\(\int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx\) [189]

Optimal result

Integrand size = 29, antiderivative size = 288 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {a^2 \log (c x)}{\sqrt {d}}-\frac {a^2 \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{\sqrt {d}}-\frac {2 i a b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\text {arccosh}(c x) \left (\log \left (1-i e^{-\text {arccosh}(c x)}\right )-\log \left (1+i e^{-\text {arccosh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-\text {arccosh}(c x)^2 \left (\log \left (1-i e^{-\text {arccosh}(c x)}\right )-\log \left (1+i e^{-\text {arccosh}(c x)}\right )\right )-2 \text {arccosh}(c x) \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )-2 \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(c x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(a^2*Log[c*x])/Sqrt[d] - (a^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/Sqrt[d 
] - ((2*I)*a*b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(ArcCosh[c*x]*(Log[1 - 
 I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]]) + PolyLog[2, (-I)/E^ArcCos 
h[c*x]] - PolyLog[2, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2] + (I*b^2*Sqrt 
[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-(ArcCosh[c*x]^2*(Log[1 - I/E^ArcCosh[c* 
x]] - Log[1 + I/E^ArcCosh[c*x]])) - 2*ArcCosh[c*x]*(PolyLog[2, (-I)/E^ArcC 
osh[c*x]] - PolyLog[2, I/E^ArcCosh[c*x]]) - 2*PolyLog[3, (-I)/E^ArcCosh[c* 
x]] + 2*PolyLog[3, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2]
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.48, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6361, 3042, 4668, 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.

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

\(\Big \downarrow \) 6361

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

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 4668

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

rule 6361

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

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 [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*arccosh(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas 
")
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + 
a^2)/(c^2*d*x^3 - d*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a+b*arccosh(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima 
")
 

Output:

-a^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/sqrt(d) + int 
egrate(b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(sqrt(-c^2*d*x^2 + d)* 
x) + 2*a*b*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(sqrt(-c^2*d*x^2 + d)*x) 
, x)
 

Giac [F]

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

integrate((a+b*arccosh(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*x),x)*a*b + int(acosh(c*x)**2/(s 
qrt( - c**2*x**2 + 1)*x),x)*b**2 + log(tan(asin(c*x)/2))*a**2)/sqrt(d)