\(\int \frac {x^2 (a+b \text {arccosh}(c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\) [203]

Optimal result

Integrand size = 29, antiderivative size = 344 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 x}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^2 (a+b \text {arccosh}(c x))}{3 c d^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.78 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\frac {a^2 c^3 x^3}{1-c^2 x^2}+a b \left (\frac {2 c^3 x^3 \text {arccosh}(c x)}{1-c^2 x^2}+\frac {\sqrt {\frac {-1+c x}{1+c x}} \left (-1+2 \left (-1+c^2 x^2\right ) \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )}{-1+c x}\right )+b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-\frac {c x \left (-1+c^2 x^2+c^2 x^2 \text {arccosh}(c x)^2\right )}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}+\text {arccosh}(c x) \left (\frac {1}{1-c^2 x^2}+\text {arccosh}(c x)+2 \log \left (1-e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(c x)}\right )\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(x^2*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]
 

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.40 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.72, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {6332, 6327, 6349, 100, 27, 87, 43, 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.

\(\displaystyle \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6332

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

\(\Big \downarrow \) 6327

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

\(\Big \downarrow \) 6349

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6328

\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {c x (a+b \text {arccosh}(c x))}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c^4}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int -i (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c^4}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {x^3 (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \int (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c^4}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 4199

\(\displaystyle \frac {x^3 (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \int -\frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^3 (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (-2 i \int \frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {x^3 (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (-2 i \left (\frac {1}{2} b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {x^3 (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (-2 i \left (\frac {1}{4} b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {x^3 (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

Input:

Int[(x^2*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]
 

Output:

(x^3*(a + b*ArcCosh[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (2*b*c*Sqrt[-1 
+ c*x]*Sqrt[1 + c*x]*((x^2*(a + b*ArcCosh[c*x]))/(2*c^2*(1 - c^2*x^2)) + ( 
b*(-(1/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (-(Sqrt[-1 + c*x]/(c^2*Sqrt[1 
 + c*x])) + ArcCosh[c*x]/c^2)/c))/(2*c) - (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^4))/(3*d^2*Sqrt[d - c^2*d*x^2])
 

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 27

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

rule 43

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a 
*d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
 

rule 87

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

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 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 6327

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( 
e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 
*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 
, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
 

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]
 

rule 6332

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

rule 6349

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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
   Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - S 
imp[b*f*(n/(2*c*(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*ArcCosh[c 
*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] 
&& GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2656\) vs. \(2(322)=644\).

Time = 0.70 (sec) , antiderivative size = 2657, normalized size of antiderivative = 7.72

method	result	size
default	\(\text {Expression too large to display}\)	\(2657\)
parts	\(\text {Expression too large to display}\)	\(2657\)

Input:

int(x^2*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+ 
1)/d^3*c^2*(c*x-1)*(c*x+1)*arccosh(c*x)*x^5+1/3*b^2*(-d*(c^2*x^2-1))^(1/2) 
/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*arccosh(c*x)^2*x^3+1/3*b 
^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3 
*arccosh(c*x)*x^3+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c 
^4*x^4-5*c^2*x^2+1)/d^3*c^4*x^7-b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^ 
6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*c^2*x^5-2/3*b^2*(c*x+1)^(1/2)*(c*x-1)^(1 
/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/c^3/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x- 
1)^(1/2)*(c*x+1)^(1/2))-2/3*b^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1 
))^(1/2)/d^3/c^3/(c^2*x^2-1)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^( 
1/2))-b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2 
+1)/d^3/c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*x^2+b^2*(-d*(c^2*x^2-1) 
)^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*c*(c*x-1)^(1/2)*( 
c*x+1)^(1/2)*arccosh(c*x)*x^4+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(c^3*x^3+(c*x 
-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-2*(c*x-1)^( 
1/2)*(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^5*c^5+2*ln( 
(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^6*c^6+6*arccosh(c*x)*c^4*x^4+2*(c 
*x-1)^(1/2)*(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^3*c^ 
3-6*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^4*c^4+c^3*x^3*(c*x-1)^(1/2 
)*(c*x+1)^(1/2)-c^4*x^4-6*c^2*x^2*arccosh(c*x)+6*ln((c*x+(c*x-1)^(1/2)*...
 

Fricas [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^2*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fric 
as")
 

Output:

integral(-(b^2*x^2*arccosh(c*x)^2 + 2*a*b*x^2*arccosh(c*x) + a^2*x^2)*sqrt 
(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
 

Sympy [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**2*(a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral(x**2*(a + b*acosh(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
 

Maxima [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^2*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxi 
ma")
 

Output:

1/3*a*b*c*(sqrt(-d)/(c^6*d^3*x^2 - c^4*d^3) - sqrt(-d)*log(c*x + 1)/(c^4*d 
^3) - sqrt(-d)*log(c*x - 1)/(c^4*d^3)) - 2/3*a*b*(x/(sqrt(-c^2*d*x^2 + d)* 
c^2*d^2) - x/((-c^2*d*x^2 + d)^(3/2)*c^2*d))*arccosh(c*x) - 1/3*a^2*(x/(sq 
rt(-c^2*d*x^2 + d)*c^2*d^2) - x/((-c^2*d*x^2 + d)^(3/2)*c^2*d)) + b^2*inte 
grate(x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(-c^2*d*x^2 + d)^(5/2), 
 x)
 

Giac [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^2*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac 
")
 

Output:

integrate((b*arccosh(c*x) + a)^2*x^2/(-c^2*d*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:

int((x^2*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(5/2),x)
 

Output:

int((x^2*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{2} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b +3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )^{2} x^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )^{2} x^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2}-a^{2} x^{3}}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int(x^2*(a+b*acosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x)
 

Output:

(6*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**2)/(sqrt( - c**2*x**2 + 1)*c* 
*4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)* 
a*b*c**2*x**2 - 6*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**2)/(sqrt( - c* 
*2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2 
*x**2 + 1)),x)*a*b + 3*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)**2*x**2)/(sq 
rt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt 
( - c**2*x**2 + 1)),x)*b**2*c**2*x**2 - 3*sqrt( - c**2*x**2 + 1)*int((acos 
h(c*x)**2*x**2)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 
1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b**2 - a**2*x**3)/(3*sqrt(d)*sqr 
t( - c**2*x**2 + 1)*d**2*(c**2*x**2 - 1))