\(\int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx\) [238]

Optimal result

Integrand size = 29, antiderivative size = 183 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {i c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {d-c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {1-c^2 x^2} \left (b^2 \left (i c x+\sqrt {1-c^2 x^2}\right ) \arcsin (c x)^2+2 b \arcsin (c x) \left (a \sqrt {1-c^2 x^2}-b c x \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+a \left (a \sqrt {1-c^2 x^2}-2 b c x \log (c x)\right )+i b^2 c x \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{x \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

-((Sqrt[1 - c^2*x^2]*(b^2*(I*c*x + Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 2*b* 
ArcSin[c*x]*(a*Sqrt[1 - c^2*x^2] - b*c*x*Log[1 - E^((2*I)*ArcSin[c*x])]) + 
 a*(a*Sqrt[1 - c^2*x^2] - 2*b*c*x*Log[c*x]) + I*b^2*c*x*PolyLog[2, E^((2*I 
)*ArcSin[c*x])]))/(x*Sqrt[d - c^2*d*x^2]))
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.75, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {5186, 5136, 3042, 25, 4200, 25, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5186

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d x}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )}{\sqrt {d-c^2 d x^2}}\)

Input:

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

Output:

-((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(d*x)) + (2*b*c*Sqrt[1 - c^2 
*x^2]*(((-1/2*I)*(a + b*ArcSin[c*x])^2)/b - (2*I)*((I/2)*(a + b*ArcSin[c*x 
])*Log[1 - E^((2*I)*ArcSin[c*x])] + (b*PolyLog[2, E^((2*I)*ArcSin[c*x])])/ 
4)))/Sqrt[d - c^2*d*x^2]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^ 
m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5136

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( 
a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
 

rule 5186

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcSin[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e*x 
^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*A 
rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 
2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (187 ) = 374\).

Time = 0.64 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.33

method	result	size
default	\(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{d x}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{\left (c^{2} x^{2}-1\right ) x d}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{d x \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )\)	\(427\)
parts	\(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{d x}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{\left (c^{2} x^{2}-1\right ) x d}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{d x \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )\)	\(427\)

Input:

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

Output:

-a^2/d/x*(-c^2*d*x^2+d)^(1/2)+b^2*(-(-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)^2/(c^2*x^2-1)/x/d+2*I*(-c^2*x^2+1)^(1/2) 
*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*(I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+ 
1)^(1/2))+I*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+arcsin(c*x)^2+polyl 
og(2,-I*c*x-(-c^2*x^2+1)^(1/2))+polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)))*c)+2* 
a*b*(2*I*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(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)/d/x/(c^2*x^2-1)-(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*x^2-1) 
*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2-1)*c)
 

Fricas [F]

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

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

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^ 
2)/(c^2*d*x^4 - d*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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