Integrand size = 23, antiderivative size = 76 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=b c d \sqrt {-1+c x} \sqrt {1+c x}-\frac {d (a+b \text {arccosh}(c x))}{x}-c^2 d x (a+b \text {arccosh}(c x))+b c d \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \] Output:
b*c*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)-d*(a+b*arccosh(c*x))/x-c^2*d*x*(a+b*arcc osh(c*x))+b*c*d*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))
Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.45 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {a d}{x}-a c^2 d x+b c d \sqrt {-1+c x} \sqrt {1+c x}-\frac {b d \text {arccosh}(c x)}{x}-b c^2 d x \text {arccosh}(c x)+\frac {b c d \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \] Input:
Integrate[((d - c^2*d*x^2)*(a + b*ArcCosh[c*x]))/x^2,x]
Output:
-((a*d)/x) - a*c^2*d*x + b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (b*d*ArcCosh [c*x])/x - b*c^2*d*x*ArcCosh[c*x] + (b*c*d*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[ -1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
Time = 0.53 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6336, 25, 27, 960, 103, 218}
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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx\) |
\(\Big \downarrow \) 6336 |
\(\displaystyle -b c \int -\frac {d \left (c^2 x^2+1\right )}{x \sqrt {c x-1} \sqrt {c x+1}}dx+c^2 (-d) x (a+b \text {arccosh}(c x))-\frac {d (a+b \text {arccosh}(c x))}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle b c \int \frac {d \left (c^2 x^2+1\right )}{x \sqrt {c x-1} \sqrt {c x+1}}dx+c^2 (-d) x (a+b \text {arccosh}(c x))-\frac {d (a+b \text {arccosh}(c x))}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b c d \int \frac {c^2 x^2+1}{x \sqrt {c x-1} \sqrt {c x+1}}dx+c^2 (-d) x (a+b \text {arccosh}(c x))-\frac {d (a+b \text {arccosh}(c x))}{x}\) |
\(\Big \downarrow \) 960 |
\(\displaystyle b c d \left (\int \frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}dx+\sqrt {c x-1} \sqrt {c x+1}\right )+c^2 (-d) x (a+b \text {arccosh}(c x))-\frac {d (a+b \text {arccosh}(c x))}{x}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle b c d \left (c \int \frac {1}{(c x-1) (c x+1) c+c}d\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\sqrt {c x-1} \sqrt {c x+1}\right )+c^2 (-d) x (a+b \text {arccosh}(c x))-\frac {d (a+b \text {arccosh}(c x))}{x}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle c^2 (-d) x (a+b \text {arccosh}(c x))-\frac {d (a+b \text {arccosh}(c x))}{x}+b c d \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\sqrt {c x-1} \sqrt {c x+1}\right )\) |
Input:
Int[((d - c^2*d*x^2)*(a + b*ArcCosh[c*x]))/x^2,x]
Output:
-((d*(a + b*ArcCosh[c*x]))/x) - c^2*d*x*(a + b*ArcCosh[c*x]) + b*c*d*(Sqrt [-1 + c*x]*Sqrt[1 + c*x] + ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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, f, m}, x] && E qQ[c^2*d + e, 0] && IGtQ[p, 0]
Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.21
method | result | size |
parts | \(-d a \left (c^{2} x +\frac {1}{x}\right )-d b c \left (c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\sqrt {c^{2} x^{2}-1}-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )\right )}{\sqrt {c^{2} x^{2}-1}}\right )\) | \(92\) |
derivativedivides | \(c \left (-d a \left (c x +\frac {1}{c x}\right )-d b \left (c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\sqrt {c^{2} x^{2}-1}-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )\right )}{\sqrt {c^{2} x^{2}-1}}\right )\right )\) | \(95\) |
default | \(c \left (-d a \left (c x +\frac {1}{c x}\right )-d b \left (c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\sqrt {c^{2} x^{2}-1}-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )\right )}{\sqrt {c^{2} x^{2}-1}}\right )\right )\) | \(95\) |
Input:
int((-c^2*d*x^2+d)*(a+b*arccosh(c*x))/x^2,x,method=_RETURNVERBOSE)
Output:
-d*a*(c^2*x+1/x)-d*b*c*(c*x*arccosh(c*x)+arccosh(c*x)/c/x-(c*x-1)^(1/2)*(c *x+1)^(1/2)/(c^2*x^2-1)^(1/2)*((c^2*x^2-1)^(1/2)-arctan(1/(c^2*x^2-1)^(1/2 ))))
Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.67 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {a c^{2} d x^{2} - 2 \, b c d x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {c^{2} x^{2} - 1} b c d x - {\left (b c^{2} + b\right )} d x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + a d + {\left (b c^{2} d x^{2} - {\left (b c^{2} + b\right )} d x + b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{x} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arccosh(c*x))/x^2,x, algorithm="fricas")
Output:
-(a*c^2*d*x^2 - 2*b*c*d*x*arctan(-c*x + sqrt(c^2*x^2 - 1)) - sqrt(c^2*x^2 - 1)*b*c*d*x - (b*c^2 + b)*d*x*log(-c*x + sqrt(c^2*x^2 - 1)) + a*d + (b*c^ 2*d*x^2 - (b*c^2 + b)*d*x + b*d)*log(c*x + sqrt(c^2*x^2 - 1)))/x
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=- d \left (\int a c^{2}\, dx + \int \left (- \frac {a}{x^{2}}\right )\, dx + \int b c^{2} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{2}}\right )\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)*(a+b*acosh(c*x))/x**2,x)
Output:
-d*(Integral(a*c**2, x) + Integral(-a/x**2, x) + Integral(b*c**2*acosh(c*x ), x) + Integral(-b*acosh(c*x)/x**2, x))
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=-a c^{2} d x - {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c d - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d - \frac {a d}{x} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arccosh(c*x))/x^2,x, algorithm="maxima")
Output:
-a*c^2*d*x - (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*c*d - (c*arcsin(1/(c *abs(x))) + arccosh(c*x)/x)*b*d - a*d/x
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arccosh(c*x))/x^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x^2} \,d x \] Input:
int(((a + b*acosh(c*x))*(d - c^2*d*x^2))/x^2,x)
Output:
int(((a + b*acosh(c*x))*(d - c^2*d*x^2))/x^2, x)
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {d \left (-\mathit {acosh} \left (c x \right ) b \,c^{2} x^{2}-\mathit {acosh} \left (c x \right ) b -2 \mathit {atan} \left (\sqrt {c^{2} x^{2}-1}+c x \right ) b c x +\sqrt {c x +1}\, \sqrt {c x -1}\, b c x -a \,c^{2} x^{2}-a \right )}{x} \] Input:
int((-c^2*d*x^2+d)*(a+b*acosh(c*x))/x^2,x)
Output:
(d*( - acosh(c*x)*b*c**2*x**2 - acosh(c*x)*b - 2*atan(sqrt(c**2*x**2 - 1) + c*x)*b*c*x + sqrt(c*x + 1)*sqrt(c*x - 1)*b*c*x - a*c**2*x**2 - a))/x