Integrand size = 16, antiderivative size = 181 \[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx=-\frac {d \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}+\frac {e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2} \]
e*Ci(2*(a+b*arcsin(c*x))/b)*cos(2*a/b)/b^2/c^2-d*cos(a/b)*Si((a+b*arcsin(c *x))/b)/b^2/c+d*Ci((a+b*arcsin(c*x))/b)*sin(a/b)/b^2/c+e*Si(2*(a+b*arcsin( c*x))/b)*sin(2*a/b)/b^2/c^2-d*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))-e*x *(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))
Time = 0.89 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82 \[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx=\frac {-\frac {b c (d+e x) \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+e \log (a+b \arcsin (c x))+c d \left (\operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )-\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )\right )+e \left (\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\log (a+b \arcsin (c x))+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )\right )}{b^2 c^2} \]
(-((b*c*(d + e*x)*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])) + e*Log[a + b*Ar cSin[c*x]] + c*d*(CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - Cos[a/b]*SinIn tegral[a/b + ArcSin[c*x]]) + e*(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^2*c^2)
Time = 0.49 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5244, 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.
| \(\displaystyle \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx\) |
| \(\Big \downarrow \) 5244 |
| \(\displaystyle \int \left (\frac {d}{(a+b \arcsin (c x))^2}+\frac {e x}{(a+b \arcsin (c x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}\) |
-((d*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) - (e*x*Sqrt[1 - c^2*x^2 ])/(b*c*(a + b*ArcSin[c*x])) + (e*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSi n[c*x]))/b])/(b^2*c^2) + (d*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/( b^2*c) - (d*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c) + (e*Sin[ (2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^2)
3.1.23.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_Sy mbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; F reeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
Time = 0.15 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.42
| method | result | size |
| derivativedivides | \(\frac {-\frac {d \left (\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\sqrt {-c^{2} x^{2}+1}\, b \right )}{\left (a +b \arcsin \left (c x \right )\right ) b^{2}}+\frac {e \left (2 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +2 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b +2 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +2 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -\sin \left (2 \arcsin \left (c x \right )\right ) b \right )}{2 c \left (a +b \arcsin \left (c x \right )\right ) b^{2}}}{c}\) | \(257\) |
| default | \(\frac {-\frac {d \left (\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\sqrt {-c^{2} x^{2}+1}\, b \right )}{\left (a +b \arcsin \left (c x \right )\right ) b^{2}}+\frac {e \left (2 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +2 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b +2 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +2 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -\sin \left (2 \arcsin \left (c x \right )\right ) b \right )}{2 c \left (a +b \arcsin \left (c x \right )\right ) b^{2}}}{c}\) | \(257\) |
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+1/2/c*e*(2*arcsin(c*x )*Si(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*b+2*arcsin(c*x)*Ci(2*arcsin(c*x)+2*a/ b)*cos(2*a/b)*b+2*Si(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*a+2*Ci(2*arcsin(c*x)+ 2*a/b)*cos(2*a/b)*a-sin(2*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^2)
\[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {e x + d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx=\int \frac {d + e x}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {e x + d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
-(sqrt(c*x + 1)*sqrt(-c*x + 1)*(e*x + d) - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate((2*c^2*e*x^2 + c^2*d*x - e)*sqrt(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)*arctan2( 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. 554 vs. \(2 (177) = 354\).
Time = 0.36 (sec) , antiderivative size = 554, normalized size of antiderivative = 3.06 \[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx=\frac {2 \, b e \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {b c d \arcsin \left (c x\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {2 \, b e \arcsin \left (c x\right ) \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^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {b c d \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {2 \, a e \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {a c d \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {2 \, a e \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^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {a c d \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {\sqrt {-c^{2} x^{2} + 1} b c e x}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {b e \arcsin \left (c x\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {\sqrt {-c^{2} x^{2} + 1} b c d}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {a e \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} \]
2*b*e*arcsin(c*x)*cos(a/b)^2*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^2* arcsin(c*x) + a*b^2*c^2) + b*c*d*arcsin(c*x)*cos_integral(a/b + arcsin(c*x ))*sin(a/b)/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 2*b*e*arcsin(c*x)*cos(a/b) *sin(a/b)*sin_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2 *c^2) - b*c*d*arcsin(c*x)*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^ 2*arcsin(c*x) + a*b^2*c^2) + 2*a*e*cos(a/b)^2*cos_integral(2*a/b + 2*arcsi n(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + a*c*d*cos_integral(a/b + arcsi n(c*x))*sin(a/b)/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 2*a*e*cos(a/b)*sin(a/ b)*sin_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - a*c*d*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b ^2*c^2) - sqrt(-c^2*x^2 + 1)*b*c*e*x/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - b *e*arcsin(c*x)*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - sqrt(-c^2*x^2 + 1)*b*c*d/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - a*e*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2)
Timed out. \[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx=\int \frac {d+e\,x}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]