Integrand size = 19, antiderivative size = 60 \[ \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx=-\frac {\text {arccosh}(a+b x)^2}{2 d}+\frac {\text {arccosh}(a+b x) \log \left (1+e^{2 \text {arccosh}(a+b x)}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a+b x)}\right )}{2 d} \]
-1/2*arccosh(b*x+a)^2/d+arccosh(b*x+a)*ln(1+(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+ 1)^(1/2))^2)/d+1/2*polylog(2,-(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))^2)/d
Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.88 \[ \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\text {arccosh}(a+b x) \left (\text {arccosh}(a+b x)+2 \log \left (1+e^{-2 \text {arccosh}(a+b x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a+b x)}\right )}{2 d} \]
(ArcCosh[a + b*x]*(ArcCosh[a + b*x] + 2*Log[1 + E^(-2*ArcCosh[a + b*x])]) - PolyLog[2, -E^(-2*ArcCosh[a + b*x])])/(2*d)
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6411, 27, 6297, 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.
\(\displaystyle \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int \frac {b \text {arccosh}(a+b x)}{d (a+b x)}d(a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\text {arccosh}(a+b x)}{a+b x}d(a+b x)}{d}\) |
\(\Big \downarrow \) 6297 |
\(\displaystyle \frac {\int \frac {\sqrt {\frac {a+b x-1}{a+b x+1}} (a+b x+1) \text {arccosh}(a+b x)}{a+b x}d\text {arccosh}(a+b x)}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -i \text {arccosh}(a+b x) \tan (i \text {arccosh}(a+b x))d\text {arccosh}(a+b x)}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {i \int \text {arccosh}(a+b x) \tan (i \text {arccosh}(a+b x))d\text {arccosh}(a+b x)}{d}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle -\frac {i \left (2 i \int \frac {e^{2 \text {arccosh}(a+b x)} \text {arccosh}(a+b x)}{1+e^{2 \text {arccosh}(a+b x)}}d\text {arccosh}(a+b x)-\frac {1}{2} i \text {arccosh}(a+b x)^2\right )}{d}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} \text {arccosh}(a+b x) \log \left (e^{2 \text {arccosh}(a+b x)}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {arccosh}(a+b x)}\right )d\text {arccosh}(a+b x)\right )-\frac {1}{2} i \text {arccosh}(a+b x)^2\right )}{d}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} \text {arccosh}(a+b x) \log \left (e^{2 \text {arccosh}(a+b x)}+1\right )-\frac {1}{4} \int e^{-2 \text {arccosh}(a+b x)} \log \left (1+e^{2 \text {arccosh}(a+b x)}\right )de^{2 \text {arccosh}(a+b x)}\right )-\frac {1}{2} i \text {arccosh}(a+b x)^2\right )}{d}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} \text {arccosh}(a+b x) \log \left (e^{2 \text {arccosh}(a+b x)}+1\right )+\frac {1}{4} \operatorname {PolyLog}(2,-a-b x)\right )-\frac {1}{2} i \text {arccosh}(a+b x)^2\right )}{d}\) |
((-I)*((-1/2*I)*ArcCosh[a + b*x]^2 + (2*I)*((ArcCosh[a + b*x]*Log[1 + E^(2 *ArcCosh[a + b*x])])/2 + PolyLog[2, -a - b*x]/4)))/d
3.3.90.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.77 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53
method | result | size |
derivativedivides | \(\frac {-\frac {b \operatorname {arccosh}\left (b x +a \right )^{2}}{2 d}+\frac {b \,\operatorname {arccosh}\left (b x +a \right ) \ln \left (1+\left (b x +a +\sqrt {b x +a -1}\, \sqrt {b x +a +1}\right )^{2}\right )}{d}+\frac {b \operatorname {polylog}\left (2, -\left (b x +a +\sqrt {b x +a -1}\, \sqrt {b x +a +1}\right )^{2}\right )}{2 d}}{b}\) | \(92\) |
default | \(\frac {-\frac {b \operatorname {arccosh}\left (b x +a \right )^{2}}{2 d}+\frac {b \,\operatorname {arccosh}\left (b x +a \right ) \ln \left (1+\left (b x +a +\sqrt {b x +a -1}\, \sqrt {b x +a +1}\right )^{2}\right )}{d}+\frac {b \operatorname {polylog}\left (2, -\left (b x +a +\sqrt {b x +a -1}\, \sqrt {b x +a +1}\right )^{2}\right )}{2 d}}{b}\) | \(92\) |
1/b*(-1/2*b/d*arccosh(b*x+a)^2+b/d*arccosh(b*x+a)*ln(1+(b*x+a+(b*x+a-1)^(1 /2)*(b*x+a+1)^(1/2))^2)+1/2*b/d*polylog(2,-(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1 )^(1/2))^2))
\[ \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int { \frac {\operatorname {arcosh}\left (b x + a\right )}{d x + \frac {a d}{b}} \,d x } \]
\[ \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {b \int \frac {\operatorname {acosh}{\left (a + b x \right )}}{a + b x}\, dx}{d} \]
\[ \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int { \frac {\operatorname {arcosh}\left (b x + a\right )}{d x + \frac {a d}{b}} \,d x } \]
\[ \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int { \frac {\operatorname {arcosh}\left (b x + a\right )}{d x + \frac {a d}{b}} \,d x } \]
Timed out. \[ \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int \frac {\mathrm {acosh}\left (a+b\,x\right )}{d\,x+\frac {a\,d}{b}} \,d x \]