Integrand size = 12, antiderivative size = 84 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=-\frac {\left (9 a^2 c+2 d\right ) \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}-\frac {d x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+c x \text {arccosh}(a x)+\frac {1}{3} d x^3 \text {arccosh}(a x) \] Output:
-1/9*(9*a^2*c+2*d)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-1/9*d*x^2*(a*x-1)^(1/2) *(a*x+1)^(1/2)/a+c*x*arccosh(a*x)+1/3*d*x^3*arccosh(a*x)
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (2 d+a^2 \left (9 c+d x^2\right )\right )}{9 a^3}+\left (c x+\frac {d x^3}{3}\right ) \text {arccosh}(a x) \] Input:
Integrate[(c + d*x^2)*ArcCosh[a*x],x]
Output:
-1/9*(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2*d + a^2*(9*c + d*x^2)))/a^3 + (c*x + (d*x^3)/3)*ArcCosh[a*x]
Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, 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.
\(\displaystyle \int \text {arccosh}(a x) \left (c+d x^2\right ) \, dx\) |
\(\Big \downarrow \) 6323 |
\(\displaystyle -a \int \frac {x \left (d x^2+3 c\right )}{3 \sqrt {a x-1} \sqrt {a x+1}}dx+c x \text {arccosh}(a x)+\frac {1}{3} d x^3 \text {arccosh}(a x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{3} a \int \frac {x \left (d x^2+3 c\right )}{\sqrt {a x-1} \sqrt {a x+1}}dx+c x \text {arccosh}(a x)+\frac {1}{3} d x^3 \text {arccosh}(a x)\) |
\(\Big \downarrow \) 960 |
\(\displaystyle -\frac {1}{3} a \left (\frac {1}{3} \left (\frac {2 d}{a^2}+9 c\right ) \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1}}dx+\frac {d x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )+c x \text {arccosh}(a x)+\frac {1}{3} d x^3 \text {arccosh}(a x)\) |
\(\Big \downarrow \) 83 |
\(\displaystyle -\frac {1}{3} a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \left (\frac {2 d}{a^2}+9 c\right )}{3 a^2}+\frac {d x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )+c x \text {arccosh}(a x)+\frac {1}{3} d x^3 \text {arccosh}(a x)\) |
Input:
Int[(c + d*x^2)*ArcCosh[a*x],x]
Output:
-1/3*(a*(((9*c + (2*d)/a^2)*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a^2) + (d*x^2 *Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a^2))) + c*x*ArcCosh[a*x] + (d*x^3*ArcCo sh[a*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 = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67
method | result | size |
parts | \(\frac {d \,x^{3} \operatorname {arccosh}\left (a x \right )}{3}+c x \,\operatorname {arccosh}\left (a x \right )-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (d \,x^{2} a^{2}+9 a^{2} c +2 d \right )}{9 a^{3}}\) | \(56\) |
derivativedivides | \(\frac {\operatorname {arccosh}\left (a x \right ) c a x +\frac {a \,\operatorname {arccosh}\left (a x \right ) d \,x^{3}}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (d \,x^{2} a^{2}+9 a^{2} c +2 d \right )}{9 a^{2}}}{a}\) | \(62\) |
default | \(\frac {\operatorname {arccosh}\left (a x \right ) c a x +\frac {a \,\operatorname {arccosh}\left (a x \right ) d \,x^{3}}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (d \,x^{2} a^{2}+9 a^{2} c +2 d \right )}{9 a^{2}}}{a}\) | \(62\) |
orering | \(\frac {x \left (5 a^{4} d^{2} x^{4}+30 a^{4} c d \,x^{2}+9 a^{4} c^{2}+2 a^{2} d^{2} x^{2}-18 a^{2} c d -4 d^{2}\right ) \operatorname {arccosh}\left (a x \right )}{9 \left (d \,x^{2}+c \right ) a^{4}}-\frac {\left (d \,x^{2} a^{2}+9 a^{2} c +2 d \right ) \left (a x -1\right ) \left (a x +1\right ) \left (2 d x \,\operatorname {arccosh}\left (a x \right )+\frac {\left (d \,x^{2}+c \right ) a}{\sqrt {a x -1}\, \sqrt {a x +1}}\right )}{9 a^{4} \left (d \,x^{2}+c \right )}\) | \(148\) |
Input:
int((d*x^2+c)*arccosh(a*x),x,method=_RETURNVERBOSE)
Output:
1/3*d*x^3*arccosh(a*x)+c*x*arccosh(a*x)-1/9/a^3*(a*x-1)^(1/2)*(a*x+1)^(1/2 )*(a^2*d*x^2+9*a^2*c+2*d)
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.85 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=\frac {3 \, {\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (a^{2} d x^{2} + 9 \, a^{2} c + 2 \, d\right )} \sqrt {a^{2} x^{2} - 1}}{9 \, a^{3}} \] Input:
integrate((d*x^2+c)*arccosh(a*x),x, algorithm="fricas")
Output:
1/9*(3*(a^3*d*x^3 + 3*a^3*c*x)*log(a*x + sqrt(a^2*x^2 - 1)) - (a^2*d*x^2 + 9*a^2*c + 2*d)*sqrt(a^2*x^2 - 1))/a^3
\[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=\int \left (c + d x^{2}\right ) \operatorname {acosh}{\left (a x \right )}\, dx \] Input:
integrate((d*x**2+c)*acosh(a*x),x)
Output:
Integral((c + d*x**2)*acosh(a*x), x)
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.88 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=-\frac {1}{9} \, {\left (\frac {\sqrt {a^{2} x^{2} - 1} d x^{2}}{a^{2}} + \frac {9 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} - 1} d}{a^{4}}\right )} a + \frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \operatorname {arcosh}\left (a x\right ) \] Input:
integrate((d*x^2+c)*arccosh(a*x),x, algorithm="maxima")
Output:
-1/9*(sqrt(a^2*x^2 - 1)*d*x^2/a^2 + 9*sqrt(a^2*x^2 - 1)*c/a^2 + 2*sqrt(a^2 *x^2 - 1)*d/a^4)*a + 1/3*(d*x^3 + 3*c*x)*arccosh(a*x)
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=\frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d}{9 \, a^{3}} - \frac {\sqrt {a^{2} x^{2} - 1} {\left (3 \, a^{2} c + d\right )}}{3 \, a^{3}} \] Input:
integrate((d*x^2+c)*arccosh(a*x),x, algorithm="giac")
Output:
1/3*(d*x^3 + 3*c*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 1/9*(a^2*x^2 - 1)^(3/2) *d/a^3 - 1/3*sqrt(a^2*x^2 - 1)*(3*a^2*c + d)/a^3
Timed out. \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=\int \mathrm {acosh}\left (a\,x\right )\,\left (d\,x^2+c\right ) \,d x \] Input:
int(acosh(a*x)*(c + d*x^2),x)
Output:
int(acosh(a*x)*(c + d*x^2), x)
Time = 0.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.95 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=\frac {9 \mathit {acosh} \left (a x \right ) a^{3} c x +3 \mathit {acosh} \left (a x \right ) a^{3} d \,x^{3}-\sqrt {a^{2} x^{2}-1}\, a^{2} d \,x^{2}-2 \sqrt {a^{2} x^{2}-1}\, d -9 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} c}{9 a^{3}} \] Input:
int((d*x^2+c)*acosh(a*x),x)
Output:
(9*acosh(a*x)*a**3*c*x + 3*acosh(a*x)*a**3*d*x**3 - sqrt(a**2*x**2 - 1)*a* *2*d*x**2 - 2*sqrt(a**2*x**2 - 1)*d - 9*sqrt(a*x + 1)*sqrt(a*x - 1)*a**2*c )/(9*a**3)