\(\int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx\) [32]

Optimal result

Integrand size = 22, antiderivative size = 59 \[ \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx=\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c d}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c d} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08 \[ \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx=\frac {-\left ((a+b \text {arccosh}(c x)) \left (\log \left (1-e^{\text {arccosh}(c x)}\right )-\log \left (1+e^{\text {arccosh}(c x)}\right )\right )\right )+b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c d} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6318, 3042, 26, 4670, 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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx\)

\(\Big \downarrow \) 6318

\(\displaystyle -\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c d}\)

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 4670

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

rule 6318

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.05

method	result	size
derivativedivides	\(\frac {\frac {a \,\operatorname {arctanh}\left (c x \right )}{d}-\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d}}{c}\)	\(180\)
default	\(\frac {\frac {a \,\operatorname {arctanh}\left (c x \right )}{d}-\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d}}{c}\)	\(180\)
parts	\(\frac {a \ln \left (c x +1\right )}{2 d c}-\frac {a \ln \left (c x -1\right )}{2 d c}-\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d c}\)	\(200\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a}{c^{2} x^{2} - 1}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \] Input:

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

Output:

-(Integral(a/(c**2*x**2 - 1), x) + Integral(b*acosh(c*x)/(c**2*x**2 - 1), 
x))/d
 

Maxima [F]

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

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

Output:

1/8*b*((4*(log(c*x + 1) - log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 
 1)) - log(c*x + 1)^2 - 2*log(c*x + 1)*log(c*x - 1))/(c*d) + 8*integrate(1 
/4*(3*c*x - 1)*log(c*x - 1)/(c^2*d*x^2 - d), x) + 8*integrate(1/2*(log(c*x 
 + 1) - log(c*x - 1))/(c^3*d*x^3 - c*d*x + (c^2*d*x^2 - d)*sqrt(c*x + 1)*s 
qrt(c*x - 1)), x)) + 1/2*a*(log(c*x + 1)/(c*d) - log(c*x - 1)/(c*d))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{d-c^2\,d\,x^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx=\frac {-2 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{2} x^{2}-1}d x \right ) b c -\mathrm {log}\left (c^{2} x -c \right ) a +\mathrm {log}\left (c^{2} x +c \right ) a}{2 c d} \] Input:

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

Output:

( - 2*int(acosh(c*x)/(c**2*x**2 - 1),x)*b*c - log(c**2*x - c)*a + log(c**2 
*x + c)*a)/(2*c*d)