3.7.69 \(\int \frac {d+e x^2}{a+b \arcsin (c x)} \, dx\) [669]

3.7.69.1 Optimal result

Integrand size = 18, antiderivative size = 179 \[ \int \frac {d+e x^2}{a+b \arcsin (c x)} \, dx=\frac {d \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b c}+\frac {e \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b c^3}-\frac {e \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b c^3}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b c}+\frac {e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b c^3}-\frac {e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b c^3} \]

output

d*Ci((a+b*arcsin(c*x))/b)*cos(a/b)/b/c+1/4*e*Ci((a+b*arcsin(c*x))/b)*cos(a 
/b)/b/c^3-1/4*e*Ci(3*(a+b*arcsin(c*x))/b)*cos(3*a/b)/b/c^3+d*Si((a+b*arcsi 
n(c*x))/b)*sin(a/b)/b/c+1/4*e*Si((a+b*arcsin(c*x))/b)*sin(a/b)/b/c^3-1/4*e 
*Si(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b/c^3
 

3.7.69.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.70 \[ \int \frac {d+e x^2}{a+b \arcsin (c x)} \, dx=\frac {\left (4 c^2 d+e\right ) \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )-e \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+4 c^2 d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{4 b c^3} \]

input

Integrate[(d + e*x^2)/(a + b*ArcSin[c*x]),x]
 

output

((4*c^2*d + e)*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - e*Cos[(3*a)/b]*Co 
sIntegral[3*(a/b + ArcSin[c*x])] + 4*c^2*d*Sin[a/b]*SinIntegral[a/b + ArcS 
in[c*x]] + e*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - e*Sin[(3*a)/b]*SinI 
ntegral[3*(a/b + ArcSin[c*x])])/(4*b*c^3)
 

3.7.69.3 Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 179, 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.

\(\displaystyle \int \frac {d+e x^2}{a+b \arcsin (c x)} \, dx\)

\(\Big \downarrow \) 5172

\(\displaystyle \int \left (\frac {d}{a+b \arcsin (c x)}+\frac {e x^2}{a+b \arcsin (c x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b c^3}-\frac {e \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b c^3}+\frac {e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b c^3}-\frac {e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b c^3}+\frac {d \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b c}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b c}\)

input

Int[(d + e*x^2)/(a + b*ArcSin[c*x]),x]
 

output

(d*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (e*Cos[a/b]*CosInt 
egral[(a + b*ArcSin[c*x])/b])/(4*b*c^3) - (e*Cos[(3*a)/b]*CosIntegral[(3*( 
a + b*ArcSin[c*x]))/b])/(4*b*c^3) + (d*Sin[a/b]*SinIntegral[(a + b*ArcSin[ 
c*x])/b])/(b*c) + (e*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b*c^3 
) - (e*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b*c^3)
 

3.7.69.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5172

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])
 

3.7.69.4 Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\frac {d \left (\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{b}+\frac {e \left (\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{4 c^{2} b}-\frac {e \left (\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )+\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )\right )}{4 c^{2} b}}{c}\)	\(148\)
default	\(\frac {\frac {d \left (\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{b}+\frac {e \left (\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{4 c^{2} b}-\frac {e \left (\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )+\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )\right )}{4 c^{2} b}}{c}\)	\(148\)

input

int((e*x^2+d)/(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
 

output

1/c*(d*(Si(arcsin(c*x)+a/b)*sin(a/b)+Ci(arcsin(c*x)+a/b)*cos(a/b))/b+1/4*e 
/c^2*(Si(arcsin(c*x)+a/b)*sin(a/b)+Ci(arcsin(c*x)+a/b)*cos(a/b))/b-1/4*e/c 
^2*(Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)+Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)) 
/b)
 

3.7.69.5 Fricas [F]

\[ \int \frac {d+e x^2}{a+b \arcsin (c x)} \, dx=\int { \frac {e x^{2} + d}{b \arcsin \left (c x\right ) + a} \,d x } \]

input

integrate((e*x^2+d)/(a+b*arcsin(c*x)),x, algorithm="fricas")
 

output

integral((e*x^2 + d)/(b*arcsin(c*x) + a), x)
 

3.7.69.6 Sympy [F]

\[ \int \frac {d+e x^2}{a+b \arcsin (c x)} \, dx=\int \frac {d + e x^{2}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]

input

integrate((e*x**2+d)/(a+b*asin(c*x)),x)
 

output

Integral((d + e*x**2)/(a + b*asin(c*x)), x)
 

3.7.69.7 Maxima [F]

\[ \int \frac {d+e x^2}{a+b \arcsin (c x)} \, dx=\int { \frac {e x^{2} + d}{b \arcsin \left (c x\right ) + a} \,d x } \]

input

integrate((e*x^2+d)/(a+b*arcsin(c*x)),x, algorithm="maxima")
 

output

integrate((e*x^2 + d)/(b*arcsin(c*x) + a), x)
 

3.7.69.8 Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.28 \[ \int \frac {d+e x^2}{a+b \arcsin (c x)} \, dx=-\frac {e \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {d \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} - \frac {e \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {d \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} + \frac {3 \, e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{3}} + \frac {e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, b c^{3}} + \frac {e \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{3}} + \frac {e \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, b c^{3}} \]

input

integrate((e*x^2+d)/(a+b*arcsin(c*x)),x, algorithm="giac")
 

output

-e*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + d*cos(a/b)*cos 
_integral(a/b + arcsin(c*x))/(b*c) - e*cos(a/b)^2*sin(a/b)*sin_integral(3* 
a/b + 3*arcsin(c*x))/(b*c^3) + d*sin(a/b)*sin_integral(a/b + arcsin(c*x))/ 
(b*c) + 3/4*e*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + 1/4*e 
*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b*c^3) + 1/4*e*sin(a/b)*sin_int 
egral(3*a/b + 3*arcsin(c*x))/(b*c^3) + 1/4*e*sin(a/b)*sin_integral(a/b + a 
rcsin(c*x))/(b*c^3)
 

3.7.69.9 Mupad [F(-1)]

Timed out. \[ \int \frac {d+e x^2}{a+b \arcsin (c x)} \, dx=\int \frac {e\,x^2+d}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]

input

int((d + e*x^2)/(a + b*asin(c*x)),x)
 

output

int((d + e*x^2)/(a + b*asin(c*x)), x)