\(\int \frac {a+b \arccos (c x)}{x (d-c^2 d x^2)} \, dx\) [5]

Optimal result

Integrand size = 25, antiderivative size = 71 \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )} \, dx=\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d}-\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d} \] Output:

2*(a+b*arccos(c*x))*arctanh((c*x+I*(-c^2*x^2+1)^(1/2))^2)/d-1/2*I*b*polylo 
g(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)/d+1/2*I*b*polylog(2,(c*x+I*(-c^2*x^2+1) 
^(1/2))^2)/d
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(71)=142\).

Time = 0.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.01 \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )} \, dx=-\frac {2 b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+2 b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )-2 b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )-2 a \log (x)+a \log \left (1-c^2 x^2\right )-2 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-2 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )+i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {5185, 4919, 3042, 4671, 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.

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

\(\Big \downarrow \) 5185

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

\(\Big \downarrow \) 4919

\(\displaystyle -\frac {2 \int (a+b \arccos (c x)) \csc (2 \arccos (c x))d\arccos (c x)}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \int (a+b \arccos (c x)) \csc (2 \arccos (c x))d\arccos (c x)}{d}\)

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(-2*(-((a + b*ArcCos[c*x])*ArcTanh[E^((2*I)*ArcCos[c*x])]) + (I/4)*b*PolyL 
og[2, -E^((2*I)*ArcCos[c*x])] - (I/4)*b*PolyLog[2, E^((2*I)*ArcCos[c*x])]) 
)/d
 

Defintions of rubi rules used

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 4671

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

rule 4919

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n 
, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
 

rule 5185

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
x_Symbol] :> Simp[-d^(-1)   Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, A 
rcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n 
, 0]
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.76

method	result	size
parts	\(-\frac {a \left (-\ln \left (x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\)	\(196\)
derivativedivides	\(-\frac {a \left (\frac {\ln \left (c x -1\right )}{2}-\ln \left (c x \right )+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\)	\(198\)
default	\(-\frac {a \left (\frac {\ln \left (c x -1\right )}{2}-\ln \left (c x \right )+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\)	\(198\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-(b*arccos(c*x) + a)/(c^2*d*x^3 - d*x), x)
 

Sympy [F]

\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a}{c^{2} x^{3} - x}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{2} x^{3} - x}\, dx}{d} \] Input:

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

Output:

-(Integral(a/(c**2*x**3 - x), x) + Integral(b*acos(c*x)/(c**2*x**3 - x), x 
))/d
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )} \, dx=\frac {-2 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) b -\mathrm {log}\left (c^{2} x -c \right ) a -\mathrm {log}\left (c^{2} x +c \right ) a +2 \,\mathrm {log}\left (x \right ) a}{2 d} \] Input:

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

Output:

( - 2*int(acos(c*x)/(c**2*x**3 - x),x)*b - log(c**2*x - c)*a - log(c**2*x 
+ c)*a + 2*log(x)*a)/(2*d)