Integrand size = 10, antiderivative size = 170 \[ \int \frac {\text {sech}^{-1}(a+b x)}{x} \, dx=\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x) \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right ) \] Output:
arcsech(b*x+a)*ln(1-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))/ (1-(-a^2+1)^(1/2)))+arcsech(b*x+a)*ln(1-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*( 1/(b*x+a)+1)^(1/2))/(1+(-a^2+1)^(1/2)))-arcsech(b*x+a)*ln(1+(1/(b*x+a)+(1/ (b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))^2)+polylog(2,a*(1/(b*x+a)+(1/(b*x+a) -1)^(1/2)*(1/(b*x+a)+1)^(1/2))/(1-(-a^2+1)^(1/2)))+polylog(2,a*(1/(b*x+a)+ (1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))/(1+(-a^2+1)^(1/2)))-1/2*polylog(2 ,-(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))^2)
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.95 \[ \int \frac {\text {sech}^{-1}(a+b x)}{x} \, dx=-4 i \arcsin \left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \text {arctanh}\left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x) \log \left (1+e^{-2 \text {sech}^{-1}(a+b x)}\right )+\text {sech}^{-1}(a+b x) \log \left (1+\frac {\left (-1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right )+2 i \arcsin \left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (-1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right )+\text {sech}^{-1}(a+b x) \log \left (1-\frac {\left (1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right )-2 i \arcsin \left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (2,-\frac {\left (-1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right )-\operatorname {PolyLog}\left (2,\frac {\left (1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right ) \] Input:
Integrate[ArcSech[a + b*x]/x,x]
Output:
(-4*I)*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*ArcTanh[((1 + a)*Tanh[ArcSech[a + b*x]/2])/Sqrt[1 - a^2]] - ArcSech[a + b*x]*Log[1 + E^(-2*ArcSech[a + b*x]) ] + ArcSech[a + b*x]*Log[1 + (-1 + Sqrt[1 - a^2])/(a*E^ArcSech[a + b*x])] + (2*I)*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 + (-1 + Sqrt[1 - a^2])/(a*E ^ArcSech[a + b*x])] + ArcSech[a + b*x]*Log[1 - (1 + Sqrt[1 - a^2])/(a*E^Ar cSech[a + b*x])] - (2*I)*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 - (1 + Sqr t[1 - a^2])/(a*E^ArcSech[a + b*x])] + PolyLog[2, -E^(-2*ArcSech[a + b*x])] /2 - PolyLog[2, -((-1 + Sqrt[1 - a^2])/(a*E^ArcSech[a + b*x]))] - PolyLog[ 2, (1 + Sqrt[1 - a^2])/(a*E^ArcSech[a + b*x])]
Result contains complex when optimal does not.
Time = 1.06 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.37, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {6875, 25, 6129, 6104, 25, 3042, 26, 4201, 2620, 2715, 2838, 6096, 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.
| \(\displaystyle \int \frac {\text {sech}^{-1}(a+b x)}{x} \, dx\) |
| \(\Big \downarrow \) 6875 |
| \(\displaystyle -\int \frac {(a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{b x}d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int -\frac {(a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{b x}d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 6129 |
| \(\displaystyle \int \frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{\frac {a}{a+b x}-1}d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 6104 |
| \(\displaystyle a \int -\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{(a+b x) \left (1-\frac {a}{a+b x}\right )}d\text {sech}^{-1}(a+b x)-\int \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)d\text {sech}^{-1}(a+b x)-a \int \frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{(a+b x) \left (1-\frac {a}{a+b x}\right )}d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a \int \frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{(a+b x) \left (1-\frac {a}{a+b x}\right )}d\text {sech}^{-1}(a+b x)-\int -i \text {sech}^{-1}(a+b x) \tan \left (i \text {sech}^{-1}(a+b x)\right )d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -a \int \frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{(a+b x) \left (1-\frac {a}{a+b x}\right )}d\text {sech}^{-1}(a+b x)+i \int \text {sech}^{-1}(a+b x) \tan \left (i \text {sech}^{-1}(a+b x)\right )d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -a \int \frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{(a+b x) \left (1-\frac {a}{a+b x}\right )}d\text {sech}^{-1}(a+b x)+i \left (2 i \int \frac {e^{2 \text {sech}^{-1}(a+b x)} \text {sech}^{-1}(a+b x)}{1+e^{2 \text {sech}^{-1}(a+b x)}}d\text {sech}^{-1}(a+b x)-\frac {1}{2} i \text {sech}^{-1}(a+b x)^2\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -a \int \frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{(a+b x) \left (1-\frac {a}{a+b x}\right )}d\text {sech}^{-1}(a+b x)+i \left (2 i \left (\frac {1}{2} \text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )d\text {sech}^{-1}(a+b x)\right )-\frac {1}{2} i \text {sech}^{-1}(a+b x)^2\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -a \int \frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{(a+b x) \left (1-\frac {a}{a+b x}\right )}d\text {sech}^{-1}(a+b x)+i \left (2 i \left (\frac {1}{2} \text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )-\frac {1}{4} \int e^{-2 \text {sech}^{-1}(a+b x)} \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )de^{2 \text {sech}^{-1}(a+b x)}\right )-\frac {1}{2} i \text {sech}^{-1}(a+b x)^2\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -a \int \frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{(a+b x) \left (1-\frac {a}{a+b x}\right )}d\text {sech}^{-1}(a+b x)+i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )+\frac {1}{2} \text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )\right )-\frac {1}{2} i \text {sech}^{-1}(a+b x)^2\right )\) |
| \(\Big \downarrow \) 6096 |
| \(\displaystyle -a \left (\int \frac {e^{\text {sech}^{-1}(a+b x)} \text {sech}^{-1}(a+b x)}{-e^{\text {sech}^{-1}(a+b x)} a-\sqrt {1-a^2}+1}d\text {sech}^{-1}(a+b x)+\int \frac {e^{\text {sech}^{-1}(a+b x)} \text {sech}^{-1}(a+b x)}{-e^{\text {sech}^{-1}(a+b x)} a+\sqrt {1-a^2}+1}d\text {sech}^{-1}(a+b x)+\frac {\text {sech}^{-1}(a+b x)^2}{2 a}\right )+i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )+\frac {1}{2} \text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )\right )-\frac {1}{2} i \text {sech}^{-1}(a+b x)^2\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -a \left (\frac {\int \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )d\text {sech}^{-1}(a+b x)}{a}+\frac {\int \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )d\text {sech}^{-1}(a+b x)}{a}-\frac {\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a}-\frac {\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a}+\frac {\text {sech}^{-1}(a+b x)^2}{2 a}\right )+i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )+\frac {1}{2} \text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )\right )-\frac {1}{2} i \text {sech}^{-1}(a+b x)^2\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -a \left (\frac {\int e^{-\text {sech}^{-1}(a+b x)} \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )de^{\text {sech}^{-1}(a+b x)}}{a}+\frac {\int e^{-\text {sech}^{-1}(a+b x)} \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )de^{\text {sech}^{-1}(a+b x)}}{a}-\frac {\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a}-\frac {\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a}+\frac {\text {sech}^{-1}(a+b x)^2}{2 a}\right )+i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )+\frac {1}{2} \text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )\right )-\frac {1}{2} i \text {sech}^{-1}(a+b x)^2\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -a \left (-\frac {\operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a}-\frac {\operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a}-\frac {\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a}-\frac {\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a}+\frac {\text {sech}^{-1}(a+b x)^2}{2 a}\right )+i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )+\frac {1}{2} \text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )\right )-\frac {1}{2} i \text {sech}^{-1}(a+b x)^2\right )\) |
Input:
Int[ArcSech[a + b*x]/x,x]
Output:
-(a*(ArcSech[a + b*x]^2/(2*a) - (ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/a - (ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/a - PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - S qrt[1 - a^2])]/a - PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])]/ a)) + I*((-1/2*I)*ArcSech[a + b*x]^2 + (2*I)*((ArcSech[a + b*x]*Log[1 + E^ (2*ArcSech[a + b*x])])/2 + PolyLog[2, -E^(2*ArcSech[a + b*x])]/4))
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[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_ .)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 - b^2, 2] + b*E^(c + d*x))) , x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x))) , x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Tanh[ c + d*x]^n, x], x] - Simp[b/a Int[(e + f*x)^m*Sinh[c + d*x]*(Tanh[c + d*x ]^(n - 1)/(a + b*Cosh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.)*(G_)[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*Sech[(c_.) + (d_.)*(x_)]), x_Symbol] :> I nt[(e + f*x)^m*Cosh[c + d*x]*F[c + d*x]^n*(G[c + d*x]^p/(b + a*Cosh[c + d*x ])), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] && IntegersQ[m, n, p]
Int[((a_.) + ArcSech[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1) Subst[Int[(a + b*x)^p*Sech[x]*T anh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 886, normalized size of antiderivative = 5.21
| method | result | size |
| derivativedivides | \(-\operatorname {arcsech}\left (b x +a \right ) \ln \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\operatorname {arcsech}\left (b x +a \right ) \ln \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\operatorname {dilog}\left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\operatorname {dilog}\left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )+\frac {\operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{2}+\frac {\operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{2}-\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{2 \left (a^{2}-1\right )}+\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{2 a^{2}-2}+\operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )+\operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )+\frac {\left (a^{2}-1-\sqrt {-a^{2}+1}\right ) \operatorname {arcsech}\left (b x +a \right ) \left (\ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right ) a^{2}+\ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right ) a^{2}-2 \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )+2 \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right ) \sqrt {-a^{2}+1}\right )}{2 a^{2} \left (a^{2}-1\right )}\) | \(886\) |
| default | \(-\operatorname {arcsech}\left (b x +a \right ) \ln \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\operatorname {arcsech}\left (b x +a \right ) \ln \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\operatorname {dilog}\left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\operatorname {dilog}\left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )+\frac {\operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{2}+\frac {\operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{2}-\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{2 \left (a^{2}-1\right )}+\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{2 a^{2}-2}+\operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )+\operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )+\frac {\left (a^{2}-1-\sqrt {-a^{2}+1}\right ) \operatorname {arcsech}\left (b x +a \right ) \left (\ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right ) a^{2}+\ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right ) a^{2}-2 \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )+2 \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right ) \sqrt {-a^{2}+1}\right )}{2 a^{2} \left (a^{2}-1\right )}\) | \(886\) |
Input:
int(arcsech(b*x+a)/x,x,method=_RETURNVERBOSE)
Output:
-arcsech(b*x+a)*ln(1+I*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2)) )-arcsech(b*x+a)*ln(1-I*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2) ))-dilog(1+I*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2)))-dilog(1- I*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2)))+1/2*arcsech(b*x+a)* ln((-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)+ 1)/(1+(-a^2+1)^(1/2)))+1/2*arcsech(b*x+a)*ln((a*(1/(b*x+a)+(1/(b*x+a)-1)^( 1/2)*(1/(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)-1)/(-1+(-a^2+1)^(1/2)))-1/2*(-a^2 +1)^(1/2)/(a^2-1)*arcsech(b*x+a)*ln((-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/ (b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)+1)/(1+(-a^2+1)^(1/2)))+1/2*(-a^2+1)^(1/2) /(a^2-1)*arcsech(b*x+a)*ln((a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1) ^(1/2))+(-a^2+1)^(1/2)-1)/(-1+(-a^2+1)^(1/2)))+dilog((a*(1/(b*x+a)+(1/(b*x +a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)-1)/(-1+(-a^2+1)^(1/2)))+d ilog((-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2 )+1)/(1+(-a^2+1)^(1/2)))+1/2*(a^2-1-(-a^2+1)^(1/2))/a^2/(a^2-1)*arcsech(b* x+a)*(ln((a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))+(-a^2+1)^( 1/2)-1)/(-1+(-a^2+1)^(1/2)))*a^2+ln((-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/ (b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)+1)/(1+(-a^2+1)^(1/2)))*a^2-2*ln((-a*(1/(b *x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)+1)/(1+(-a^2+ 1)^(1/2)))+2*ln((-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))+(- a^2+1)^(1/2)+1)/(1+(-a^2+1)^(1/2)))*(-a^2+1)^(1/2))
\[ \int \frac {\text {sech}^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )}{x} \,d x } \] Input:
integrate(arcsech(b*x+a)/x,x, algorithm="fricas")
Output:
integral(arcsech(b*x + a)/x, x)
\[ \int \frac {\text {sech}^{-1}(a+b x)}{x} \, dx=\int \frac {\operatorname {asech}{\left (a + b x \right )}}{x}\, dx \] Input:
integrate(asech(b*x+a)/x,x)
Output:
Integral(asech(a + b*x)/x, x)
\[ \int \frac {\text {sech}^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )}{x} \,d x } \] Input:
integrate(arcsech(b*x+a)/x,x, algorithm="maxima")
Output:
integrate(arcsech(b*x + a)/x, x)
\[ \int \frac {\text {sech}^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )}{x} \,d x } \] Input:
integrate(arcsech(b*x+a)/x,x, algorithm="giac")
Output:
integrate(arcsech(b*x + a)/x, x)
Timed out. \[ \int \frac {\text {sech}^{-1}(a+b x)}{x} \, dx=\int \frac {\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}{x} \,d x \] Input:
int(acosh(1/(a + b*x))/x,x)
Output:
int(acosh(1/(a + b*x))/x, x)
\[ \int \frac {\text {sech}^{-1}(a+b x)}{x} \, dx=\int \frac {\mathit {asech} \left (b x +a \right )}{x}d x \] Input:
int(asech(b*x+a)/x,x)
Output:
int(asech(a + b*x)/x,x)