\(\int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [186]

Optimal result

Integrand size = 29, antiderivative size = 227 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {b^2 x \sqrt {d-c^2 d x^2}}{4 c^2 d}-\frac {b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{4 c^3 d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 c d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c^3 d \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/4*b^2*x*(-c^2*d*x^2+d)^(1/2)/c^2/d-1/4*b^2*(-c^2*d*x^2+d)^(1/2)*arccosh 
(c*x)/c^3/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*b*x^2*(-c^2*d*x^2+d)^(1/2)*(a+ 
b*arccosh(c*x))/c/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*x*(-c^2*d*x^2+d)^(1/2) 
*(a+b*arccosh(c*x))^2/c^2/d-1/6*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^3/ 
b/c^3/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-\frac {12 a^2 c x \sqrt {d-c^2 d x^2}}{d}-\frac {12 a^2 \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{\sqrt {d}}+\frac {b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (4 \text {arccosh}(c x)^3-6 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))+\left (3+6 \text {arccosh}(c x)^2\right ) \sinh (2 \text {arccosh}(c x))\right )}{\sqrt {d-c^2 d x^2}}+\frac {6 a b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (-\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)+\sinh (2 \text {arccosh}(c x))))}{\sqrt {d-c^2 d x^2}}}{24 c^3} \] Input:

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

Output:

((-12*a^2*c*x*Sqrt[d - c^2*d*x^2])/d - (12*a^2*ArcTan[(c*x*Sqrt[d - c^2*d* 
x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/Sqrt[d] + (b^2*Sqrt[(-1 + c*x)/(1 + c*x)] 
*(1 + c*x)*(4*ArcCosh[c*x]^3 - 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] + (3 + 
6*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2] + (6*a*b*Sqrt 
[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*( 
ArcCosh[c*x] + Sinh[2*ArcCosh[c*x]])))/Sqrt[d - c^2*d*x^2])/(24*c^3)
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {6353, 6298, 101, 43, 6307}

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.

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6307

\(\displaystyle -\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\)

Input:

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

Output:

-1/2*(x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^2*d) + (Sqrt[-1 + c 
*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2]) - 
(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((x^2*(a + b*ArcCosh[c*x]))/2 - (b*c*((x*S 
qrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ArcCosh[c*x]/(2*c^3)))/2))/(c*Sqrt[ 
d - c^2*d*x^2])
 

Defintions of rubi rules used

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 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 6298

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

rule 6307

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(562\) vs. \(2(195)=390\).

Time = 0.36 (sec) , antiderivative size = 563, normalized size of antiderivative = 2.48

method	result	size
default	\(-\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{3}}{6 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(563\)
parts	\(-\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{3}}{6 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(563\)

Input:

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

Output:

-1/2*a^2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2*a^2/c^2/(c^2*d)^(1/2)*arctan((c^ 
2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-1/6*(-d*(c^2*x^2-1))^(1/2)*(c*x-1 
)^(1/2)*(c*x+1)^(1/2)/d/c^3/(c^2*x^2-1)*arccosh(c*x)^3-1/16*(-d*(c^2*x^2-1 
))^(1/2)*(2*c^3*x^3-2*c*x+2*(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*arccosh(c*x)^2-2*arccosh(c*x)+1)/d/c^3/(c^2*x^2-1)-1 
/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x 
^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*(2*arccosh(c*x)^2+2*arccosh(c*x)+1)/ 
d/c^3/(c^2*x^2-1))+2*a*b*(-1/4*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1 
)^(1/2)/d/c^3/(c^2*x^2-1)*arccosh(c*x)^2-1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^ 
3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1 
/2))*(-1+2*arccosh(c*x))/d/c^3/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(-2 
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2) 
-2*c*x)*(1+2*arccosh(c*x))/d/c^3/(c^2*x^2-1))
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*a^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt(d))) + in 
tegrate(b^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/sqrt(-c^2*d*x^2 + 
 d) + 2*a*b*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d 
), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(asin(c*x)*a**2 - sqrt( - c**2*x**2 + 1)*a**2*c*x + 4*int((acosh(c*x)*x**2 
)/sqrt( - c**2*x**2 + 1),x)*a*b*c**3 + 2*int((acosh(c*x)**2*x**2)/sqrt( - 
c**2*x**2 + 1),x)*b**2*c**3)/(2*sqrt(d)*c**3)