\(\int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx\) [23]

Optimal result

Integrand size = 16, antiderivative size = 150 \[ \int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx=2 b^2 d x+\frac {1}{4} b^2 e x^2-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c}-\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 e}-\frac {e (a+b \text {arccosh}(c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \text {arccosh}(c x))^2}{2 e} \] Output:

2*b^2*d*x+1/4*b^2*e*x^2-2*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x 
))/c-1/2*b*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))/c-1/2*d^2*(a 
+b*arccosh(c*x))^2/e-1/4*e*(a+b*arccosh(c*x))^2/c^2+1/2*(e*x+d)^2*(a+b*arc 
cosh(c*x))^2/e
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.16 \[ \int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx=\frac {c \left (2 a^2 c x (2 d+e x)-2 a b \sqrt {-1+c x} \sqrt {1+c x} (4 d+e x)+b^2 c x (8 d+e x)\right )-2 b c \left (-2 a c x (2 d+e x)+b \sqrt {-1+c x} \sqrt {1+c x} (4 d+e x)\right ) \text {arccosh}(c x)+b^2 \left (4 c^2 d x+e \left (-1+2 c^2 x^2\right )\right ) \text {arccosh}(c x)^2-2 a b e \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{4 c^2} \] Input:

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

Output:

(c*(2*a^2*c*x*(2*d + e*x) - 2*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d + e*x) 
 + b^2*c*x*(8*d + e*x)) - 2*b*c*(-2*a*c*x*(2*d + e*x) + b*Sqrt[-1 + c*x]*S 
qrt[1 + c*x]*(4*d + e*x))*ArcCosh[c*x] + b^2*(4*c^2*d*x + e*(-1 + 2*c^2*x^ 
2))*ArcCosh[c*x]^2 - 2*a*b*e*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(4*c 
^2)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6378, 6390, 2009}

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.

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

\(\Big \downarrow \) 6378

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

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 6390

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( 
d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand 
Integrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, 
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && 
 EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[ 
d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1 
] || (EqQ[m, 2] && LtQ[p, -2]))
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.49

method	result	size
parts	\(a^{2} \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b^{2} \left (\frac {e \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4 c}+d \left (\operatorname {arccosh}\left (c x \right )^{2} c x -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) c x d +\frac {c \,\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\, e c x +\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) e \right )}{4 c \sqrt {c^{2} x^{2}-1}}\right )}{c}\)	\(224\)
derivativedivides	\(\frac {\frac {a^{2} \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (\operatorname {arccosh}\left (c x \right )^{2} c x -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {e \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4}\right )}{c}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\, e c x +\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) e \right )}{4 \sqrt {c^{2} x^{2}-1}}\right )}{c}}{c}\)	\(236\)
default	\(\frac {\frac {a^{2} \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (\operatorname {arccosh}\left (c x \right )^{2} c x -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {e \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4}\right )}{c}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\, e c x +\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) e \right )}{4 \sqrt {c^{2} x^{2}-1}}\right )}{c}}{c}\)	\(236\)
orering	\(\frac {\left (7 c^{2} e^{3} x^{4}+33 c^{2} d \,e^{2} x^{3}+20 c^{2} d^{2} e \,x^{2}+8 c^{2} d^{3} x -6 x^{2} e^{3}-30 d \,e^{2} x -10 d^{2} e \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{8 c^{2} \left (e x +d \right )^{2}}-\frac {\left (3 c^{2} e^{2} x^{4}+17 c^{2} d e \,x^{3}-4 e^{2} x^{2}-26 d e x -8 d^{2}\right ) \left (e \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}+\frac {2 \left (e x +d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{8 \left (e x +d \right )^{2} c^{2}}+\frac {x \left (e x +8 d \right ) \left (c x -1\right ) \left (c x +1\right ) \left (\frac {4 e \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 b^{2} c^{2} \left (e x +d \right )}{\left (c x -1\right ) \left (c x +1\right )}-\frac {\left (e x +d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {\left (e x +d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{8 c^{2} \left (e x +d \right )}\)	\(337\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.23 \[ \int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx=\frac {{\left (2 \, a^{2} + b^{2}\right )} c^{2} e x^{2} + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d x + {\left (2 \, b^{2} c^{2} e x^{2} + 4 \, b^{2} c^{2} d x - b^{2} e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 2 \, {\left (2 \, a b c^{2} e x^{2} + 4 \, a b c^{2} d x - a b e - {\left (b^{2} c e x + 4 \, b^{2} c d\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, {\left (a b c e x + 4 \, a b c d\right )} \sqrt {c^{2} x^{2} - 1}}{4 \, c^{2}} \] Input:

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

Output:

1/4*((2*a^2 + b^2)*c^2*e*x^2 + 4*(a^2 + 2*b^2)*c^2*d*x + (2*b^2*c^2*e*x^2 
+ 4*b^2*c^2*d*x - b^2*e)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 2*(2*a*b*c^2*e*x 
^2 + 4*a*b*c^2*d*x - a*b*e - (b^2*c*e*x + 4*b^2*c*d)*sqrt(c^2*x^2 - 1))*lo 
g(c*x + sqrt(c^2*x^2 - 1)) - 2*(a*b*c*e*x + 4*a*b*c*d)*sqrt(c^2*x^2 - 1))/ 
c^2
 

Sympy [F]

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

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

Output:

Integral((a + b*acosh(c*x))**2*(d + e*x), x)
 

Maxima [F]

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

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

Output:

b^2*d*x*arccosh(c*x)^2 + 1/2*a^2*e*x^2 + 1/2*(2*x^2*arccosh(c*x) - c*(sqrt 
(c^2*x^2 - 1)*x/c^2 + log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3))*a*b*e + 1 
/2*(x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 - 2*integrate((c^3*x^4 + 
sqrt(c*x + 1)*sqrt(c*x - 1)*c^2*x^3 - c*x^2)*log(c*x + sqrt(c*x + 1)*sqrt( 
c*x - 1))/(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x), x)) 
*b^2*e + 2*b^2*d*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*d*x + 2*(c*x 
*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a*b*d/c
 

Giac [F(-2)]

Exception generated. \[ \int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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