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

Optimal result

Integrand size = 12, antiderivative size = 146 \[ \int (d+e x)^2 \text {arccosh}(c x) \, dx=-\frac {\left (16 c^2 d^2+5 c d e+4 e^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}{18 c^3}-\frac {5 d e (-1+c x)^{3/2} \sqrt {1+c x}}{18 c^2}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{9 c}-\frac {1}{6} d \left (\frac {2 d^2}{e}+\frac {3 e}{c^2}\right ) \text {arccosh}(c x)+\frac {(d+e x)^3 \text {arccosh}(c x)}{3 e} \] Output:

-1/18*(16*c^2*d^2+5*c*d*e+4*e^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-5/18*d*e* 
(c*x-1)^(3/2)*(c*x+1)^(1/2)/c^2-1/9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(e*x+d)^2/ 
c-1/6*d*(2*d^2/e+3*e/c^2)*arccosh(c*x)+1/3*(e*x+d)^3*arccosh(c*x)/e
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.77 \[ \int (d+e x)^2 \text {arccosh}(c x) \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )-6 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \text {arccosh}(c x)+9 c d e \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{18 c^3} \] Input:

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

Output:

-1/18*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2 
*x^2)) - 6*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcCosh[c*x] + 9*c*d*e*Log[c* 
x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/c^3
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.

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

\(\Big \downarrow \) 6378

\(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^3}{3 e}-\frac {c \int \frac {(d+e x)^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 e}\)

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 164

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

\(\Big \downarrow \) 43

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

Input:

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

Output:

((d + e*x)^3*ArcCosh[c*x])/(3*e) - (c*((e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d 
+ e*x)^2)/(3*c^2) + ((e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*(4*c^2*d^2 + e^2) 
+ 5*c^2*d*e*x))/(2*c^2) + (3*d*(2*c^2*d^2 + 3*e^2)*ArcCosh[c*x])/(2*c))/(3 
*c^2)))/(3*e)
 

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 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]
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.29

method	result	size
orering	\(\frac {\left (10 c^{4} e^{3} x^{4}+42 c^{4} d \,e^{2} x^{3}+72 c^{4} d^{2} e \,x^{2}+18 c^{4} d^{3} x +4 c^{2} e^{3} x^{2}-27 c^{2} d \,e^{2} x -45 c^{2} e \,d^{2}-8 e^{3}\right ) \operatorname {arccosh}\left (c x \right )}{18 \left (e x +d \right ) c^{4}}-\frac {\left (2 c^{2} e^{2} x^{2}+9 c^{2} d e x +18 c^{2} d^{2}+4 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 \left (e x +d \right ) \operatorname {arccosh}\left (c x \right ) e +\frac {\left (e x +d \right )^{2} c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{18 c^{4} \left (e x +d \right )^{2}}\)	\(189\)
derivativedivides	\(\frac {\frac {c \,\operatorname {arccosh}\left (c x \right ) d^{3}}{3 e}+\operatorname {arccosh}\left (c x \right ) d^{2} c x +c \,\operatorname {arccosh}\left (c x \right ) d e \,x^{2}+\frac {c \,e^{2} \operatorname {arccosh}\left (c x \right ) x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 c^{3} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+18 c^{2} d^{2} e \sqrt {c^{2} x^{2}-1}+9 c^{2} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+2 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 c d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+4 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{18 c^{2} e \sqrt {c^{2} x^{2}-1}}}{c}\)	\(213\)
default	\(\frac {\frac {c \,\operatorname {arccosh}\left (c x \right ) d^{3}}{3 e}+\operatorname {arccosh}\left (c x \right ) d^{2} c x +c \,\operatorname {arccosh}\left (c x \right ) d e \,x^{2}+\frac {c \,e^{2} \operatorname {arccosh}\left (c x \right ) x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 c^{3} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+18 c^{2} d^{2} e \sqrt {c^{2} x^{2}-1}+9 c^{2} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+2 e^{3} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 c d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+4 e^{3} \sqrt {c^{2} x^{2}-1}\right )}{18 c^{2} e \sqrt {c^{2} x^{2}-1}}}{c}\)	\(213\)
parts	\(\frac {\operatorname {arccosh}\left (c x \right ) e^{2} x^{3}}{3}+\operatorname {arccosh}\left (c x \right ) e d \,x^{2}+\operatorname {arccosh}\left (c x \right ) d^{2} x +\frac {\operatorname {arccosh}\left (c x \right ) d^{3}}{3 e}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 \,\operatorname {csgn}\left (c \right ) c^{2} e^{3} x^{2} \sqrt {c^{2} x^{2}-1}+9 \,\operatorname {csgn}\left (c \right ) \sqrt {c^{2} x^{2}-1}\, c^{2} d \,e^{2} x +18 \,\operatorname {csgn}\left (c \right ) \sqrt {c^{2} x^{2}-1}\, c^{2} d^{2} e +6 \ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \operatorname {csgn}\left (c \right )+c x \right ) \operatorname {csgn}\left (c \right )\right ) c^{3} d^{3}+4 \,\operatorname {csgn}\left (c \right ) \sqrt {c^{2} x^{2}-1}\, e^{3}+9 \ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \operatorname {csgn}\left (c \right )+c x \right ) \operatorname {csgn}\left (c \right )\right ) c d \,e^{2}\right ) \operatorname {csgn}\left (c \right )}{18 e \,c^{3} \sqrt {c^{2} x^{2}-1}}\)	\(227\)

Input:

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

Output:

1/18*(10*c^4*e^3*x^4+42*c^4*d*e^2*x^3+72*c^4*d^2*e*x^2+18*c^4*d^3*x+4*c^2* 
e^3*x^2-27*c^2*d*e^2*x-45*c^2*d^2*e-8*e^3)/(e*x+d)/c^4*arccosh(c*x)-1/18*( 
2*c^2*e^2*x^2+9*c^2*d*e*x+18*c^2*d^2+4*e^2)/c^4*(c*x-1)*(c*x+1)/(e*x+d)^2* 
(2*(e*x+d)*arccosh(c*x)*e+(e*x+d)^2*c/(c*x-1)^(1/2)/(c*x+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.73 \[ \int (d+e x)^2 \text {arccosh}(c x) \, dx=\frac {3 \, {\left (2 \, c^{3} e^{2} x^{3} + 6 \, c^{3} d e x^{2} + 6 \, c^{3} d^{2} x - 3 \, c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, c^{2} e^{2} x^{2} + 9 \, c^{2} d e x + 18 \, c^{2} d^{2} + 4 \, e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{18 \, c^{3}} \] Input:

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

Output:

1/18*(3*(2*c^3*e^2*x^3 + 6*c^3*d*e*x^2 + 6*c^3*d^2*x - 3*c*d*e)*log(c*x + 
sqrt(c^2*x^2 - 1)) - (2*c^2*e^2*x^2 + 9*c^2*d*e*x + 18*c^2*d^2 + 4*e^2)*sq 
rt(c^2*x^2 - 1))/c^3
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.96 \[ \int (d+e x)^2 \text {arccosh}(c x) \, dx=-\frac {1}{18} \, {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} e^{2} x^{2}}{c^{2}} + \frac {9 \, \sqrt {c^{2} x^{2} - 1} d e x}{c^{2}} + \frac {18 \, \sqrt {c^{2} x^{2} - 1} d^{2}}{c^{2}} + \frac {9 \, d e \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} e^{2}}{c^{4}}\right )} c + \frac {1}{3} \, {\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )} \operatorname {arcosh}\left (c x\right ) \] Input:

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

Output:

-1/18*(2*sqrt(c^2*x^2 - 1)*e^2*x^2/c^2 + 9*sqrt(c^2*x^2 - 1)*d*e*x/c^2 + 1 
8*sqrt(c^2*x^2 - 1)*d^2/c^2 + 9*d*e*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c 
^3 + 4*sqrt(c^2*x^2 - 1)*e^2/c^4)*c + 1/3*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)* 
arccosh(c*x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.91 \[ \int (d+e x)^2 \text {arccosh}(c x) \, dx=\frac {{\left (e x + d\right )}^{3} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{3 \, e} - \frac {\sqrt {c^{2} x^{2} - 1} {\left ({\left (\frac {2 \, e^{3} x}{c} + \frac {9 \, d e^{2}}{c}\right )} x + \frac {2 \, {\left (9 \, c^{3} d^{2} e + 2 \, c e^{3}\right )}}{c^{4}}\right )} - \frac {3 \, {\left (2 \, c^{2} d^{3} + 3 \, d e^{2}\right )} \log \left ({\left | -x {\left | c \right |} + \sqrt {c^{2} x^{2} - 1} \right |}\right )}{c {\left | c \right |}}}{18 \, e} \] Input:

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

Output:

1/3*(e*x + d)^3*log(c*x + sqrt(c^2*x^2 - 1))/e - 1/18*(sqrt(c^2*x^2 - 1)*( 
(2*e^3*x/c + 9*d*e^2/c)*x + 2*(9*c^3*d^2*e + 2*c*e^3)/c^4) - 3*(2*c^2*d^3 
+ 3*d*e^2)*log(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c*abs(c)))/e
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.97 \[ \int (d+e x)^2 \text {arccosh}(c x) \, dx=\frac {18 \mathit {acosh} \left (c x \right ) c^{3} d^{2} x +18 \mathit {acosh} \left (c x \right ) c^{3} d e \,x^{2}+6 \mathit {acosh} \left (c x \right ) c^{3} e^{2} x^{3}-9 \sqrt {c^{2} x^{2}-1}\, c^{2} d e x -2 \sqrt {c^{2} x^{2}-1}\, c^{2} e^{2} x^{2}-4 \sqrt {c^{2} x^{2}-1}\, e^{2}-18 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} d^{2}-9 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) c d e}{18 c^{3}} \] Input:

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

Output:

(18*acosh(c*x)*c**3*d**2*x + 18*acosh(c*x)*c**3*d*e*x**2 + 6*acosh(c*x)*c* 
*3*e**2*x**3 - 9*sqrt(c**2*x**2 - 1)*c**2*d*e*x - 2*sqrt(c**2*x**2 - 1)*c* 
*2*e**2*x**2 - 4*sqrt(c**2*x**2 - 1)*e**2 - 18*sqrt(c*x + 1)*sqrt(c*x - 1) 
*c**2*d**2 - 9*log(sqrt(c**2*x**2 - 1) + c*x)*c*d*e)/(18*c**3)