Integrand size = 38, antiderivative size = 133 \[ \int \frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=-\frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{2 b c}-\frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \log \left (1+e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \]
-1/2*(a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/b/c-(a+b*arccosh((-c*x+ 1)^(1/2)/(c*x+1)^(1/2)))*ln(1+1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1 /2)/(c*x+1)^(1/2)-1)^(1/2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2)/c+1/ 2*b*polylog(2,-1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/ 2)-1)^(1/2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2)/c
\[ \int \frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int \frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx \]
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {7232, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{1-c^2 x^2} \, dx\) |
| \(\Big \downarrow \) 7232 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c x+1} \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{\sqrt {1-c x}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle -\frac {\int -\left (\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \tanh \left (\frac {a}{b}-\frac {\sqrt {1-c x}}{b \sqrt {c x+1}}\right )\right )d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \tanh \left (\frac {a}{b}-\frac {\sqrt {1-c x}}{b \sqrt {c x+1}}\right )d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -i \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \tan \left (\frac {i a}{b}-\frac {i \sqrt {1-c x}}{b \sqrt {c x+1}}\right )d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \tan \left (\frac {i a}{b}-\frac {i \sqrt {1-c x}}{b \sqrt {c x+1}}\right )d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {i \left (2 i \int \frac {e^{\frac {2 \left (a-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}} \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{1+e^{\frac {2 \left (a-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}}}d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {i (1-c x)}{2 (c x+1)}\right )}{b c}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (e^{\frac {2 \left (a-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}}+1\right )\right )-\frac {i (1-c x)}{2 (c x+1)}\right )}{b c}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int \frac {\sqrt {c x+1} \log \left (1+e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{\sqrt {1-c x}}de^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}-\frac {1}{2} b \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (e^{\frac {2 \left (a-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}}+1\right )\right )-\frac {i (1-c x)}{2 (c x+1)}\right )}{b c}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}\left (2,-a-b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (e^{\frac {2 \left (a-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}}+1\right )\right )-\frac {i (1-c x)}{2 (c x+1)}\right )}{b c}\) |
((-I)*(((-1/2*I)*(1 - c*x))/(1 + c*x) + (2*I)*(-1/2*(b*(a + b*ArcCosh[Sqrt [1 - c*x]/Sqrt[1 + c*x]])*Log[1 + E^((2*(a - Sqrt[1 - c*x]/Sqrt[1 + c*x])) /b)]) + (b^2*PolyLog[2, -a - b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]]])/4))) /(b*c)
3.3.71.3.1 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]
Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.) *(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e*(g/(C*(e*f - d *g))) Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]
Time = 0.69 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.56
| method | result | size |
| default | \(\frac {a \ln \left (c x +1\right )}{2 c}-\frac {a \ln \left (c x -1\right )}{2 c}+\frac {b \operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}-\frac {b \,\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}+1\right )}{c}-\frac {b \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}\) | \(207\) |
| parts | \(\frac {a \ln \left (c x +1\right )}{2 c}-\frac {a \ln \left (c x -1\right )}{2 c}+\frac {b \operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}-\frac {b \,\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}+1\right )}{c}-\frac {b \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}\) | \(207\) |
1/2*a/c*ln(c*x+1)-1/2*a/c*ln(c*x-1)+1/2*b/c*arccosh((-c*x+1)^(1/2)/(c*x+1) ^(1/2))^2-b/c*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2))*ln(((-c*x+1)^(1/2)/(c* x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)^(1/2)*((-c*x+1)^(1/2)/(c*x+1)^ (1/2)+1)^(1/2))^2+1)-1/2*b/c*polylog(2,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c *x+1)^(1/2)/(c*x+1)^(1/2)-1)^(1/2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2)) ^2)
\[ \int \frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \]
\[ \int \frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=- \int \frac {a}{c^{2} x^{2} - 1}\, dx - \int \frac {b \operatorname {acosh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
-Integral(a/(c**2*x**2 - 1), x) - Integral(b*acosh(sqrt(-c*x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1), x)
\[ \int \frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \]
-1/8*b*((2*(log(c*x + 1) - log(-c*x + 1))*log(c*x + 1) - log(c*x + 1)^2 + 2*log(c*x + 1)*log(-c*x + 1) - log(-c*x + 1)^2 - 4*(log(c*x + 1) - log(-c* x + 1))*log(sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqr t(-c*x + 1)) + sqrt(-c*x + 1)))/c + 8*integrate(1/2*(c*x + 1)*sqrt(-c*x + 1)*(log(c*x + 1) - log(-c*x + 1))/((c^2*x^2 - 1)*(c*x + 1)*sqrt(-c*x + 1) - (c^2*x^2 - 1)*(-c*x + 1)^(3/2) + ((c^2*x^2 - 1)*(c*x + 1) + (c^2*x^2 - 1 )*(c*x - 1))*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sq rt(-c*x + 1))), x) + 8*integrate(-1/4*sqrt(c*x + 1)*(log(c*x + 1) - log(-c *x + 1))/((c^2*x^2 - 1)*sqrt(c*x + 1) + (c^2*x^2 - 1)*sqrt(-c*x + 1)), x) - 8*integrate(1/4*sqrt(c*x + 1)*(log(c*x + 1) - log(-c*x + 1))/((c^2*x^2 - 1)*sqrt(c*x + 1) - (c^2*x^2 - 1)*sqrt(-c*x + 1)), x)) + 1/2*a*(log(c*x + 1)/c - log(c*x - 1)/c)
Exception generated. \[ \int \frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int -\frac {a+b\,\mathrm {acosh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )}{c^2\,x^2-1} \,d x \]