3.6.90 \(\int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d+c d x} \sqrt {e-c e x}} \, dx\) [590]

3.6.90.1 Optimal result

Integrand size = 35, antiderivative size = 214 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d+c d x} \sqrt {e-c e x}} \, dx=-\frac {i c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {d+c d x} \sqrt {e-c e x}} \]

output

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

3.6.90.2 Mathematica [A] (verified)

Time = 2.67 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\frac {b^2 \left (-1+c^2 x^2-i c x \sqrt {1-c^2 x^2}\right ) \arcsin (c x)^2+2 b \arcsin (c x) \left (-a+a c^2 x^2+b c x \sqrt {1-c^2 x^2} \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+a \left (-a+a c^2 x^2+2 b c x \sqrt {1-c^2 x^2} \log (c x)\right )-i b^2 c x \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{x \sqrt {d+c d x} \sqrt {e-c e x}} \]

input

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

output

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

3.6.90.3 Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.66, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5238, 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.

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

\(\Big \downarrow \) 5238

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

\(\Big \downarrow \) 5186

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.6.90.3.1 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]
 

rule 5238

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[((-d^2)*(g/e))^In 
tPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPar 
t[q])   Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n 
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] & 
& EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

3.6.90.4 Maple [F]

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

input

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

output

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

3.6.90.5 Fricas [F]

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

input

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

output

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

3.6.90.6 Sympy [F]

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

input

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

output

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

3.6.90.7 Maxima [F(-2)]

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

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.6.90.8 Giac [F]

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

input

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

output

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

3.6.90.9 Mupad [F(-1)]

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

input

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

output

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