3.1.41 \(\int \frac {a+b \text {arccosh}(c x)}{(d-c^2 d x^2)^2} \, dx\) [41]

3.1.41.1 Optimal result

Integrand size = 22, antiderivative size = 120 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d^2}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 c d^2}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 c d^2} \]

output

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

3.1.41.2 Mathematica [A] (warning: unable to verify)

Time = 1.05 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.58 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\frac {-2 a c x-2 b \sqrt {\frac {-1+c x}{1+c x}}-2 b c x \sqrt {\frac {-1+c x}{1+c x}}-2 b \text {arccosh}(c x) \left (c x+\left (-1+c^2 x^2\right ) \log \left (1-e^{\text {arccosh}(c x)}\right )+\left (1-c^2 x^2\right ) \log \left (1+e^{\text {arccosh}(c x)}\right )\right )+\left (a-a c^2 x^2\right ) \log (1-c x)-a \log (1+c x)+a c^2 x^2 \log (1+c x)}{-1+c^2 x^2}+2 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-2 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 c d^2} \]

input

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

output

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

3.1.41.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6316, 27, 83, 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.

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

\(\Big \downarrow \) 6316

\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{d \left (1-c^2 x^2\right )}dx}{2 d}+\frac {b c \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{2 d^2}+\frac {b c \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 6318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

-1/2*b/(c*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCosh[c*x]))/(2* 
d^2*(1 - c^2*x^2)) - ((I/2)*((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^2)
 

3.1.41.3.1 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 27

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

rule 83

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f 
*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 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 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 6316

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 
 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b* 
ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + 
 c*x)^p*(-1 + c*x)^p)]   Int[x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a 
+ b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2* 
d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
 

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]
 

3.1.41.4 Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(\frac {\frac {a \left (-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}}{c}\)	\(192\)
default	\(\frac {\frac {a \left (-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}}{c}\)	\(192\)
parts	\(\frac {a \left (-\frac {1}{4 c \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4 c}-\frac {1}{4 c \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4 c}\right )}{d^{2}}+\frac {b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2} c}\)	\(203\)

input

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

output

1/c*(a/d^2*(-1/4/(c*x+1)+1/4*ln(c*x+1)-1/4/(c*x-1)-1/4*ln(c*x-1))+b/d^2*(- 
1/2*(c*x*arccosh(c*x)+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c^2*x^2-1)-1/2*arccosh 
(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-1/2*polylog(2,c*x+(c*x-1)^(1/2 
)*(c*x+1)^(1/2))+1/2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+1/ 
2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))))
 

3.1.41.5 Fricas [F]

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

input

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

output

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

3.1.41.6 Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]

input

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

output

(Integral(a/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*acosh(c*x)/(c** 
4*x**4 - 2*c**2*x**2 + 1), x))/d**2
 

3.1.41.7 Maxima [F]

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

input

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

output

1/64*(192*c^3*integrate(1/8*x^3*log(c*x - 1)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 
+ d^2), x) - 8*c^2*(2*x/(c^4*d^2*x^2 - c^2*d^2) + log(c*x + 1)/(c^3*d^2) - 
 log(c*x - 1)/(c^3*d^2)) - 64*c^2*integrate(1/8*x^2*log(c*x - 1)/(c^4*d^2* 
x^4 - 2*c^2*d^2*x^2 + d^2), x) + 3*(c*(2/(c^4*d^2*x - c^3*d^2) - log(c*x + 
 1)/(c^3*d^2) + log(c*x - 1)/(c^3*d^2)) + 4*log(c*x - 1)/(c^4*d^2*x^2 - c^ 
2*d^2))*c - 4*((c^2*x^2 - 1)*log(c*x + 1)^2 + 2*(c^2*x^2 - 1)*log(c*x + 1) 
*log(c*x - 1) + 4*(2*c*x - (c^2*x^2 - 1)*log(c*x + 1) + (c^2*x^2 - 1)*log( 
c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^3*d^2*x^2 - c*d^2) + 
64*integrate(-1/4*(2*c*x - (c^2*x^2 - 1)*log(c*x + 1) + (c^2*x^2 - 1)*log( 
c*x - 1))/(c^5*d^2*x^5 - 2*c^3*d^2*x^3 + c*d^2*x + (c^4*d^2*x^4 - 2*c^2*d^ 
2*x^2 + d^2)*sqrt(c*x + 1)*sqrt(c*x - 1)), x) + 64*integrate(1/8*log(c*x - 
 1)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x))*b - 1/4*a*(2*x/(c^2*d^2*x^2 - 
 d^2) - log(c*x + 1)/(c*d^2) + log(c*x - 1)/(c*d^2))
 

3.1.41.8 Giac [F]

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

input

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

output

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

3.1.41.9 Mupad [F(-1)]

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

input

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

output

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