\(\int (d+e x) \text {arccosh}(c x)^2 \, dx\) [10]

Optimal result

Integrand size = 12, antiderivative size = 122 \[ \int (d+e x) \text {arccosh}(c x)^2 \, dx=2 d x+\frac {e x^2}{4}-\frac {2 d \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{c}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{2 c}-\frac {d^2 \text {arccosh}(c x)^2}{2 e}-\frac {e \text {arccosh}(c x)^2}{4 c^2}+\frac {(d+e x)^2 \text {arccosh}(c x)^2}{2 e} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.86 \[ \int (d+e x) \text {arccosh}(c x)^2 \, dx=2 d x+\frac {e x^2}{4}-\frac {2 d \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{c}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{2 c}+d x \text {arccosh}(c x)^2+\frac {e \left (-1+2 c^2 x^2\right ) \text {arccosh}(c x)^2}{4 c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.17, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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.

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

\(\Big \downarrow \) 6378

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

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((d + e*x)^2*ArcCosh[c*x]^2)/(2*e) - (c*((-2*d*e*x)/c - (e^2*x^2)/(4*c) + 
(2*d*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c^2 + (e^2*x*Sqrt[-1 + c 
*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(2*c^2) + (d^2*ArcCosh[c*x]^2)/(2*c) + (e^ 
2*ArcCosh[c*x]^2)/(4*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 = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {\frac {e \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4 c}+d \left (\operatorname {arccosh}\left (c x \right )^{2} c x -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )}{c}\)	\(100\)
default	\(\frac {\frac {e \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4 c}+d \left (\operatorname {arccosh}\left (c x \right )^{2} c x -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )}{c}\)	\(100\)
orering	\(\frac {\left (7 c^{2} e^{3} x^{4}+33 c^{2} d \,e^{2} x^{3}+20 c^{2} d^{2} e \,x^{2}+8 c^{2} d^{3} x -6 x^{2} e^{3}-30 d \,e^{2} x -10 d^{2} e \right ) \operatorname {arccosh}\left (c x \right )^{2}}{8 c^{2} \left (e x +d \right )^{2}}-\frac {\left (3 c^{2} e^{2} x^{4}+17 c^{2} d e \,x^{3}-4 e^{2} x^{2}-26 d e x -8 d^{2}\right ) \left (e \operatorname {arccosh}\left (c x \right )^{2}+\frac {2 \left (e x +d \right ) \operatorname {arccosh}\left (c x \right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{8 \left (e x +d \right )^{2} c^{2}}+\frac {x \left (e x +8 d \right ) \left (c x -1\right ) \left (c x +1\right ) \left (\frac {4 e \,\operatorname {arccosh}\left (c x \right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 c^{2} \left (e x +d \right )}{\left (c x -1\right ) \left (c x +1\right )}-\frac {\left (e x +d \right ) \operatorname {arccosh}\left (c x \right ) c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {\left (e x +d \right ) \operatorname {arccosh}\left (c x \right ) c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{8 c^{2} \left (e x +d \right )}\)	\(306\)

Input:

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

Output:

1/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))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80 \[ \int (d+e x) \text {arccosh}(c x)^2 \, dx=\frac {c^{2} e x^{2} + 8 \, c^{2} d x + {\left (2 \, c^{2} e x^{2} + 4 \, c^{2} d x - e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (c e x + 4 \, c d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, c^{2}} \] Input:

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

Output:

1/4*(c^2*e*x^2 + 8*c^2*d*x + (2*c^2*e*x^2 + 4*c^2*d*x - e)*log(c*x + sqrt( 
c^2*x^2 - 1))^2 - 2*sqrt(c^2*x^2 - 1)*(c*e*x + 4*c*d)*log(c*x + sqrt(c^2*x 
^2 - 1)))/c^2
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*(e*x^2 + 2*d*x)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 - integrate(( 
c^3*e*x^4 + 2*c^3*d*x^3 - c*e*x^2 - 2*c*d*x + (c^2*e*x^3 + 2*c^2*d*x^2)*sq 
rt(c*x + 1)*sqrt(c*x - 1))*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)
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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