\(\int \frac {\arccos (a+b x)}{\frac {a d}{b}+d x} \, dx\) [22]

Optimal result

Integrand size = 19, antiderivative size = 68 \[ \int \frac {\arccos (a+b x)}{\frac {a d}{b}+d x} \, dx=-\frac {i \arccos (a+b x)^2}{2 d}+\frac {\arccos (a+b x) \log \left (1+e^{2 i \arccos (a+b x)}\right )}{d}-\frac {i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a+b x)}\right )}{2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int \frac {\arccos (a+b x)}{\frac {a d}{b}+d x} \, dx=-\frac {i \left (\arccos (a+b x) \left (\arccos (a+b x)+2 i \log \left (1+e^{2 i \arccos (a+b x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \arccos (a+b x)}\right )\right )}{2 d} \] Input:

Integrate[ArcCos[a + b*x]/((a*d)/b + d*x),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.42 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5305, 27, 5137, 3042, 4202, 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.

\(\displaystyle \int \frac {\arccos (a+b x)}{\frac {a d}{b}+d x} \, dx\)

\(\Big \downarrow \) 5305

\(\displaystyle \frac {\int \frac {b \arccos (a+b x)}{d (a+b x)}d(a+b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\arccos (a+b x)}{a+b x}d(a+b x)}{d}\)

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \arccos (a+b x) \tan (\arccos (a+b x))d\arccos (a+b x)}{d}\)

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcCos[a + b*x]/((a*d)/b + d*x),x]
 

Output:

-(((I/2)*ArcCos[a + b*x]^2 - (2*I)*((-1/2*I)*ArcCos[a + b*x]*Log[1 + E^((2 
*I)*ArcCos[a + b*x])] - PolyLog[2, -a - b*x]/4))/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 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 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 5137

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

rule 5305

Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.35

method	result	size
derivativedivides	\(\frac {-\frac {i b \arccos \left (b x +a \right )^{2}}{2 d}+\frac {b \arccos \left (b x +a \right ) \ln \left (1+\left (b x +a +i \sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )}{d}-\frac {i b \operatorname {polylog}\left (2, -\left (b x +a +i \sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )}{2 d}}{b}\)	\(92\)
default	\(\frac {-\frac {i b \arccos \left (b x +a \right )^{2}}{2 d}+\frac {b \arccos \left (b x +a \right ) \ln \left (1+\left (b x +a +i \sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )}{d}-\frac {i b \operatorname {polylog}\left (2, -\left (b x +a +i \sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )}{2 d}}{b}\)	\(92\)

Input:

int(arccos(b*x+a)/(a*d/b+d*x),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

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

Output:

integral(b*arccos(b*x + a)/(b*d*x + a*d), x)
 

Sympy [F]

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

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

Output:

b*Integral(acos(a + b*x)/(a + b*x), x)/d
 

Maxima [F]

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

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

Output:

integrate(arccos(b*x + a)/(d*x + a*d/b), x)
 

Giac [F]

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

integrate(arccos(b*x+a)/(a*d/b+d*x),x, algorithm="giac")
 

Output:

integrate(arccos(b*x + a)/(d*x + a*d/b), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a+b x)}{\frac {a d}{b}+d x} \, dx=\int \frac {\mathrm {acos}\left (a+b\,x\right )}{d\,x+\frac {a\,d}{b}} \,d x \] Input:

int(acos(a + b*x)/(d*x + (a*d)/b),x)
                                                                                    
                                                                                    
 

Output:

int(acos(a + b*x)/(d*x + (a*d)/b), x)
 

Reduce [F]

\[ \int \frac {\arccos (a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\left (\int \frac {\mathit {acos} \left (b x +a \right )}{b x +a}d x \right ) b}{d} \] Input:

int(acos(b*x+a)/(a*d/b+d*x),x)
 

Output:

(int(acos(a + b*x)/(a + b*x),x)*b)/d