Integrand size = 14, antiderivative size = 63 \[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\frac {c^2 \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{2 b} \]
-1/2*c^2*cosh(2*a/b)*Shi(2*a/b+2*arcsech(c*x))/b+1/2*c^2*Chi(2*a/b+2*arcse ch(c*x))*sinh(2*a/b)/b
Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\frac {c^2 \left (\text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )\right )}{2 b} \]
(c^2*(CoshIntegral[(2*a)/b + 2*ArcSech[c*x]]*Sinh[(2*a)/b] - Cosh[(2*a)/b] *SinhIntegral[(2*a)/b + 2*ArcSech[c*x]]))/(2*b)
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6839, 5971, 27, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}
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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx\) |
| \(\Big \downarrow \) 6839 |
| \(\displaystyle -c^2 \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle -c^2 \int \frac {\sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 \left (a+b \text {sech}^{-1}(c x)\right )}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} c^2 \int \frac {\sinh \left (2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{2} c^2 \int -\frac {i \sin \left (2 i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{2} i c^2 \int \frac {\sin \left (2 i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\cosh \left (\frac {2 a}{b}\right ) \int \frac {i \sinh \left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{2} i c^2 \left (i \cosh \left (\frac {2 a}{b}\right ) \int \frac {\sinh \left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} i c^2 \left (i \cosh \left (\frac {2 a}{b}\right ) \int -\frac {i \sin \left (\frac {2 i a}{b}+2 i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i a}{b}+2 i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i a}{b}+2 i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i a}{b}+2 i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b}-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i a}{b}+2 i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b}-\frac {i \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b}\right )\) |
(I/2)*c^2*(((-I)*CoshIntegral[(2*a)/b + 2*ArcSech[c*x]]*Sinh[(2*a)/b])/b + (I*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSech[c*x]])/b)
3.1.55.3.1 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[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ -(c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, A rcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G tQ[n, 0] || LtQ[m, -1])
Time = 0.68 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (\frac {2 a}{b}+2 \,\operatorname {arcsech}\left (c x \right )\right )}{4 b}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{4 b}\right )\) | \(60\) |
| default | \(c^{2} \left (-\frac {{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (\frac {2 a}{b}+2 \,\operatorname {arcsech}\left (c x \right )\right )}{4 b}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{4 b}\right )\) | \(60\) |
c^2*(-1/4/b*exp(2*a/b)*Ei(1,2*a/b+2*arcsech(c*x))+1/4/b*exp(-2*a/b)*Ei(1,- 2*arcsech(c*x)-2*a/b))
\[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}} \,d x } \]
\[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\int \frac {1}{x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}} \,d x } \]
\[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\int \frac {1}{x^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )} \,d x \]