+e x^2}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {e x^{2} + d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {d+e x^2}{(a+b \arcsin (c x))^2} \, dx=\int \frac {d + e x^{2}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {d+e x^2}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {e x^{2} + d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
-((e*x^2 + d)*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate((3*c^2*e*x^3 + (c^2*d - 2*e)*x)*sq rt(c*x + 1)*sqrt(-c*x + 1)/(a*b*c^3*x^2 - a*b*c + (b^2*c^3*x^2 - b^2*c)*ar ctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x))/(b^2*c*arctan2(c*x, sqrt(c* x + 1)*sqrt(-c*x + 1)) + a*b*c)
Leaf count of result is larger than twice the leaf count of optimal. 891 vs. \(2 (237) = 474\).
Time = 0.36 (sec) , antiderivative size = 891, normalized size of antiderivative = 3.58 \[ \int \frac {d+e x^2}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \]
-3*b*e*arcsin(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b) /(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + b*c^2*d*arcsin(c*x)*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 3*b*e*arcsin(c *x)*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - b*c^2*d*arcsin(c*x)*cos(a/b)*sin_integral(a/b + arcsin(c*x))/ (b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 3*a*e*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + a*c^2*d*cos_in tegral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 3*a *e*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a *b^2*c^3) - a*c^2*d*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^3*arcs in(c*x) + a*b^2*c^3) + 3/4*b*e*arcsin(c*x)*cos_integral(3*a/b + 3*arcsin(c *x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/4*b*e*arcsin(c*x)*cos_ integral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 9 /4*b*e*arcsin(c*x)*cos(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^3*a rcsin(c*x) + a*b^2*c^3) - 1/4*b*e*arcsin(c*x)*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - sqrt(-c^2*x^2 + 1)*b*c^2* d/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 3/4*a*e*cos_integral(3*a/b + 3*arcsi n(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/4*a*e*cos_integral( a/b + arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 9/4*a*e*co s(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2...
Timed out. \[ \int \frac {d+e x^2}{(a+b \arcsin (c x))^2} \, dx=\int \frac {e\,x^2+d}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]