Integrand size = 19, antiderivative size = 75 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))+b c d \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
-d*(a+b*arccosh(c*x))/x+e*x*(a+b*arccosh(c*x))+b*c*d*arctan((c*x-1)^(1/2)* (c*x+1)^(1/2))-b*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.40 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {a d}{x}+a e x-\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {b d \text {arccosh}(c x)}{x}+b e x \text {arccosh}(c x)+\frac {b c d \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
-((a*d)/x) + a*e*x - (b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c - (b*d*ArcCosh[c *x])/x + b*e*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 = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6371, 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.
\(\displaystyle \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx\) |
\(\Big \downarrow \) 6371 |
\(\displaystyle b c \int \frac {d-e x^2}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 960 |
\(\displaystyle b c \left (d \int \frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {e \sqrt {c x-1} \sqrt {c x+1}}{c^2}\right )-\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 103 |
\(\displaystyle b c \left (c d \int \frac {1}{(c x-1) (c x+1) c+c}d\left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {e \sqrt {c x-1} \sqrt {c x+1}}{c^2}\right )-\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))+b c \left (d \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {e \sqrt {c x-1} \sqrt {c x+1}}{c^2}\right )\) |
-((d*(a + b*ArcCosh[c*x]))/x) + e*x*(a + b*ArcCosh[c*x]) + b*c*(-((e*Sqrt[ -1 + c*x]*Sqrt[1 + c*x])/c^2) + d*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])
3.5.67.3.1 Defintions of rubi rules used
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), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*ArcCosh[c*x])/(f*(m + 1))) , x] + (Simp[e*(f*x)^(m + 3)*((a + b*ArcCosh[c*x])/(f^3*(m + 3))), x] - Sim p[b*(c/(f*(m + 1)*(m + 3))) Int[(f*x)^(m + 1)*((d*(m + 3) + e*(m + 1)*x^2 )/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && NeQ[m, -1] && NeQ[m, -3]
Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.36
method | result | size |
parts | \(a \left (e x -\frac {d}{x}\right )+b c \left (\frac {\operatorname {arccosh}\left (c x \right ) e x}{c}-\frac {\operatorname {arccosh}\left (c x \right ) d}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (d \,c^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {c^{2} x^{2}-1}}\right )\) | \(102\) |
derivativedivides | \(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) e c x -\frac {\operatorname {arccosh}\left (c x \right ) d c}{x}+\frac {\left (-d \,c^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-e \sqrt {c^{2} x^{2}-1}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{\sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\right )\) | \(105\) |
default | \(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) e c x -\frac {\operatorname {arccosh}\left (c x \right ) d c}{x}+\frac {\left (-d \,c^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-e \sqrt {c^{2} x^{2}-1}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{\sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\right )\) | \(105\) |
a*(e*x-d/x)+b*c*(1/c*arccosh(c*x)*e*x-arccosh(c*x)*d/c/x-1/c^2*(c*x-1)^(1/ 2)*(c*x+1)^(1/2)*(d*c^2*arctan(1/(c^2*x^2-1)^(1/2))+e*(c^2*x^2-1)^(1/2))/( c^2*x^2-1)^(1/2))
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.76 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {2 \, b c^{2} d x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + a c e x^{2} - \sqrt {c^{2} x^{2} - 1} b e x - a c d + {\left (b c d - b c e\right )} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c e x^{2} - b c d + {\left (b c d - b c e\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{c x} \]
(2*b*c^2*d*x*arctan(-c*x + sqrt(c^2*x^2 - 1)) + a*c*e*x^2 - sqrt(c^2*x^2 - 1)*b*e*x - a*c*d + (b*c*d - b*c*e)*x*log(-c*x + sqrt(c^2*x^2 - 1)) + (b*c *e*x^2 - b*c*d + (b*c*d - b*c*e)*x)*log(c*x + sqrt(c^2*x^2 - 1)))/(c*x)
\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{2}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=-{\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d + a e x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b e}{c} - \frac {a d}{x} \]
-(c*arcsin(1/(c*abs(x))) + arccosh(c*x)/x)*b*d + a*e*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*e/c - a*d/x
\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^2} \,d x \]