Integrand size = 27, antiderivative size = 158 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log (x)}{d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \]
(-a-b*arccosh(c*x))/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2*x*(a+b*arccosh(c*x))/d/ (-c^2*d*x^2+d)^(1/2)+b*c*ln(x)*(-c^2*d*x^2+d)^(1/2)/d^2/(c*x-1)^(1/2)/(c*x +1)^(1/2)+1/2*b*c*ln(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/d^2/(c*x-1)^(1/2)/(c *x+1)^(1/2)
Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-2 a+4 a c^2 x^2+2 b \left (-1+2 c^2 x^2\right ) \text {arccosh}(c x)-2 b c x \sqrt {-1+c x} \sqrt {1+c x} \log (x)-b c x \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 d x \sqrt {d-c^2 d x^2}} \]
(-2*a + 4*a*c^2*x^2 + 2*b*(-1 + 2*c^2*x^2)*ArcCosh[c*x] - 2*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x] - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c ^2*x^2])/(2*d*x*Sqrt[d - c^2*d*x^2])
Time = 0.40 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6337, 25, 27, 354, 86, 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.
\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6337 |
\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {1-2 c^2 x^2}{d^2 x \left (1-c^2 x^2\right )}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {1-2 c^2 x^2}{d^2 x \left (1-c^2 x^2\right )}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {1-2 c^2 x^2}{x \left (1-c^2 x^2\right )}dx}{d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {1-2 c^2 x^2}{x^2 \left (1-c^2 x^2\right )}dx^2}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \left (\frac {c^2}{c^2 x^2-1}+\frac {1}{x^2}\right )dx^2}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}}\) |
-((a + b*ArcCosh[c*x])/(d*x*Sqrt[d - c^2*d*x^2])) + (2*c^2*x*(a + b*ArcCos h[c*x]))/(d*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*(Log[x^2] + Lo g[1 - c^2*x^2]))/(2*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.2.21.3.1 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[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(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 = 1.22 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.68
method | result | size |
default | \(a \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{3} c^{3}+2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{4} c^{4}+\sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x c -2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{2} c^{2}+\operatorname {arccosh}\left (c x \right )\right ) \left (2 c^{2} x^{2}-1+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{d^{2} \left (c^{2} x^{2}-1\right ) x}\) | \(266\) |
parts | \(a \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{3} c^{3}+2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{4} c^{4}+\sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x c -2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{2} c^{2}+\operatorname {arccosh}\left (c x \right )\right ) \left (2 c^{2} x^{2}-1+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{d^{2} \left (c^{2} x^{2}-1\right ) x}\) | \(266\) |
a*(-1/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2/d*x/(-c^2*d*x^2+d)^(1/2))-b*(-2*(c*x- 1)^(1/2)*(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^4-1)*x^3*c^3+2 *ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^4-1)*x^4*c^4+(c*x-1)^(1/2)*(c*x+1)^( 1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^4-1)*x*c-2*ln((c*x+(c*x-1)^(1/2) *(c*x+1)^(1/2))^4-1)*x^2*c^2+arccosh(c*x))*(2*c^2*x^2-1+2*(c*x-1)^(1/2)*(c *x+1)^(1/2)*c*x)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)/x
\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^4*d^2*x^6 - 2*c^2*d^ 2*x^4 + d^2*x^2), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {1}{2} \, b c {\left (\frac {\sqrt {-d} \log \left (c x + 1\right )}{d^{2}} + \frac {\sqrt {-d} \log \left (c x - 1\right )}{d^{2}} + \frac {2 \, \sqrt {-d} \log \left (x\right )}{d^{2}}\right )} + {\left (\frac {2 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x}\right )} b \operatorname {arcosh}\left (c x\right ) + {\left (\frac {2 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x}\right )} a \]
1/2*b*c*(sqrt(-d)*log(c*x + 1)/d^2 + sqrt(-d)*log(c*x - 1)/d^2 + 2*sqrt(-d )*log(x)/d^2) + (2*c^2*x/(sqrt(-c^2*d*x^2 + d)*d) - 1/(sqrt(-c^2*d*x^2 + d )*d*x))*b*arccosh(c*x) + (2*c^2*x/(sqrt(-c^2*d*x^2 + d)*d) - 1/(sqrt(-c^2* d*x^2 + d)*d*x))*a
\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]