Integrand size = 18, antiderivative size = 109 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {1}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^2}-\frac {\operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^2} \]
1/2*x*arccosh(a*x)/c^2/(-a^2*x^2+1)+arccosh(a*x)*arctanh(a*x+(a*x-1)^(1/2) *(a*x+1)^(1/2))/a/c^2+1/2*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^ 2-1/2*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2-1/2/a/c^2/(a*x-1)^( 1/2)/(a*x+1)^(1/2)
Time = 0.69 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.10 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {-\frac {2 \left (\sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\text {arccosh}(a x) \left (a x+\left (-1+a^2 x^2\right ) \log \left (1-e^{\text {arccosh}(a x)}\right )+\left (1-a^2 x^2\right ) \log \left (1+e^{\text {arccosh}(a x)}\right )\right )\right )}{-1+a^2 x^2}+2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{4 a c^2} \]
((-2*(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + ArcCosh[a*x]*(a*x + (-1 + a^2 *x^2)*Log[1 - E^ArcCosh[a*x]] + (1 - a^2*x^2)*Log[1 + E^ArcCosh[a*x]])))/( -1 + a^2*x^2) + 2*PolyLog[2, -E^ArcCosh[a*x]] - 2*PolyLog[2, E^ArcCosh[a*x ]])/(4*a*c^2)
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6316, 27, 83, 6318, 3042, 26, 4670, 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.
| \(\displaystyle \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {\int \frac {\text {arccosh}(a x)}{c \left (1-a^2 x^2\right )}dx}{2 c}+\frac {a \int \frac {x}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{2 c^2}+\frac {x \text {arccosh}(a x)}{2 c^2 \left (1-a^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\text {arccosh}(a x)}{1-a^2 x^2}dx}{2 c^2}+\frac {a \int \frac {x}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{2 c^2}+\frac {x \text {arccosh}(a x)}{2 c^2 \left (1-a^2 x^2\right )}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {\int \frac {\text {arccosh}(a x)}{1-a^2 x^2}dx}{2 c^2}+\frac {x \text {arccosh}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac {1}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle -\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {\frac {a x-1}{a x+1}} (a x+1)}d\text {arccosh}(a x)}{2 a c^2}+\frac {x \text {arccosh}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac {1}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int i \text {arccosh}(a x) \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a c^2}+\frac {x \text {arccosh}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac {1}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int \text {arccosh}(a x) \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a c^2}+\frac {x \text {arccosh}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac {1}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {i \left (i \int \log \left (1-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-i \int \log \left (1+e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a c^2}+\frac {x \text {arccosh}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac {1}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {i \left (i \int e^{-\text {arccosh}(a x)} \log \left (1-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-i \int e^{-\text {arccosh}(a x)} \log \left (1+e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}+2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a c^2}+\frac {x \text {arccosh}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac {1}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {x \text {arccosh}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac {i \left (2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )}{2 a c^2}-\frac {1}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}\) |
-1/2*1/(a*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x])/(2*c^2*(1 - a^2*x^2)) - ((I/2)*((2*I)*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]] + I*PolyLo g[2, -E^ArcCosh[a*x]] - I*PolyLog[2, E^ArcCosh[a*x]]))/(a*c^2)
3.1.56.3.1 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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 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[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Time = 0.48 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.48
| method | result | size |
| derivativedivides | \(\frac {-\frac {a x \,\operatorname {arccosh}\left (a x \right )+\sqrt {a x -1}\, \sqrt {a x +1}}{2 \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {\operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {\operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}}{a}\) | \(161\) |
| default | \(\frac {-\frac {a x \,\operatorname {arccosh}\left (a x \right )+\sqrt {a x -1}\, \sqrt {a x +1}}{2 \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {\operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {\operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}}{a}\) | \(161\) |
1/a*(-1/2*(a*x*arccosh(a*x)+(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a^2*x^2-1)/c^2-1 /2/c^2*arccosh(a*x)*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-1/2/c^2*polylog( 2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+1/2/c^2*arccosh(a*x)*ln(1+a*x+(a*x-1)^( 1/2)*(a*x+1)^(1/2))+1/2/c^2*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2)))
\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {acosh}{\left (a x \right )}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]
-1/16*((a^2*x^2 - 1)*log(a*x + 1)^2 + 2*(a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1) - (a^2*x^2 - 1)*log(a*x - 1)^2 + 4*a*x + 4*(2*a*x - (a^2*x^2 - 1)*lo g(a*x + 1) + (a^2*x^2 - 1)*log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)) - 2*(a^2*x^2 - 1)*log(a*x - 1))/(a^3*c^2*x^2 - a*c^2) + 1/4*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/(a*c^2) - 1/8*log(a*x + 1)/(a*c^2) + integrate(-1/4*(2*a*x - (a^2*x^2 - 1)*log(a*x + 1) + (a^2*x^2 - 1)*log(a*x - 1))/(a^5*c^2*x^5 - 2*a^3*c^2*x^3 + a*c^2*x + (a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*sqrt(a*x + 1)*sqrt(a*x - 1)), x)
\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]