\(\int (c+d x^2)^2 \text {arccosh}(a x) \, dx\) [124]

Optimal result

Integrand size = 14, antiderivative size = 147 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=-\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \sqrt {-1+a x} \sqrt {1+a x}}{15 a^5}-\frac {2 d \left (5 a^2 c+3 d\right ) (-1+a x)^{3/2} (1+a x)^{3/2}}{45 a^5}-\frac {d^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{25 a^5}+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x) \] Output:

-1/15*(15*a^4*c^2+10*a^2*c*d+3*d^2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5-2/45*d 
*(5*a^2*c+3*d)*(a*x-1)^(3/2)*(a*x+1)^(3/2)/a^5-1/25*d^2*(a*x-1)^(5/2)*(a*x 
+1)^(5/2)/a^5+c^2*x*arccosh(a*x)+2/3*c*d*x^3*arccosh(a*x)+1/5*d^2*x^5*arcc 
osh(a*x)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.70 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (24 d^2+4 a^2 d \left (25 c+3 d x^2\right )+a^4 \left (225 c^2+50 c d x^2+9 d^2 x^4\right )\right )}{225 a^5}+\left (c^2 x+\frac {2}{3} c d x^3+\frac {d^2 x^5}{5}\right ) \text {arccosh}(a x) \] Input:

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

Output:

-1/225*(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(24*d^2 + 4*a^2*d*(25*c + 3*d*x^2) + 
a^4*(225*c^2 + 50*c*d*x^2 + 9*d^2*x^4)))/a^5 + (c^2*x + (2*c*d*x^3)/3 + (d 
^2*x^5)/5)*ArcCosh[a*x]
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6323, 27, 1905, 1576, 1140, 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.

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

\(\Big \downarrow \) 6323

\(\displaystyle -a \int \frac {x \left (3 d^2 x^4+10 c d x^2+15 c^2\right )}{15 \sqrt {a x-1} \sqrt {a x+1}}dx+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{15} a \int \frac {x \left (3 d^2 x^4+10 c d x^2+15 c^2\right )}{\sqrt {a x-1} \sqrt {a x+1}}dx+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\)

\(\Big \downarrow \) 1905

\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \frac {x \left (3 d^2 x^4+10 c d x^2+15 c^2\right )}{\sqrt {a^2 x^2-1}}dx}{15 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\)

\(\Big \downarrow \) 1576

\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \frac {3 d^2 x^4+10 c d x^2+15 c^2}{\sqrt {a^2 x^2-1}}dx^2}{30 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\)

\(\Big \downarrow \) 1140

\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \left (\frac {3 \left (a^2 x^2-1\right )^{3/2} d^2}{a^4}+\frac {2 \left (5 c a^2+3 d\right ) \sqrt {a^2 x^2-1} d}{a^4}+\frac {15 c^2 a^4+10 c d a^2+3 d^2}{a^4 \sqrt {a^2 x^2-1}}\right )dx^2}{30 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \left (\frac {4 d \left (a^2 x^2-1\right )^{3/2} \left (5 a^2 c+3 d\right )}{3 a^6}+\frac {6 d^2 \left (a^2 x^2-1\right )^{5/2}}{5 a^6}+\frac {2 \sqrt {a^2 x^2-1} \left (15 a^4 c^2+10 a^2 c d+3 d^2\right )}{a^6}\right )}{30 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \text {arccosh}(a x)+\frac {2}{3} c d x^3 \text {arccosh}(a x)+\frac {1}{5} d^2 x^5 \text {arccosh}(a x)\)

Input:

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

Output:

-1/30*(a*Sqrt[-1 + a^2*x^2]*((2*(15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*Sqrt[-1 
+ a^2*x^2])/a^6 + (4*d*(5*a^2*c + 3*d)*(-1 + a^2*x^2)^(3/2))/(3*a^6) + (6* 
d^2*(-1 + a^2*x^2)^(5/2))/(5*a^6)))/(Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + c^2*x 
*ArcCosh[a*x] + (2*c*d*x^3*ArcCosh[a*x])/3 + (d^2*x^5*ArcCosh[a*x])/5
 

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 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
 

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

rule 1905

Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) 
*(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x 
_Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ 
q]/(d1*d2 + e1*e2*x^n)^FracPart[q])   Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a 
+ b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, 
q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
 

rule 2009

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

rule 6323

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb 
ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCosh[c*x])   u, 
 x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] 
, x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] 
|| ILtQ[p + 1/2, 0])
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.72

method	result	size
parts	\(\frac {d^{2} x^{5} \operatorname {arccosh}\left (a x \right )}{5}+\frac {2 c d \,x^{3} \operatorname {arccosh}\left (a x \right )}{3}+c^{2} x \,\operatorname {arccosh}\left (a x \right )-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (9 a^{4} d^{2} x^{4}+50 a^{4} c d \,x^{2}+225 a^{4} c^{2}+12 a^{2} d^{2} x^{2}+100 a^{2} c d +24 d^{2}\right )}{225 a^{5}}\)	\(106\)
derivativedivides	\(\frac {\operatorname {arccosh}\left (a x \right ) c^{2} a x +\frac {2 a \,\operatorname {arccosh}\left (a x \right ) c d \,x^{3}}{3}+\frac {a \,\operatorname {arccosh}\left (a x \right ) d^{2} x^{5}}{5}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (9 a^{4} d^{2} x^{4}+50 a^{4} c d \,x^{2}+225 a^{4} c^{2}+12 a^{2} d^{2} x^{2}+100 a^{2} c d +24 d^{2}\right )}{225 a^{4}}}{a}\)	\(113\)
default	\(\frac {\operatorname {arccosh}\left (a x \right ) c^{2} a x +\frac {2 a \,\operatorname {arccosh}\left (a x \right ) c d \,x^{3}}{3}+\frac {a \,\operatorname {arccosh}\left (a x \right ) d^{2} x^{5}}{5}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (9 a^{4} d^{2} x^{4}+50 a^{4} c d \,x^{2}+225 a^{4} c^{2}+12 a^{2} d^{2} x^{2}+100 a^{2} c d +24 d^{2}\right )}{225 a^{4}}}{a}\)	\(113\)
orering	\(\frac {x \left (81 a^{6} d^{3} x^{6}+395 a^{6} c \,d^{2} x^{4}+1275 a^{6} c^{2} d \,x^{2}+12 a^{4} d^{3} x^{4}+225 a^{6} c^{3}+200 a^{4} c \,d^{2} x^{2}-900 a^{4} c^{2} d +48 a^{2} d^{3} x^{2}-400 a^{2} c \,d^{2}-96 d^{3}\right ) \operatorname {arccosh}\left (a x \right )}{225 \left (d \,x^{2}+c \right ) a^{6}}-\frac {\left (9 a^{4} d^{2} x^{4}+50 a^{4} c d \,x^{2}+225 a^{4} c^{2}+12 a^{2} d^{2} x^{2}+100 a^{2} c d +24 d^{2}\right ) \left (a x -1\right ) \left (a x +1\right ) \left (4 \left (d \,x^{2}+c \right ) \operatorname {arccosh}\left (a x \right ) d x +\frac {\left (d \,x^{2}+c \right )^{2} a}{\sqrt {a x -1}\, \sqrt {a x +1}}\right )}{225 a^{6} \left (d \,x^{2}+c \right )^{2}}\)	\(240\)

Input:

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

Output:

1/5*d^2*x^5*arccosh(a*x)+2/3*c*d*x^3*arccosh(a*x)+c^2*x*arccosh(a*x)-1/225 
/a^5*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(9*a^4*d^2*x^4+50*a^4*c*d*x^2+225*a^4*c^2 
+12*a^2*d^2*x^2+100*a^2*c*d+24*d^2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.82 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=\frac {15 \, {\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (9 \, a^{4} d^{2} x^{4} + 225 \, a^{4} c^{2} + 100 \, a^{2} c d + 2 \, {\left (25 \, a^{4} c d + 6 \, a^{2} d^{2}\right )} x^{2} + 24 \, d^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{225 \, a^{5}} \] Input:

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

Output:

1/225*(15*(3*a^5*d^2*x^5 + 10*a^5*c*d*x^3 + 15*a^5*c^2*x)*log(a*x + sqrt(a 
^2*x^2 - 1)) - (9*a^4*d^2*x^4 + 225*a^4*c^2 + 100*a^2*c*d + 2*(25*a^4*c*d 
+ 6*a^2*d^2)*x^2 + 24*d^2)*sqrt(a^2*x^2 - 1))/a^5
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.05 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=-\frac {1}{225} \, {\left (\frac {9 \, \sqrt {a^{2} x^{2} - 1} d^{2} x^{4}}{a^{2}} + \frac {50 \, \sqrt {a^{2} x^{2} - 1} c d x^{2}}{a^{2}} + \frac {225 \, \sqrt {a^{2} x^{2} - 1} c^{2}}{a^{2}} + \frac {12 \, \sqrt {a^{2} x^{2} - 1} d^{2} x^{2}}{a^{4}} + \frac {100 \, \sqrt {a^{2} x^{2} - 1} c d}{a^{4}} + \frac {24 \, \sqrt {a^{2} x^{2} - 1} d^{2}}{a^{6}}\right )} a + \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \operatorname {arcosh}\left (a x\right ) \] Input:

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

Output:

-1/225*(9*sqrt(a^2*x^2 - 1)*d^2*x^4/a^2 + 50*sqrt(a^2*x^2 - 1)*c*d*x^2/a^2 
 + 225*sqrt(a^2*x^2 - 1)*c^2/a^2 + 12*sqrt(a^2*x^2 - 1)*d^2*x^2/a^4 + 100* 
sqrt(a^2*x^2 - 1)*c*d/a^4 + 24*sqrt(a^2*x^2 - 1)*d^2/a^6)*a + 1/15*(3*d^2* 
x^5 + 10*c*d*x^3 + 15*c^2*x)*arccosh(a*x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.91 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=\frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{15 \, a^{5}} - \frac {50 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{2} c d + 9 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} d^{2} + 30 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d^{2}}{225 \, a^{5}} \] Input:

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

Output:

1/15*(3*d^2*x^5 + 10*c*d*x^3 + 15*c^2*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 1/ 
15*(15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*sqrt(a^2*x^2 - 1)/a^5 - 1/225*(50*(a^ 
2*x^2 - 1)^(3/2)*a^2*c*d + 9*(a^2*x^2 - 1)^(5/2)*d^2 + 30*(a^2*x^2 - 1)^(3 
/2)*d^2)/a^5
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.10 \[ \int \left (c+d x^2\right )^2 \text {arccosh}(a x) \, dx=\frac {225 \mathit {acosh} \left (a x \right ) a^{5} c^{2} x +150 \mathit {acosh} \left (a x \right ) a^{5} c d \,x^{3}+45 \mathit {acosh} \left (a x \right ) a^{5} d^{2} x^{5}-50 \sqrt {a^{2} x^{2}-1}\, a^{4} c d \,x^{2}-9 \sqrt {a^{2} x^{2}-1}\, a^{4} d^{2} x^{4}-100 \sqrt {a^{2} x^{2}-1}\, a^{2} c d -12 \sqrt {a^{2} x^{2}-1}\, a^{2} d^{2} x^{2}-24 \sqrt {a^{2} x^{2}-1}\, d^{2}-225 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{4} c^{2}}{225 a^{5}} \] Input:

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

Output:

(225*acosh(a*x)*a**5*c**2*x + 150*acosh(a*x)*a**5*c*d*x**3 + 45*acosh(a*x) 
*a**5*d**2*x**5 - 50*sqrt(a**2*x**2 - 1)*a**4*c*d*x**2 - 9*sqrt(a**2*x**2 
- 1)*a**4*d**2*x**4 - 100*sqrt(a**2*x**2 - 1)*a**2*c*d - 12*sqrt(a**2*x**2 
 - 1)*a**2*d**2*x**2 - 24*sqrt(a**2*x**2 - 1)*d**2 - 225*sqrt(a*x + 1)*sqr 
t(a*x - 1)*a**4*c**2)/(225*a**5)