\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^2} \, dx\) [58]

Optimal result

Integrand size = 27, antiderivative size = 110 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^2} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}-\frac {c \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log (x)}{\sqrt {1-c^2 x^2}} \] Output:

-(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x-1/2*c*(-c^2*d*x^2+d)^(1/2)*(a+b* 
arcsin(c*x))^2/b/(-c^2*x^2+1)^(1/2)+b*c*(-c^2*d*x^2+d)^(1/2)*ln(x)/(-c^2*x 
^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^2} \, dx=-\frac {a \sqrt {-d \left (-1+c^2 x^2\right )}}{x}+a c \sqrt {d} \arctan \left (\frac {c x \sqrt {-d \left (-1+c^2 x^2\right )}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-\frac {b c \sqrt {d \left (1-c^2 x^2\right )} \left (\frac {2 \sqrt {1-c^2 x^2} \arcsin (c x)}{c x}+\arcsin (c x)^2-2 \log (c x)\right )}{2 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

-((a*Sqrt[-(d*(-1 + c^2*x^2))])/x) + a*c*Sqrt[d]*ArcTan[(c*x*Sqrt[-(d*(-1 
+ c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))] - (b*c*Sqrt[d*(1 - c^2*x^2)]*((2*S 
qrt[1 - c^2*x^2]*ArcSin[c*x])/(c*x) + ArcSin[c*x]^2 - 2*Log[c*x]))/(2*Sqrt 
[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5196, 14, 5152}

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.

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

\(\Big \downarrow \) 5196

\(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \int \frac {1}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}\)

\(\Big \downarrow \) 14

\(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {c \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

-((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x) - (c*Sqrt[d - c^2*d*x^2]*(a 
 + b*ArcSin[c*x])^2)/(2*b*Sqrt[1 - c^2*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Lo 
g[x])/Sqrt[1 - c^2*x^2]
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a 
 + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d 
+ e, 0] && NeQ[n, -1]
 

rule 5196

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
(e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS 
in[c*x])^n/(f*(m + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*Simp[Sqrt[d + e*x^ 
2]/Sqrt[1 - c^2*x^2]]   Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], 
x] + Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int 
[(f*x)^(m + 2)*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[ 
{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.71

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}-a \,c^{2} x \sqrt {-c^{2} d \,x^{2}+d}-\frac {a \,c^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} c}{2 c^{2} x^{2}-2}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c}{c^{2} x^{2}-1}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{\left (c^{2} x^{2}-1\right ) x}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{c^{2} x^{2}-1}\right )\)	\(298\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}-a \,c^{2} x \sqrt {-c^{2} d \,x^{2}+d}-\frac {a \,c^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} c}{2 c^{2} x^{2}-2}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c}{c^{2} x^{2}-1}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{\left (c^{2} x^{2}-1\right ) x}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{c^{2} x^{2}-1}\right )\)	\(298\)

Input:

int((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^2,x,method=_RETURNVERBOSE)
 

Output:

-a/d/x*(-c^2*d*x^2+d)^(3/2)-a*c^2*x*(-c^2*d*x^2+d)^(1/2)-a*c^2*d/(c^2*d)^( 
1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(1/2*(-d*(c^2*x^2-1))^ 
(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)^2*c+2*I*(-c^2*x^2+1)^(1/2 
)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)*arcsin(c*x)*c-(-d*(c^2*x^2-1))^(1/2)* 
(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*arcsin(c*x)/(c^2*x^2-1)/x-(-d*(c^2*x^ 
2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2 
-1)*c)
 

Fricas [F]

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

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^2,x, algorithm="fricas" 
)
 

Output:

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

Sympy [F]

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

integrate((-c**2*d*x**2+d)**(1/2)*(a+b*asin(c*x))/x**2,x)
 

Output:

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))/x**2, x)
 

Maxima [F]

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

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^2,x, algorithm="maxima" 
)
 

Output:

b*sqrt(d)*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1 
)*sqrt(-c*x + 1))/x^2, x) - (c*sqrt(d)*arcsin(c*x) + sqrt(-c^2*d*x^2 + d)/ 
x)*a
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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