\(\int \frac {x^2 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx\) [30]

Optimal result

Integrand size = 25, antiderivative size = 102 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^2 d}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^3 d}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^3 d}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^3 d} \] Output:

b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d-x*(a+b*arccosh(c*x))/c^2/d+2*(a+b*arcc 
osh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^3/d+b*polylog(2,-c*x- 
(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^3/d-b*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^( 
1/2))/c^3/d
 

Mathematica [A] (warning: unable to verify)

Time = 0.20 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.52 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {-2 a c x+2 b \sqrt {\frac {-1+c x}{1+c x}}+2 b c x \sqrt {\frac {-1+c x}{1+c x}}-2 b c x \text {arccosh}(c x)+b \text {arccosh}(c x)^2+2 b \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )-2 b \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )-a \log (1-c x)+a \log (1+c x)-2 b \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )-2 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 c^3 d} \] Input:

Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^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*c*x*ArcCosh[c*x] + b*ArcCosh[c*x]^2 + 2*b*ArcCosh[c*x]*Log[1 + 
 E^(-ArcCosh[c*x])] - 2*b*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] - a*Log[1 - 
 c*x] + a*Log[1 + c*x] - 2*b*PolyLog[2, -E^(-ArcCosh[c*x])] - 2*b*PolyLog[ 
2, E^ArcCosh[c*x]])/(2*c^3*d)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.82 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6353, 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.

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 6318

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3 d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3 d}\)

\(\Big \downarrow \) 4670

\(\displaystyle -\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3 d}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3 d}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^2 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3 d}\)

Input:

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

Output:

(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c^3*d) - (x*(a + b*ArcCosh[c*x]))/(c^2*d 
) - (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^3*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 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 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]
 

rule 6353

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcCosh[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2 
*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] 
- Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x) 
^p)]   Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*Ar 
cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* 
d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.77

method	result	size
derivativedivides	\(\frac {-\frac {a \left (c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}}{c^{3}}\)	\(181\)
default	\(\frac {-\frac {a \left (c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}}{c^{3}}\)	\(181\)
parts	\(-\frac {a \left (\frac {x}{c^{2}}-\frac {\ln \left (c x +1\right )}{2 c^{3}}+\frac {\ln \left (c x -1\right )}{2 c^{3}}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) x}{d \,c^{2}}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{c^{3} d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{3}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{3}}+\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{3} d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{3} d}\)	\(202\)

Input:

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

Output:

1/c^3*(-a/d*(c*x+1/2*ln(c*x-1)-1/2*ln(c*x+1))+b/d*(c*x-1)^(1/2)*(c*x+1)^(1 
/2)-b/d*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+b/d*arccosh(c*x 
)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-b/d*arccosh(c*x)*c*x-b/d*polylog(2 
,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+b/d*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1) 
^(1/2)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {-2 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{c^{2} x^{2}-1}d x \right ) b \,c^{3}-\mathrm {log}\left (c^{2} x -c \right ) a +\mathrm {log}\left (c^{2} x +c \right ) a -2 a c x}{2 c^{3} d} \] Input:

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

Output:

( - 2*int((acosh(c*x)*x**2)/(c**2*x**2 - 1),x)*b*c**3 - log(c**2*x - c)*a 
+ log(c**2*x + c)*a - 2*a*c*x)/(2*c**3*d)