Integrand size = 16, antiderivative size = 94 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (9 c^2 d+2 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c}+d x (a+b \text {arccosh}(c x))+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x)) \] Output:
-1/9*b*(9*c^2*d+2*e)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-1/9*b*e*x^2*(c*x-1)^( 1/2)*(c*x+1)^(1/2)/c+d*x*(a+b*arccosh(c*x))+1/3*e*x^3*(a+b*arccosh(c*x))
Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{9} \left (3 a x \left (3 d+e x^2\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (2 e+c^2 \left (9 d+e x^2\right )\right )}{c^3}+3 b x \left (3 d+e x^2\right ) \text {arccosh}(c x)\right ) \] Input:
Integrate[(d + e*x^2)*(a + b*ArcCosh[c*x]),x]
Output:
(3*a*x*(3*d + e*x^2) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*e + c^2*(9*d + e *x^2)))/c^3 + 3*b*x*(3*d + e*x^2)*ArcCosh[c*x])/9
Time = 0.47 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6323, 27, 960, 83}
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.
\(\displaystyle \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx\) |
\(\Big \downarrow \) 6323 |
\(\displaystyle -b c \int \frac {x \left (e x^2+3 d\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}dx+d x (a+b \text {arccosh}(c x))+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{3} b c \int \frac {x \left (e x^2+3 d\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+d x (a+b \text {arccosh}(c x))+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 960 |
\(\displaystyle -\frac {1}{3} b c \left (\frac {1}{3} \left (\frac {2 e}{c^2}+9 d\right ) \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {e x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )+d x (a+b \text {arccosh}(c x))+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 83 |
\(\displaystyle d x (a+b \text {arccosh}(c x))+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 e}{c^2}+9 d\right )}{3 c^2}+\frac {e x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\) |
Input:
Int[(d + e*x^2)*(a + b*ArcCosh[c*x]),x]
Output:
-1/3*(b*c*(((9*d + (2*e)/c^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2) + (e*x ^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2))) + d*x*(a + b*ArcCosh[c*x]) + (e *x^3*(a + b*ArcCosh[c*x]))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) *(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n *(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ (b1*b2*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n /2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
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])
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81
method | result | size |
parts | \(a \left (\frac {1}{3} e \,x^{3}+d x \right )+\frac {b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) e \,x^{3}}{3}+\operatorname {arccosh}\left (c x \right ) c x d -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9 c^{2}}\right )}{c}\) | \(76\) |
derivativedivides | \(\frac {\frac {a \left (c^{3} d x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c}\) | \(90\) |
default | \(\frac {\frac {a \left (c^{3} d x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c}\) | \(90\) |
orering | \(\frac {x \left (5 c^{4} e^{2} x^{4}+30 c^{4} d e \,x^{2}+9 c^{4} d^{2}+2 c^{2} e^{2} x^{2}-18 c^{2} d e -4 e^{2}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{9 \left (e \,x^{2}+d \right ) c^{4}}-\frac {\left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right ) \left (c x -1\right ) \left (c x +1\right ) \left (\frac {\left (e \,x^{2}+d \right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}+2 e x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )\right )}{9 c^{4} \left (e \,x^{2}+d \right )}\) | \(157\) |
Input:
int((a+b*arccosh(c*x))*(e*x^2+d),x,method=_RETURNVERBOSE)
Output:
a*(1/3*e*x^3+d*x)+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 = 94, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {3 \, a c^{3} e x^{3} + 9 \, a c^{3} d x + 3 \, {\left (b c^{3} e x^{3} + 3 \, b c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} e x^{2} + 9 \, b c^{2} d + 2 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c^{3}} \] Input:
integrate((e*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="fricas")
Output:
1/9*(3*a*c^3*e*x^3 + 9*a*c^3*d*x + 3*(b*c^3*e*x^3 + 3*b*c^3*d*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^2*e*x^2 + 9*b*c^2*d + 2*b*e)*sqrt(c^2*x^2 - 1)) /c^3
\[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \] Input:
integrate((e*x**2+d)*(a+b*acosh(c*x)),x)
Output:
Integral((a + b*acosh(c*x))*(d + e*x**2), x)
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{3} \, a e x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b e + a d x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d}{c} \] Input:
integrate((e*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="maxima")
Output:
1/3*a*e*x^3 + 1/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*s qrt(c^2*x^2 - 1)/c^4))*b*e + a*d*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1) )*b*d/c
Exception generated. \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*x^2+d)*(a+b*arccosh(c*x)),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. \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \] Input:
int((a + b*acosh(c*x))*(d + e*x^2),x)
Output:
int((a + b*acosh(c*x))*(d + e*x^2), x)
Time = 0.18 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {9 \mathit {acosh} \left (c x \right ) b \,c^{3} d x +3 \mathit {acosh} \left (c x \right ) b \,c^{3} e \,x^{3}-\sqrt {c^{2} x^{2}-1}\, b \,c^{2} e \,x^{2}-2 \sqrt {c^{2} x^{2}-1}\, b e -9 \sqrt {c x +1}\, \sqrt {c x -1}\, b \,c^{2} d +9 a \,c^{3} d x +3 a \,c^{3} e \,x^{3}}{9 c^{3}} \] Input:
int((e*x^2+d)*(a+b*acosh(c*x)),x)
Output:
(9*acosh(c*x)*b*c**3*d*x + 3*acosh(c*x)*b*c**3*e*x**3 - sqrt(c**2*x**2 - 1 )*b*c**2*e*x**2 - 2*sqrt(c**2*x**2 - 1)*b*e - 9*sqrt(c*x + 1)*sqrt(c*x - 1 )*b*c**2*d + 9*a*c**3*d*x + 3*a*c**3*e*x**3)/(9*c**3)