\(\int \frac {a+b \arcsin (c x)}{x^2 (d-c^2 d x^2)} \, dx\) [34]

Optimal result

Integrand size = 25, antiderivative size = 116 \[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=-\frac {a+b \arcsin (c x)}{d x}-\frac {2 i c (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d}-\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d}+\frac {i b c \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d}-\frac {i b c \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d} \] Output:

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(259\) vs. \(2(116)=232\).

Time = 0.33 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.23 \[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=-\frac {2 a+2 b \arcsin (c x)+i b c \pi x \arcsin (c x)+2 b c x \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-b c \pi x \log \left (1-i e^{i \arcsin (c x)}\right )-2 b c x \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-b c \pi x \log \left (1+i e^{i \arcsin (c x)}\right )+2 b c x \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+a c x \log (1-c x)-a c x \log (1+c x)+b c \pi x \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+b c \pi x \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i b c x \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+2 i b c x \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 d x} \] Input:

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

Output:

-1/2*(2*a + 2*b*ArcSin[c*x] + I*b*c*Pi*x*ArcSin[c*x] + 2*b*c*x*ArcTanh[Sqr 
t[1 - c^2*x^2]] - b*c*Pi*x*Log[1 - I*E^(I*ArcSin[c*x])] - 2*b*c*x*ArcSin[c 
*x]*Log[1 - I*E^(I*ArcSin[c*x])] - b*c*Pi*x*Log[1 + I*E^(I*ArcSin[c*x])] + 
 2*b*c*x*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + a*c*x*Log[1 - c*x] - a 
*c*x*Log[1 + c*x] + b*c*Pi*x*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + b*c*Pi*x* 
Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*b*c*x*PolyLog[2, (-I)*E^(I*ArcSin 
[c*x])] + (2*I)*b*c*x*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d*x)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5204, 27, 243, 73, 221, 5164, 3042, 4669, 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.

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {c^2 \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{d}-\frac {b \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c d}-\frac {a+b \arcsin (c x)}{d x}\)

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 4669

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

rule 5164

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] 
/; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

rule 5204

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[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], 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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, 
 c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.76

method	result	size
parts	\(-\frac {a \left (\frac {c \ln \left (c x -1\right )}{2}-\frac {c \ln \left (c x +1\right )}{2}+\frac {1}{x}\right )}{d}-\frac {b c \left (\frac {\arcsin \left (c x \right )}{c x}+\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}\)	\(204\)
derivativedivides	\(c \left (-\frac {a \left (-\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}+\frac {1}{c x}\right )}{d}-\frac {b \left (\frac {\arcsin \left (c x \right )}{c x}+\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}\right )\)	\(207\)
default	\(c \left (-\frac {a \left (-\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}+\frac {1}{c x}\right )}{d}-\frac {b \left (\frac {\arcsin \left (c x \right )}{c x}+\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}\right )\)	\(207\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a}{c^{2} x^{4} - x^{2}}\, dx + \int \frac {b \operatorname {asin}{\left (c x \right )}}{c^{2} x^{4} - x^{2}}\, dx}{d} \] Input:

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

Output:

-(Integral(a/(c**2*x**4 - x**2), x) + Integral(b*asin(c*x)/(c**2*x**4 - x* 
*2), x))/d
 

Maxima [F]

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

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

Output:

1/2*a*(c*log(c*x + 1)/d - c*log(c*x - 1)/d - 2/(d*x)) + 1/2*(c*x*arctan2(c 
*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - c*x*arctan2(c*x, sqrt(c*x 
 + 1)*sqrt(-c*x + 1))*log(-c*x + 1) + 2*d*x*integrate(1/2*(c^2*x*log(c*x + 
 1) - c^2*x*log(-c*x + 1) - 2*c)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^2*d*x^3 - 
 d*x), x) - 2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*b/(d*x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=\frac {-2 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{2} x^{4}-x^{2}}d x \right ) b x -\mathrm {log}\left (c^{2} x -c \right ) a c x +\mathrm {log}\left (c^{2} x +c \right ) a c x -2 a}{2 d x} \] Input:

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

Output:

( - 2*int(asin(c*x)/(c**2*x**4 - x**2),x)*b*x - log(c**2*x - c)*a*c*x + lo 
g(c**2*x + c)*a*c*x - 2*a)/(2*d*x)