\(\int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx\) [23]

Optimal result

Integrand size = 16, antiderivative size = 181 \[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx=-\frac {d \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}+\frac {e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2} \] Output:

-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))+e*cos(2*a/b)*Ci(2*(a+b*arcsin(c*x))/b)/b^2/c^2+d*Ci((a+b*ar 
csin(c*x))/b)*sin(a/b)/b^2/c-d*cos(a/b)*Si((a+b*arcsin(c*x))/b)/b^2/c+e*si 
n(2*a/b)*Si(2*(a+b*arcsin(c*x))/b)/b^2/c^2
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82 \[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx=\frac {-\frac {b c (d+e x) \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+e \log (a+b \arcsin (c x))+c d \left (\operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )-\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )\right )+e \left (\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\log (a+b \arcsin (c x))+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )\right )}{b^2 c^2} \] Input:

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

Output:

(-((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)
 

Rubi [A] (verified)

Time = 0.54 (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

\(\displaystyle \frac {e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}\)

Input:

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

Output:

-((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)
 

Defintions of rubi rules used

rule 2009

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

rule 5244

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]
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(\frac {\frac {d \left (\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b -\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a -\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\sqrt {-c^{2} x^{2}+1}\, b \right )}{\left (a +b \arcsin \left (c x \right )\right ) b^{2}}-\frac {\sin \left (2 \arcsin \left (c x \right )\right ) e}{2 b c \left (a +b \arcsin \left (c x \right )\right )}+\frac {e \left (\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )+\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )\right )}{c \,b^{2}}}{c}\)	\(202\)
default	\(\frac {\frac {d \left (\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b -\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a -\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\sqrt {-c^{2} x^{2}+1}\, b \right )}{\left (a +b \arcsin \left (c x \right )\right ) b^{2}}-\frac {\sin \left (2 \arcsin \left (c x \right )\right ) e}{2 b c \left (a +b \arcsin \left (c x \right )\right )}+\frac {e \left (\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )+\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )\right )}{c \,b^{2}}}{c}\)	\(202\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(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)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (177) = 354\).

Time = 0.18 (sec) , antiderivative size = 554, normalized size of antiderivative = 3.06 \[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {d+e x}{(a+b \arcsin (c x))^2} \, dx=\left (\int \frac {x}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) e +\left (\int \frac {1}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) d \] Input:

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

Output:

int(x/(asin(c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)*e + int(1/(asin(c*x) 
**2*b**2 + 2*asin(c*x)*a*b + a**2),x)*d