\(\int \arccos (c e^{a+b x}) \, dx\) [84]

Optimal result

Integrand size = 10, antiderivative size = 84 \[ \int \arccos \left (c e^{a+b x}\right ) \, dx=-\frac {i \arccos \left (c e^{a+b x}\right )^2}{2 b}+\frac {\arccos \left (c e^{a+b x}\right ) \log \left (1+e^{2 i \arccos \left (c e^{a+b x}\right )}\right )}{b}-\frac {i \operatorname {PolyLog}\left (2,-e^{2 i \arccos \left (c e^{a+b x}\right )}\right )}{2 b} \] Output:

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

Mathematica [F]

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

Integrate[ArcCos[c*E^(a + b*x)],x]
 

Output:

Integrate[ArcCos[c*E^(a + b*x)], x]
 

Rubi [A] (warning: unable to verify)

Time = 0.45 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {2720, 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.

\(\displaystyle \int \arccos \left (c e^{a+b x}\right ) \, dx\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\int e^{-a-b x} \arccos \left (c e^{a+b x}\right )de^{a+b x}}{b}\)

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \arccos \left (c e^{a+b x}\right ) \tan \left (\arccos \left (c e^{a+b x}\right )\right )d\arccos \left (c e^{a+b x}\right )}{b}\)

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcCos[c*E^(a + b*x)],x]
 

Output:

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

Defintions of rubi rules used

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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

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 
]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {-\frac {i \arccos \left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2}+\arccos \left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+\left (c \,{\mathrm e}^{b x +a}+i \sqrt {-c^{2} {\mathrm e}^{2 b x +2 a}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c \,{\mathrm e}^{b x +a}+i \sqrt {-c^{2} {\mathrm e}^{2 b x +2 a}+1}\right )^{2}\right )}{2}}{b}\)	\(102\)
default	\(\frac {-\frac {i \arccos \left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2}+\arccos \left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+\left (c \,{\mathrm e}^{b x +a}+i \sqrt {-c^{2} {\mathrm e}^{2 b x +2 a}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c \,{\mathrm e}^{b x +a}+i \sqrt {-c^{2} {\mathrm e}^{2 b x +2 a}+1}\right )^{2}\right )}{2}}{b}\)	\(102\)

Input:

int(arccos(c*exp(b*x+a)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \arccos \left (c e^{a+b x}\right ) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(arccos(c*exp(b*x+a)),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

integrate(acos(c*exp(b*x+a)),x)
 

Output:

Integral(acos(c*exp(a + b*x)), x)
 

Maxima [F]

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

integrate(arccos(c*exp(b*x+a)),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

-1/2*(2*I*b^2*c^2*integrate(x*e^(2*b*x + 2*a)/(c^4*e^(4*b*x + 4*a) - c^2*e 
^(2*b*x + 2*a) + (c^2*e^(2*b*x + 2*a) - 1)*e^(log(c*e^(b*x + a) + 1) + log 
(-c*e^(b*x + a) + 1))), x) + 2*b^2*c*integrate(x*e^(b*x + a + 1/2*log(c*e^ 
(b*x + a) + 1) + 1/2*log(-c*e^(b*x + a) + 1))/(c^4*e^(4*b*x + 4*a) - c^2*e 
^(2*b*x + 2*a) + (c^2*e^(2*b*x + 2*a) - 1)*e^(log(c*e^(b*x + a) + 1) + log 
(-c*e^(b*x + a) + 1))), x) - 2*b*x*arctan(sqrt(c*e^(b*x + a) + 1)*sqrt(-c* 
e^(b*x + a) + 1)*e^(-b*x - a)/c) - I*b*x*log(c*e^(b*x + a) + 1) - I*b*x*lo 
g(-c*e^(b*x + a) + 1) - I*dilog(c*e^(b*x + a)) - I*dilog(-c*e^(b*x + a)))/ 
b
 

Giac [F]

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

integrate(arccos(c*exp(b*x+a)),x, algorithm="giac")
 

Output:

integrate(arccos(c*e^(b*x + a)), x)
 

Mupad [F(-1)]

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

int(acos(c*exp(a + b*x)),x)
 

Output:

int(acos(c*exp(a + b*x)), x)
 

Reduce [F]

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

int(acos(c*exp(b*x+a)),x)
 

Output:

int(acos(e**(a + b*x)*c),x)