\(\int \frac {(a+b \text {arccosh}(c x))^2}{d-c^2 d x^2} \, dx\) [10]

Optimal result

Integrand size = 24, antiderivative size = 118 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{d-c^2 d x^2} \, dx=\frac {2 (a+b \text {arccosh}(c x))^2 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d}+\frac {2 b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c d}-\frac {2 b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c d}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )}{c d}+\frac {2 b^2 \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )}{c d} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{d-c^2 d x^2} \, dx=\frac {-(a+b \text {arccosh}(c x))^2 \log \left (1-e^{\text {arccosh}(c x)}\right )+(a+b \text {arccosh}(c x))^2 \log \left (1+e^{\text {arccosh}(c x)}\right )+2 b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-2 b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )-2 b^2 \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )+2 b^2 \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )}{c d} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6318, 3042, 26, 4670, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 6318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

((-I)*((2*I)*(a + b*ArcCosh[c*x])^2*ArcTanh[E^ArcCosh[c*x]] - (2*I)*b*(-(( 
a + b*ArcCosh[c*x])*PolyLog[2, -E^ArcCosh[c*x]]) + b*PolyLog[3, -E^ArcCosh 
[c*x]]) + (2*I)*b*(-((a + b*ArcCosh[c*x])*PolyLog[2, E^ArcCosh[c*x]]) + b* 
PolyLog[3, 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 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 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.01

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \operatorname {arctanh}\left (c x \right )}{d}-\frac {b^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {2 a b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d}}{c}\)	\(355\)
default	\(\frac {\frac {a^{2} \operatorname {arctanh}\left (c x \right )}{d}-\frac {b^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {2 a b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d}}{c}\)	\(355\)
parts	\(\frac {a^{2} \ln \left (c x +1\right )}{2 d c}-\frac {a^{2} \ln \left (c x -1\right )}{2 d c}-\frac {b^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d c}-\frac {2 a b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d c}\)	\(380\)

Input:

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

Output:

1/c*(a^2/d*arctanh(c*x)-b^2/d*(arccosh(c*x)^2*ln(1-c*x-(c*x-1)^(1/2)*(c*x+ 
1)^(1/2))+2*arccosh(c*x)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-2*poly 
log(3,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-arccosh(c*x)^2*ln(1+c*x+(c*x-1)^(1/ 
2)*(c*x+1)^(1/2))-2*arccosh(c*x)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2 
))+2*polylog(3,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2)))-2*a*b/d*(-arctanh(c*x)*a 
rccosh(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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*a^2*(log(c*x + 1)/(c*d) - log(c*x - 1)/(c*d)) + 1/2*(b^2*log(c*x + 1) 
- b^2*log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(c*d) - integ 
rate((2*a*b*c*x + (b^2*c*x*log(c*x + 1) - b^2*c*x*log(c*x - 1) + 2*a*b)*sq 
rt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^2*x^2 - b^2)*log(c*x + 1) - (b^2*c^2*x^ 
2 - b^2)*log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^3*d*x^3 - 
 c*d*x + (c^2*d*x^2 - d)*sqrt(c*x + 1)*sqrt(c*x - 1)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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