3.1.31 \(\int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx\) [31]

3.1.31.1 Optimal result

Integrand size = 23, antiderivative size = 74 \[ \int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {(a+b \text {arccosh}(c x))^2}{2 b c^2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^2 d}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^2 d} \]

output

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

3.1.31.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.15 \[ \int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {(a+b \text {arccosh}(c x)) \left (a+b \text {arccosh}(c x)-2 b \log \left (1-e^{\text {arccosh}(c x)}\right )-2 b \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-2 b^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-2 b^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 b c^2 d} \]

input

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

output

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

3.1.31.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.44 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6328, 3042, 26, 4199, 25, 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.

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

\(\Big \downarrow \) 6328

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \int (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c^2 d}\)

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.1.31.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(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*fz*x))/(1 + E^(2*((-I)*e + f*fz*x 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6328

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

3.1.31.4 Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.85

method	result	size
parts	\(-\frac {a \ln \left (c^{2} x^{2}-1\right )}{2 d \,c^{2}}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \,c^{2}}\)	\(137\)
derivativedivides	\(\frac {-\frac {a \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}}{c^{2}}\)	\(142\)
default	\(\frac {-\frac {a \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}}{c^{2}}\)	\(142\)

input

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

output

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

3.1.31.5 Fricas [F]

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

input

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

output

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

3.1.31.6 Sympy [F]

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

input

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

output

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

3.1.31.7 Maxima [F]

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

input

integrate(x*(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) - log(c*x - 1)^2)/(c^ 
2*d) + 8*integrate(1/2*(log(c*x + 1) + log(c*x - 1))/(c^4*d*x^3 - c^2*d*x 
+ (c^3*d*x^2 - c*d)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x)) - 1/2*a* 
log(c^2*d*x^2 - d)/(c^2*d)
 

3.1.31.8 Giac [F]

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

input

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

output

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

3.1.31.9 Mupad [F(-1)]

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

input

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

output

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