Integrand size = 18, antiderivative size = 244 \[ \int \frac {(d+e x)^2}{a+b \arcsin (c x)} \, dx=\frac {d^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b c^3}-\frac {e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b c^3}-\frac {d e \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )}{b c^2}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b c^3}+\frac {d e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{b c^2}-\frac {e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b c^3} \]
d^2*Ci(a/b+arcsin(c*x))*cos(a/b)/b/c+1/4*e^2*Ci(a/b+arcsin(c*x))*cos(a/b)/ b/c^3-1/4*e^2*Ci(3*a/b+3*arcsin(c*x))*cos(3*a/b)/b/c^3+d*e*cos(2*a/b)*Si(2 *a/b+2*arcsin(c*x))/b/c^2+d^2*Si(a/b+arcsin(c*x))*sin(a/b)/b/c+1/4*e^2*Si( a/b+arcsin(c*x))*sin(a/b)/b/c^3-d*e*Ci(2*a/b+2*arcsin(c*x))*sin(2*a/b)/b/c ^2-1/4*e^2*Si(3*a/b+3*arcsin(c*x))*sin(3*a/b)/b/c^3
Time = 0.40 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^2}{a+b \arcsin (c x)} \, dx=\frac {\left (4 c^2 d^2+e^2\right ) \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )-e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-4 c d e \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )+4 c^2 d^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+4 c d e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{4 b c^3} \]
((4*c^2*d^2 + e^2)*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - e^2*Cos[(3*a) /b]*CosIntegral[3*(a/b + ArcSin[c*x])] - 4*c*d*e*CosIntegral[2*(a/b + ArcS in[c*x])]*Sin[(2*a)/b] + 4*c^2*d^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + e^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 4*c*d*e*Cos[(2*a)/b]*SinI ntegral[2*(a/b + ArcSin[c*x])] - e^2*Sin[(3*a)/b]*SinIntegral[3*(a/b + Arc Sin[c*x])])/(4*b*c^3)
Time = 0.87 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5246, 7293, 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.
| \(\displaystyle \int \frac {(d+e x)^2}{a+b \arcsin (c x)} \, dx\) |
| \(\Big \downarrow \) 5246 |
| \(\displaystyle \frac {\int \frac {(c d+c e x)^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}d\arcsin (c x)}{c^3}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {c^2 \sqrt {1-c^2 x^2} d^2}{a+b \arcsin (c x)}+\frac {c e \sin (2 \arcsin (c x)) d}{a+b \arcsin (c x)}+\frac {c^2 e^2 x^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}\right )d\arcsin (c x)}{c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {c^2 d^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b}+\frac {c^2 d^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b}-\frac {c d e \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{b}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b}-\frac {e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b}+\frac {c d e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{b}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b}-\frac {e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b}}{c^3}\) |
((c^2*d^2*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/b + (e^2*Cos[a/b]*CosIn tegral[a/b + ArcSin[c*x]])/(4*b) - (e^2*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSin[c*x]])/(4*b) - (c*d*e*CosIntegral[(2*a)/b + 2*ArcSin[c*x]]*Sin[( 2*a)/b])/b + (c^2*d^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/b + (e^2*Si n[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(4*b) + (c*d*e*Cos[(2*a)/b]*SinInte gral[(2*a)/b + 2*ArcSin[c*x]])/b - (e^2*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(4*b))/c^3
3.1.17.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_.) + (e_.)*(x_))^(m_.), x_S ymbol] :> Simp[1/c^(m + 1) Subst[Int[(a + b*x)^n*Cos[x]*(c*d + e*Sin[x])^ m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
Time = 0.20 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {4 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{2} d^{2}+4 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{2} d^{2}+4 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) c d e -4 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) c d e -\sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) e^{2}-\cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) e^{2}+\sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) e^{2}+\cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) e^{2}}{4 c^{3} b}\) | \(206\) |
| default | \(\frac {4 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{2} d^{2}+4 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{2} d^{2}+4 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) c d e -4 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) c d e -\sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) e^{2}-\cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) e^{2}+\sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) e^{2}+\cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) e^{2}}{4 c^{3} b}\) | \(206\) |
1/4/c^3*(4*Si(arcsin(c*x)+a/b)*sin(a/b)*c^2*d^2+4*Ci(arcsin(c*x)+a/b)*cos( a/b)*c^2*d^2+4*cos(2*a/b)*Si(2*arcsin(c*x)+2*a/b)*c*d*e-4*Ci(2*arcsin(c*x) +2*a/b)*sin(2*a/b)*c*d*e-sin(3*a/b)*Si(3*arcsin(c*x)+3*a/b)*e^2-cos(3*a/b) *Ci(3*arcsin(c*x)+3*a/b)*e^2+sin(a/b)*Si(arcsin(c*x)+a/b)*e^2+cos(a/b)*Ci( arcsin(c*x)+a/b)*e^2)/b
\[ \int \frac {(d+e x)^2}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{b \arcsin \left (c x\right ) + a} \,d x } \]
\[ \int \frac {(d+e x)^2}{a+b \arcsin (c x)} \, dx=\int \frac {\left (d + e x\right )^{2}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]
\[ \int \frac {(d+e x)^2}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{b \arcsin \left (c x\right ) + a} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^2}{a+b \arcsin (c x)} \, dx=-\frac {e^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {d^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} - \frac {2 \, d e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c^{2}} - \frac {e^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {2 \, d e \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c^{2}} + \frac {d^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} + \frac {3 \, e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{3}} + \frac {e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, b c^{3}} + \frac {e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{3}} - \frac {d e \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c^{2}} + \frac {e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, b c^{3}} \]
-e^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + d^2*cos(a/b) *cos_integral(a/b + arcsin(c*x))/(b*c) - 2*d*e*cos(a/b)*cos_integral(2*a/b + 2*arcsin(c*x))*sin(a/b)/(b*c^2) - e^2*cos(a/b)^2*sin(a/b)*sin_integral( 3*a/b + 3*arcsin(c*x))/(b*c^3) + 2*d*e*cos(a/b)^2*sin_integral(2*a/b + 2*a rcsin(c*x))/(b*c^2) + d^2*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c) + 3/4*e^2*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + 1/4*e^2*co s(a/b)*cos_integral(a/b + arcsin(c*x))/(b*c^3) + 1/4*e^2*sin(a/b)*sin_inte gral(3*a/b + 3*arcsin(c*x))/(b*c^3) - d*e*sin_integral(2*a/b + 2*arcsin(c* x))/(b*c^2) + 1/4*e^2*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c^3)
Timed out. \[ \int \frac {(d+e x)^2}{a+b \arcsin (c x)} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]