Integrand size = 12, antiderivative size = 55 \[ \int \frac {a+b \text {arccosh}(c x)}{x} \, dx=\frac {(a+b \text {arccosh}(c x))^2}{2 b}+(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]
1/2*(a+b*arccosh(c*x))^2/b+(a+b*arccosh(c*x))*ln(1+1/(c*x+(c*x-1)^(1/2)*(c *x+1)^(1/2))^2)-1/2*b*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \text {arccosh}(c x)}{x} \, dx=a \log (x)+\frac {1}{2} b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]
a*Log[x] + (b*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])] ) - PolyLog[2, -E^(-2*ArcCosh[c*x])]))/2
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.33, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6297, 25, 3042, 26, 4201, 2620, 2715, 2838}
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.
\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x} \, dx\) |
\(\Big \downarrow \) 6297 |
\(\displaystyle \frac {\int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle \frac {i \left (2 i \int \frac {e^{-2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1+e^{-2 \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{-2 \text {arccosh}(c x)}\right )d(a+b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {i \left (2 i \left (-\frac {1}{4} b^2 \int e^{2 \text {arccosh}(c x)} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )de^{-2 \text {arccosh}(c x)}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}\) |
(I*((-1/2*I)*(a + b*ArcCosh[c*x])^2 + (2*I)*(-1/2*(b*(a + b*ArcCosh[c*x])* Log[1 + E^(-2*ArcCosh[c*x])]) + (b^2*PolyLog[2, -a - b*ArcCosh[c*x]])/4))) /b
3.2.37.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.33
method | result | size |
parts | \(a \ln \left (x \right )+b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\) | \(73\) |
derivativedivides | \(a \ln \left (c x \right )+b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\) | \(75\) |
default | \(a \ln \left (c x \right )+b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\) | \(75\) |
a*ln(x)+b*(-1/2*arccosh(c*x)^2+arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1 )^(1/2))^2)+1/2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2))
\[ \int \frac {a+b \text {arccosh}(c x)}{x} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{x} \,d x } \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x}\, dx \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{x} \,d x } \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x} \,d x \]