Integrand size = 18, antiderivative size = 168 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=2 b^2 d x+\frac {4 b^2 e x}{9 c^2}+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b e \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3}-\frac {2 b e x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c}+d x (a+b \text {arccosh}(c x))^2+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))^2 \] Output:
2*b^2*d*x+4/9*b^2*e*x/c^2+2/27*b^2*e*x^3-2*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2) *(a+b*arccosh(c*x))/c-4/9*b*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x ))/c^3-2/9*b*e*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))/c+d*x*(a +b*arccosh(c*x))^2+1/3*e*x^3*(a+b*arccosh(c*x))^2
Time = 0.16 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.04 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\frac {9 a^2 c^3 x \left (3 d+e x^2\right )-6 a b \sqrt {-1+c x} \sqrt {1+c x} \left (2 e+c^2 \left (9 d+e x^2\right )\right )+2 b^2 c x \left (6 e+c^2 \left (27 d+e x^2\right )\right )-6 b \left (-3 a c^3 x \left (3 d+e x^2\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (2 e+c^2 \left (9 d+e x^2\right )\right )\right ) \text {arccosh}(c x)+9 b^2 c^3 x \left (3 d+e x^2\right ) \text {arccosh}(c x)^2}{27 c^3} \] Input:
Integrate[(d + e*x^2)*(a + b*ArcCosh[c*x])^2,x]
Output:
(9*a^2*c^3*x*(3*d + e*x^2) - 6*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*e + c^2 *(9*d + e*x^2)) + 2*b^2*c*x*(6*e + c^2*(27*d + e*x^2)) - 6*b*(-3*a*c^3*x*( 3*d + e*x^2) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*e + c^2*(9*d + e*x^2)))*A rcCosh[c*x] + 9*b^2*c^3*x*(3*d + e*x^2)*ArcCosh[c*x]^2)/(27*c^3)
Time = 0.89 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6324, 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.
| \(\displaystyle \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6324 |
| \(\displaystyle \int \left (d (a+b \text {arccosh}(c x))^2+e x^2 (a+b \text {arccosh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 b e \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{9 c^3}+d x (a+b \text {arccosh}(c x))^2-\frac {2 b d \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c}+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))^2-\frac {2 b e x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{9 c}+\frac {4 b^2 e x}{9 c^2}+2 b^2 d x+\frac {2}{27} b^2 e x^3\) |
Input:
Int[(d + e*x^2)*(a + b*ArcCosh[c*x])^2,x]
Output:
2*b^2*d*x + (4*b^2*e*x)/(9*c^2) + (2*b^2*e*x^3)/27 - (2*b*d*Sqrt[-1 + c*x] *Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (4*b*e*Sqrt[-1 + c*x]*Sqrt[1 + c* x]*(a + b*ArcCosh[c*x]))/(9*c^3) - (2*b*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] *(a + b*ArcCosh[c*x]))/(9*c) + d*x*(a + b*ArcCosh[c*x])^2 + (e*x^3*(a + b* ArcCosh[c*x])^2)/3
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.21
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{3} e \,x^{3}+d x \right )+\frac {b^{2} \left (\frac {e \left (9 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}-6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27 c^{2}}+d \left (\operatorname {arccosh}\left (c x \right )^{2} c x -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) e \,x^{3}}{3}+\operatorname {arccosh}\left (c x \right ) c x d -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9 c^{2}}\right )}{c}\) | \(203\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (c^{3} d x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} c x -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {e \left (9 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}-6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c}\) | \(217\) |
| default | \(\frac {\frac {a^{2} \left (c^{3} d x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} c x -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {e \left (9 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}-6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c}\) | \(217\) |
| orering | \(\frac {x \left (19 c^{4} e^{3} x^{6}+209 c^{4} d \,e^{2} x^{4}+9 c^{4} d^{2} e \,x^{2}+24 c^{2} e^{3} x^{4}+27 c^{4} d^{3}-232 c^{2} d \,e^{2} x^{2}-48 e^{3} x^{2}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{27 \left (e \,x^{2}+d \right )^{2} c^{4}}-\frac {\left (6 c^{4} e^{2} x^{6}+110 c^{4} d e \,x^{4}+17 c^{2} e^{2} x^{4}-138 c^{2} d e \,x^{2}-27 c^{2} d^{2}-30 e^{2} x^{2}-6 d e \right ) \left (2 e x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}+\frac {2 \left (e \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{27 \left (e \,x^{2}+d \right )^{2} c^{4}}+\frac {x \left (c^{2} e \,x^{2}+27 c^{2} d +6 e \right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 e \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}+\frac {8 e x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 b^{2} c^{2} \left (e \,x^{2}+d \right )}{\left (c x -1\right ) \left (c x +1\right )}-\frac {\left (e \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {\left (e \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{27 c^{4} \left (e \,x^{2}+d \right )}\) | \(411\) |
Input:
int((e*x^2+d)*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
a^2*(1/3*e*x^3+d*x)+b^2/c*(1/27*e*(9*arccosh(c*x)^2*c^3*x^3-6*arccosh(c*x) *(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^2*x^2-12*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1) ^(1/2)+2*c^3*x^3+12*c*x)/c^2+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*(1/3*c*arccosh(c*x)*e*x^3+arccosh(c*x) *c*x*d-1/9/c^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(c^2*e*x^2+9*c^2*d+2*e))
Time = 0.10 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.24 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e x^{3} + 9 \, {\left (b^{2} c^{3} e x^{3} + 3 \, b^{2} c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 3 \, {\left (9 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d + 4 \, b^{2} c e\right )} x + 6 \, {\left (3 \, a b c^{3} e x^{3} + 9 \, a b c^{3} d x - {\left (b^{2} c^{2} e x^{2} + 9 \, b^{2} c^{2} d + 2 \, b^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (a b c^{2} e x^{2} + 9 \, a b c^{2} d + 2 \, a b e\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, c^{3}} \] Input:
integrate((e*x^2+d)*(a+b*arccosh(c*x))^2,x, algorithm="fricas")
Output:
1/27*((9*a^2 + 2*b^2)*c^3*e*x^3 + 9*(b^2*c^3*e*x^3 + 3*b^2*c^3*d*x)*log(c* x + sqrt(c^2*x^2 - 1))^2 + 3*(9*(a^2 + 2*b^2)*c^3*d + 4*b^2*c*e)*x + 6*(3* a*b*c^3*e*x^3 + 9*a*b*c^3*d*x - (b^2*c^2*e*x^2 + 9*b^2*c^2*d + 2*b^2*e)*sq rt(c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1)) - 6*(a*b*c^2*e*x^2 + 9*a*b*c ^2*d + 2*a*b*e)*sqrt(c^2*x^2 - 1))/c^3
\[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )\, dx \] Input:
integrate((e*x**2+d)*(a+b*acosh(c*x))**2,x)
Output:
Integral((a + b*acosh(c*x))**2*(d + e*x**2), x)
Time = 0.04 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.30 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\frac {1}{3} \, b^{2} e x^{3} \operatorname {arcosh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} e x^{3} + b^{2} d x \operatorname {arcosh}\left (c x\right )^{2} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a b e - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} e + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b d}{c} \] Input:
integrate((e*x^2+d)*(a+b*arccosh(c*x))^2,x, algorithm="maxima")
Output:
1/3*b^2*e*x^3*arccosh(c*x)^2 + 1/3*a^2*e*x^3 + b^2*d*x*arccosh(c*x)^2 + 2/ 9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1) /c^4))*a*b*e - 2/27*(3*c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/ c^4)*arccosh(c*x) - (c^2*x^3 + 6*x)/c^2)*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
Exception generated. \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*x^2+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
Timed out. \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \] Input:
int((a + b*acosh(c*x))^2*(d + e*x^2),x)
Output:
int((a + b*acosh(c*x))^2*(d + e*x^2), x)
\[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\frac {18 \mathit {acosh} \left (c x \right ) a b \,c^{3} d x +6 \mathit {acosh} \left (c x \right ) a b \,c^{3} e \,x^{3}-2 \sqrt {c^{2} x^{2}-1}\, a b \,c^{2} e \,x^{2}-4 \sqrt {c^{2} x^{2}-1}\, a b e -18 \sqrt {c x +1}\, \sqrt {c x -1}\, a b \,c^{2} d +9 \left (\int \mathit {acosh} \left (c x \right )^{2}d x \right ) b^{2} c^{3} d +9 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3} e +9 a^{2} c^{3} d x +3 a^{2} c^{3} e \,x^{3}}{9 c^{3}} \] Input:
int((e*x^2+d)*(a+b*acosh(c*x))^2,x)
Output:
(18*acosh(c*x)*a*b*c**3*d*x + 6*acosh(c*x)*a*b*c**3*e*x**3 - 2*sqrt(c**2*x **2 - 1)*a*b*c**2*e*x**2 - 4*sqrt(c**2*x**2 - 1)*a*b*e - 18*sqrt(c*x + 1)* sqrt(c*x - 1)*a*b*c**2*d + 9*int(acosh(c*x)**2,x)*b**2*c**3*d + 9*int(acos h(c*x)**2*x**2,x)*b**2*c**3*e + 9*a**2*c**3*d*x + 3*a**2*c**3*e*x**3)/(9*c **3)