Integrand size = 21, antiderivative size = 160 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {b e \left (6 c^2 d+e\right ) \left (1-c^2 x^2\right )}{3 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (1-c^2 x^2\right )^2}{9 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}+2 d e x (a+b \text {arccosh}(c x))+\frac {1}{3} e^2 x^3 (a+b \text {arccosh}(c x))+b c d^2 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
-d^2*(a+b*arccosh(c*x))/x+2*d*e*x*(a+b*arccosh(c*x))+1/3*e^2*x^3*(a+b*arcc osh(c*x))+b*c*d^2*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))+1/3*b*e*(6*c^2*d+e)* (-c^2*x^2+1)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/9*b*e^2*(-c^2*x^2+1)^2/c^3/ (c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.17 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.80 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{3} \left (-\frac {3 a d^2}{x}+6 a d e x+a e^2 x^3-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \left (2 e+c^2 \left (18 d+e x^2\right )\right )}{3 c^3}+\frac {b \left (-3 d^2+6 d e x^2+e^2 x^4\right ) \text {arccosh}(c x)}{x}-3 b c d^2 \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right ) \]
((-3*a*d^2)/x + 6*a*d*e*x + a*e^2*x^3 - (b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]* (2*e + c^2*(18*d + e*x^2)))/(3*c^3) + (b*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)*Ar cCosh[c*x])/x - 3*b*c*d^2*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])])/3
Time = 0.52 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6373, 27, 1905, 1578, 1192, 1467, 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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx\) |
\(\Big \downarrow \) 6373 |
\(\displaystyle -b c \int -\frac {-e^2 x^4-6 d e x^2+3 d^2}{3 x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d^2 (a+b \text {arccosh}(c x))}{x}+2 d e x (a+b \text {arccosh}(c x))+\frac {1}{3} e^2 x^3 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} b c \int \frac {-e^2 x^4-6 d e x^2+3 d^2}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {d^2 (a+b \text {arccosh}(c x))}{x}+2 d e x (a+b \text {arccosh}(c x))+\frac {1}{3} e^2 x^3 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 1905 |
\(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \int \frac {-e^2 x^4-6 d e x^2+3 d^2}{x \sqrt {c^2 x^2-1}}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}+2 d e x (a+b \text {arccosh}(c x))+\frac {1}{3} e^2 x^3 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \int \frac {-e^2 x^4-6 d e x^2+3 d^2}{x^2 \sqrt {c^2 x^2-1}}dx^2}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}+2 d e x (a+b \text {arccosh}(c x))+\frac {1}{3} e^2 x^3 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 1192 |
\(\displaystyle \frac {b \sqrt {c^2 x^2-1} \int \frac {-e^2 x^8-2 e \left (3 d c^2+e\right ) x^4+3 c^4 d^2-e^2-6 c^2 d e}{x^4+1}d\sqrt {c^2 x^2-1}}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}+2 d e x (a+b \text {arccosh}(c x))+\frac {1}{3} e^2 x^3 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 1467 |
\(\displaystyle \frac {b \sqrt {c^2 x^2-1} \int \left (\frac {3 d^2 c^4}{x^4+1}-e^2 x^4-e \left (6 d c^2+e\right )\right )d\sqrt {c^2 x^2-1}}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}+2 d e x (a+b \text {arccosh}(c x))+\frac {1}{3} e^2 x^3 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d^2 (a+b \text {arccosh}(c x))}{x}+2 d e x (a+b \text {arccosh}(c x))+\frac {1}{3} e^2 x^3 (a+b \text {arccosh}(c x))+\frac {b \sqrt {c^2 x^2-1} \left (3 c^4 d^2 \arctan \left (\sqrt {c^2 x^2-1}\right )-e \sqrt {c^2 x^2-1} \left (6 c^2 d+e\right )-\frac {1}{3} e^2 x^6\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
-((d^2*(a + b*ArcCosh[c*x]))/x) + 2*d*e*x*(a + b*ArcCosh[c*x]) + (e^2*x^3* (a + b*ArcCosh[c*x]))/3 + (b*Sqrt[-1 + c^2*x^2]*(-1/3*(e^2*x^6) - e*(6*c^2 *d + e)*Sqrt[-1 + c^2*x^2] + 3*c^4*d^2*ArcTan[Sqrt[-1 + c^2*x^2]]))/(3*c^3 *Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.5.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 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]}, Sim p[(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] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le Q[m + p, 0]))
Time = 0.61 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.11
method | result | size |
parts | \(a \left (\frac {e^{2} x^{3}}{3}+2 d e x -\frac {d^{2}}{x}\right )+b c \left (\frac {\operatorname {arccosh}\left (c x \right ) x^{3} e^{2}}{3 c}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) x d e}{c}-\frac {\operatorname {arccosh}\left (c x \right ) d^{2}}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} d^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+18 d \,c^{2} e \sqrt {c^{2} x^{2}-1}+e^{2} \sqrt {c^{2} x^{2}-1}\, c^{2} x^{2}+2 e^{2} \sqrt {c^{2} x^{2}-1}\right )}{9 c^{4} \sqrt {c^{2} x^{2}-1}}\right )\) | \(178\) |
derivativedivides | \(c \left (\frac {a \left (2 c^{3} x d e +\frac {c^{3} x^{3} e^{2}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {b \left (2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{3} e x +\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {\operatorname {arccosh}\left (c x \right ) c^{3} d^{2}}{x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} d^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+18 d \,c^{2} e \sqrt {c^{2} x^{2}-1}+e^{2} \sqrt {c^{2} x^{2}-1}\, c^{2} x^{2}+2 e^{2} \sqrt {c^{2} x^{2}-1}\right )}{9 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\right )\) | \(191\) |
default | \(c \left (\frac {a \left (2 c^{3} x d e +\frac {c^{3} x^{3} e^{2}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {b \left (2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{3} e x +\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {\operatorname {arccosh}\left (c x \right ) c^{3} d^{2}}{x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} d^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+18 d \,c^{2} e \sqrt {c^{2} x^{2}-1}+e^{2} \sqrt {c^{2} x^{2}-1}\, c^{2} x^{2}+2 e^{2} \sqrt {c^{2} x^{2}-1}\right )}{9 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\right )\) | \(191\) |
a*(1/3*e^2*x^3+2*d*e*x-d^2/x)+b*c*(1/3/c*arccosh(c*x)*x^3*e^2+2/c*arccosh( c*x)*x*d*e-arccosh(c*x)*d^2/c/x-1/9/c^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(9*c^4 *d^2*arctan(1/(c^2*x^2-1)^(1/2))+18*d*c^2*e*(c^2*x^2-1)^(1/2)+e^2*(c^2*x^2 -1)^(1/2)*c^2*x^2+2*e^2*(c^2*x^2-1)^(1/2))/(c^2*x^2-1)^(1/2))
Time = 0.29 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.48 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {3 \, a c^{3} e^{2} x^{4} + 18 \, b c^{4} d^{2} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 18 \, a c^{3} d e x^{2} - 9 \, a c^{3} d^{2} + 3 \, {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 3 \, {\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2} + {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} e^{2} x^{3} + 2 \, {\left (9 \, b c^{2} d e + b e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c^{3} x} \]
1/9*(3*a*c^3*e^2*x^4 + 18*b*c^4*d^2*x*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 1 8*a*c^3*d*e*x^2 - 9*a*c^3*d^2 + 3*(3*b*c^3*d^2 - 6*b*c^3*d*e - b*c^3*e^2)* x*log(-c*x + sqrt(c^2*x^2 - 1)) + 3*(b*c^3*e^2*x^4 + 6*b*c^3*d*e*x^2 - 3*b *c^3*d^2 + (3*b*c^3*d^2 - 6*b*c^3*d*e - b*c^3*e^2)*x)*log(c*x + sqrt(c^2*x ^2 - 1)) - (b*c^2*e^2*x^3 + 2*(9*b*c^2*d*e + b*e^2)*x)*sqrt(c^2*x^2 - 1))/ (c^3*x)
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{2}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{3} \, a e^{2} x^{3} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{2} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b e^{2} + 2 \, a d e x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d e}{c} - \frac {a d^{2}}{x} \]
1/3*a*e^2*x^3 - (c*arcsin(1/(c*abs(x))) + arccosh(c*x)/x)*b*d^2 + 1/9*(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)) *b*e^2 + 2*a*d*e*x + 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d*e/c - a* d^2/x
\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2}{x^2} \,d x \]