\(\int x^2 \text {arccosh}(a+b x) \, dx\) [2]

Optimal result

Integrand size = 10, antiderivative size = 130 \[ \int x^2 \text {arccosh}(a+b x) \, dx=-\frac {\left (4-5 a+16 a^2\right ) \sqrt {-1+a+b x} \sqrt {1+a+b x}}{18 b^3}-\frac {x^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{9 b}+\frac {5 a (-1+a+b x)^{3/2} \sqrt {1+a+b x}}{18 b^3}+\frac {a \left (3+2 a^2\right ) \text {arccosh}(a+b x)}{6 b^3}+\frac {1}{3} x^3 \text {arccosh}(a+b x) \] Output:

-1/18*(16*a^2-5*a+4)*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/b^3-1/9*x^2*(b*x+a-1) 
^(1/2)*(b*x+a+1)^(1/2)/b+5/18*a*(b*x+a-1)^(3/2)*(b*x+a+1)^(1/2)/b^3+1/6*a* 
(2*a^2+3)*arccosh(b*x+a)/b^3+1/3*x^3*arccosh(b*x+a)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.78 \[ \int x^2 \text {arccosh}(a+b x) \, dx=\frac {-\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (4+11 a^2-5 a b x+2 b^2 x^2\right )+6 b^3 x^3 \text {arccosh}(a+b x)+\left (9 a+6 a^3\right ) \log \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{18 b^3} \] Input:

Integrate[x^2*ArcCosh[a + b*x],x]
 

Output:

(-(Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(4 + 11*a^2 - 5*a*b*x + 2*b^2*x^2) 
) + 6*b^3*x^3*ArcCosh[a + b*x] + (9*a + 6*a^3)*Log[a + b*x + Sqrt[-1 + a + 
 b*x]*Sqrt[1 + a + b*x]])/(18*b^3)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6411, 27, 6378, 111, 164, 43}

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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6378

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

\(\Big \downarrow \) 111

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{3} \int -\frac {b x \left (3 a^2-5 (a+b x) a+2\right )}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)-\frac {1}{3} b^2 x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{3} b^3 x^3 \text {arccosh}(a+b x)}{b^3}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{3} \left (\frac {3}{2} a \left (2 a^2+3\right ) \int \frac {1}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)-\frac {1}{2} \sqrt {a+b x-1} \sqrt {a+b x+1} \left (4 \left (4 a^2+1\right )-5 a (a+b x)\right )\right )-\frac {1}{3} b^2 x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{3} b^3 x^3 \text {arccosh}(a+b x)}{b^3}\)

\(\Big \downarrow \) 43

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{3} \left (\frac {3}{2} a \left (2 a^2+3\right ) \text {arccosh}(a+b x)-\frac {1}{2} \sqrt {a+b x-1} \sqrt {a+b x+1} \left (4 \left (4 a^2+1\right )-5 a (a+b x)\right )\right )-\frac {1}{3} b^2 x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{3} b^3 x^3 \text {arccosh}(a+b x)}{b^3}\)

Input:

Int[x^2*ArcCosh[a + b*x],x]
 

Output:

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

Defintions of rubi rules used

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 111

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

rule 164

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

rule 6378

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

rule 6411

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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.13

method	result	size
orering	\(\frac {\left (10 b^{4} x^{4}-2 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+40 a^{3} b x +22 a^{4}+4 b^{2} x^{2}+35 a b x -14 a^{2}-8\right ) \operatorname {arccosh}\left (b x +a \right )}{18 b^{4} x}-\frac {\left (2 b^{2} x^{2}-5 a b x +11 a^{2}+4\right ) \left (b x +a -1\right ) \left (b x +a +1\right ) \left (2 x \,\operatorname {arccosh}\left (b x +a \right )+\frac {x^{2} b}{\sqrt {b x +a -1}\, \sqrt {b x +a +1}}\right )}{18 b^{4} x^{2}}\)	\(147\)
derivativedivides	\(\frac {-\frac {\operatorname {arccosh}\left (b x +a \right ) a^{3}}{3}+\operatorname {arccosh}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arccosh}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arccosh}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (6 a^{3} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-18 a^{2} \sqrt {\left (b x +a \right )^{2}-1}+9 \left (b x +a \right ) a \sqrt {\left (b x +a \right )^{2}-1}-2 \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+9 a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-4 \sqrt {\left (b x +a \right )^{2}-1}\right )}{18 \sqrt {\left (b x +a \right )^{2}-1}}}{b^{3}}\)	\(203\)
default	\(\frac {-\frac {\operatorname {arccosh}\left (b x +a \right ) a^{3}}{3}+\operatorname {arccosh}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arccosh}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arccosh}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (6 a^{3} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-18 a^{2} \sqrt {\left (b x +a \right )^{2}-1}+9 \left (b x +a \right ) a \sqrt {\left (b x +a \right )^{2}-1}-2 \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+9 a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-4 \sqrt {\left (b x +a \right )^{2}-1}\right )}{18 \sqrt {\left (b x +a \right )^{2}-1}}}{b^{3}}\)	\(203\)
parts	\(\frac {x^{3} \operatorname {arccosh}\left (b x +a \right )}{3}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (2 \,\operatorname {csgn}\left (b \right ) b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-5 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a b x +11 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}-6 \ln \left (\left (\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+b x +a \right ) \operatorname {csgn}\left (b \right )\right ) a^{3}+4 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-9 \ln \left (\left (\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+b x +a \right ) \operatorname {csgn}\left (b \right )\right ) a \right ) \operatorname {csgn}\left (b \right )}{18 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\)	\(230\)

Input:

int(x^2*arccosh(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/18*(10*b^4*x^4-2*a*b^3*x^3+6*a^2*b^2*x^2+40*a^3*b*x+22*a^4+4*b^2*x^2+35* 
a*b*x-14*a^2-8)/b^4/x*arccosh(b*x+a)-1/18*(2*b^2*x^2-5*a*b*x+11*a^2+4)/b^4 
*(b*x+a-1)*(b*x+a+1)/x^2*(2*x*arccosh(b*x+a)+x^2*b/(b*x+a-1)^(1/2)/(b*x+a+ 
1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.70 \[ \int x^2 \text {arccosh}(a+b x) \, dx=\frac {3 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{18 \, b^{3}} \] Input:

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

Output:

1/18*(3*(2*b^3*x^3 + 2*a^3 + 3*a)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a 
^2 - 1)) - (2*b^2*x^2 - 5*a*b*x + 11*a^2 + 4)*sqrt(b^2*x^2 + 2*a*b*x + a^2 
 - 1))/b^3
 

Sympy [F]

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

integrate(x**2*acosh(b*x+a),x)
 

Output:

Integral(x**2*acosh(a + b*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.63 \[ \int x^2 \text {arccosh}(a+b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcosh}\left (b x + a\right ) - \frac {1}{18} \, b {\left (\frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} x^{2}}{b^{2}} - \frac {15 \, a^{3} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{4}} - \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a x}{b^{3}} + \frac {9 \, {\left (a^{2} - 1\right )} a \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{4}} + \frac {15 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a^{2}}{b^{4}} - \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )}}{b^{4}}\right )} \] Input:

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

Output:

1/3*x^3*arccosh(b*x + a) - 1/18*b*(2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*x^2 
/b^2 - 15*a^3*log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*b) 
/b^4 - 5*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*a*x/b^3 + 9*(a^2 - 1)*a*log(2*b 
^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*b)/b^4 + 15*sqrt(b^2*x^ 
2 + 2*a*b*x + a^2 - 1)*a^2/b^4 - 4*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(a^2 
- 1)/b^4)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.02 \[ \int x^2 \text {arccosh}(a+b x) \, dx=\frac {1}{3} \, x^{3} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right ) - \frac {1}{18} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (x {\left (\frac {2 \, x}{b^{2}} - \frac {5 \, a}{b^{3}}\right )} + \frac {11 \, a^{2} b + 4 \, b}{b^{5}}\right )} + \frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \log \left ({\left | -a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} {\left | b \right |} \right |}\right )}{b^{3} {\left | b \right |}}\right )} b \] Input:

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

Output:

1/3*x^3*log(b*x + a + sqrt((b*x + a)^2 - 1)) - 1/18*(sqrt(b^2*x^2 + 2*a*b* 
x + a^2 - 1)*(x*(2*x/b^2 - 5*a/b^3) + (11*a^2*b + 4*b)/b^5) + 3*(2*a^3 + 3 
*a)*log(abs(-a*b - (x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))*abs(b))) 
/(b^3*abs(b)))*b
 

Mupad [F(-1)]

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

int(x^2*acosh(a + b*x),x)
 

Output:

int(x^2*acosh(a + b*x), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.29 \[ \int x^2 \text {arccosh}(a+b x) \, dx=\frac {6 \mathit {acosh} \left (b x +a \right ) b^{3} x^{3}-11 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}+5 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a b x -2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, b^{2} x^{2}-4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+6 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+a +b x \right ) a^{3}+9 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+a +b x \right ) a}{18 b^{3}} \] Input:

int(x^2*acosh(b*x+a),x)
 

Output:

(6*acosh(a + b*x)*b**3*x**3 - 11*sqrt(a**2 + 2*a*b*x + b**2*x**2 - 1)*a**2 
 + 5*sqrt(a**2 + 2*a*b*x + b**2*x**2 - 1)*a*b*x - 2*sqrt(a**2 + 2*a*b*x + 
b**2*x**2 - 1)*b**2*x**2 - 4*sqrt(a**2 + 2*a*b*x + b**2*x**2 - 1) + 6*log( 
sqrt(a**2 + 2*a*b*x + b**2*x**2 - 1) + a + b*x)*a**3 + 9*log(sqrt(a**2 + 2 
*a*b*x + b**2*x**2 - 1) + a + b*x)*a)/(18*b**3)