Integrand size = 27, antiderivative size = 248 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}-\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log (x)}{d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{6 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
(-a-b*arccosh(c*x))/d/x/(-c^2*d*x^2+d)^(3/2)+4/3*c^2*x*(a+b*arccosh(c*x))/ d/(-c^2*d*x^2+d)^(3/2)+8/3*c^2*x*(a+b*arccosh(c*x))/d^2/(-c^2*d*x^2+d)^(1/ 2)-1/6*b*c*(-c^2*d*x^2+d)^(1/2)/d^3/(-c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/ 2)+b*c*ln(x)*(-c^2*d*x^2+d)^(1/2)/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/6*b*c* ln(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.26 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.70 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (\frac {a+b \text {arccosh}(c x)}{x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 \sqrt {-1+c x} \sqrt {1+c x}}-b c \left (\frac {1}{6 \left (-1+c^2 x^2\right )}+\log (x)+\frac {5}{6} \log \left (1-c^2 x^2\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/(x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (4*c^2*x*(a + b*ArcCosh[c*x]))/(3*(-1 + c*x)^(3/2)*(1 + c *x)^(3/2)) + (8*c^2*x*(a + b*ArcCosh[c*x]))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x ]) - b*c*(1/(6*(-1 + c^2*x^2)) + Log[x] + (5*Log[1 - c^2*x^2])/6)))/(d^2*S qrt[d - c^2*d*x^2])
Time = 0.51 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.70, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6337, 27, 1578, 1195, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6337 |
| \(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {8 c^4 x^4-12 c^2 x^2+3}{3 d^3 x \left (1-c^2 x^2\right )^2}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {8 c^4 x^4-12 c^2 x^2+3}{x \left (1-c^2 x^2\right )^2}dx}{3 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {8 c^4 x^4-12 c^2 x^2+3}{x^2 \left (1-c^2 x^2\right )^2}dx^2}{6 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \left (\frac {5 c^2}{c^2 x^2-1}-\frac {c^2}{\left (c^2 x^2-1\right )^2}+\frac {3}{x^2}\right )dx^2}{6 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+3 \log \left (x^2\right )\right )}{6 d^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
-((a + b*ArcCosh[c*x])/(d*x*(d - c^2*d*x^2)^(3/2))) + (4*c^2*x*(a + b*ArcC osh[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (8*c^2*x*(a + b*ArcCosh[c*x]))/(3 *d^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*((-1 + c^2*x^2)^(-1) + 3*Log[x^2] + 5*Log[1 - c^2*x^2]))/(6*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.2.31.3.1 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] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
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[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCo sh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c *x])] Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b , c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Time = 1.26 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.75
| method | result | size |
| default | \(a \left (-\frac {1}{d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+4 c^{2} \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (16 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+16 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}-6 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{5} c^{5}-10 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-32 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+12 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+20 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}-c^{3} x^{3}+6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+16 c x \,\operatorname {arccosh}\left (c x \right )-6 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -10 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x c +c x \right )}{6 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) x}\) | \(434\) |
| parts | \(a \left (-\frac {1}{d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+4 c^{2} \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (16 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+16 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}-6 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{5} c^{5}-10 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-32 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+12 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+20 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}-c^{3} x^{3}+6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+16 c x \,\operatorname {arccosh}\left (c x \right )-6 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -10 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x c +c x \right )}{6 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) x}\) | \(434\) |
a*(-1/d/x/(-c^2*d*x^2+d)^(3/2)+4*c^2*(1/3/d*x/(-c^2*d*x^2+d)^(3/2)+2/3/d^2 *x/(-c^2*d*x^2+d)^(1/2)))-1/6*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+ 1)^(1/2)*(16*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^4*x^4+16*arccosh(c *x)*c^5*x^5-6*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^5*c^5-10*ln((c*x +(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^5*c^5-24*(c*x+1)^(1/2)*arccosh(c*x)*( c*x-1)^(1/2)*c^2*x^2-32*c^3*x^3*arccosh(c*x)+12*ln(1+(c*x+(c*x-1)^(1/2)*(c *x+1)^(1/2))^2)*x^3*c^3+20*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^3*c ^3-c^3*x^3+6*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+16*c*x*arccosh(c*x)- 6*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x*c-10*ln((c*x+(c*x-1)^(1/2)*( c*x+1)^(1/2))^2-1)*x*c+c*x)/d^3/(c^6*x^6-3*c^4*x^4+3*c^2*x^2-1)/x
\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^6*d^3*x^8 - 3*c^4*d ^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
1/3*a*(8*c^2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 4*c^2*x/((-c^2*d*x^2 + d)^(3/2 )*d) - 3/((-c^2*d*x^2 + d)^(3/2)*d*x)) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/((-c^2*d*x^2 + d)^(5/2)*x^2), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]