Integrand size = 23, antiderivative size = 141 \[ \int \frac {(c e+d e x)^2}{a+b \arcsin (c+d x)} \, dx=\frac {e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{4 b d}-\frac {e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{4 b d}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{4 b d}-\frac {e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{4 b d} \]
1/4*e^2*Ci((a+b*arcsin(d*x+c))/b)*cos(a/b)/b/d-1/4*e^2*Ci(3*(a+b*arcsin(d* x+c))/b)*cos(3*a/b)/b/d+1/4*e^2*Si((a+b*arcsin(d*x+c))/b)*sin(a/b)/b/d-1/4 *e^2*Si(3*(a+b*arcsin(d*x+c))/b)*sin(3*a/b)/b/d
Time = 0.39 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.72 \[ \int \frac {(c e+d e x)^2}{a+b \arcsin (c+d x)} \, dx=\frac {e^2 \left (\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right )-\cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )-\sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )}{4 b d} \]
(e^2*(Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] - Cos[(3*a)/b]*CosIntegr al[3*(a/b + ArcSin[c + d*x])] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x] ] - Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])]))/(4*b*d)
Time = 0.43 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.82, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5304, 27, 5146, 4906, 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.
| \(\displaystyle \int \frac {(c e+d e x)^2}{a+b \arcsin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {e^2 (c+d x)^2}{a+b \arcsin (c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int \frac {(c+d x)^2}{a+b \arcsin (c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5146 |
| \(\displaystyle \frac {e^2 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b d}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {e^2 \int \left (\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{4 (a+b \arcsin (c+d x))}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{4 (a+b \arcsin (c+d x))}\right )d(a+b \arcsin (c+d x))}{b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{4} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )-\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{4} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )-\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )\right )}{b d}\) |
(e^2*((Cos[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/4 - (Cos[(3*a)/b]* CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/4 + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/4 - (Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/4))/(b*d)
3.3.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[1 /(b*c^(m + 1)) Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.32 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.73
| method | result | size |
| derivativedivides | \(-\frac {e^{2} \left (\operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )+\operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-\operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )-\operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{4 d b}\) | \(103\) |
| default | \(-\frac {e^{2} \left (\operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )+\operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-\operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )-\operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{4 d b}\) | \(103\) |
-1/4/d*e^2*(Ci(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)+Si(3*arcsin(d*x+c)+3*a/b) *sin(3*a/b)-Si(arcsin(d*x+c)+a/b)*sin(a/b)-Ci(arcsin(d*x+c)+a/b)*cos(a/b)) /b
\[ \int \frac {(c e+d e x)^2}{a+b \arcsin (c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{b \arcsin \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {(c e+d e x)^2}{a+b \arcsin (c+d x)} \, dx=e^{2} \left (\int \frac {c^{2}}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx\right ) \]
e**2*(Integral(c**2/(a + b*asin(c + d*x)), x) + Integral(d**2*x**2/(a + b* asin(c + d*x)), x) + Integral(2*c*d*x/(a + b*asin(c + d*x)), x))
\[ \int \frac {(c e+d e x)^2}{a+b \arcsin (c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{b \arcsin \left (d x + c\right ) + a} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.44 \[ \int \frac {(c e+d e x)^2}{a+b \arcsin (c+d x)} \, dx=-\frac {e^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{b d} - \frac {e^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{b d} + \frac {3 \, e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} + \frac {e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{4 \, b d} + \frac {e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} + \frac {e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{4 \, b d} \]
-e^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(d*x + c))/(b*d) - e^2*cos(a/ b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b*d) + 3/4*e^2*cos( a/b)*cos_integral(3*a/b + 3*arcsin(d*x + c))/(b*d) + 1/4*e^2*cos(a/b)*cos_ integral(a/b + arcsin(d*x + c))/(b*d) + 1/4*e^2*sin(a/b)*sin_integral(3*a/ b + 3*arcsin(d*x + c))/(b*d) + 1/4*e^2*sin(a/b)*sin_integral(a/b + arcsin( d*x + c))/(b*d)
Timed out. \[ \int \frac {(c e+d e x)^2}{a+b \arcsin (c+d x)} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]