Integrand size = 14, antiderivative size = 46 \[ \int \frac {1}{x^2 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=-\frac {c \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{b}+\frac {c \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{b} \]
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^2 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=-\frac {c \left (\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )\right )}{b} \]
-((c*(Cosh[a/b]*CoshIntegral[a/b + ArcCsch[c*x]] - Sinh[a/b]*SinhIntegral[ a/b + ArcCsch[c*x]]))/b)
Time = 0.44 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6840, 3042, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx\) |
\(\Big \downarrow \) 6840 |
\(\displaystyle -c \int \frac {\sqrt {1+\frac {1}{c^2 x^2}}}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c \int \frac {\sin \left (i \text {csch}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle -c \left (\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)+i \sinh \left (\frac {a}{b}\right ) \int \frac {i \sinh \left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -c \left (\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)-\sinh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c \left (\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {csch}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)-\sinh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i a}{b}+i \text {csch}^{-1}(c x)\right )}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -c \left (\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {csch}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)+i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {csch}^{-1}(c x)\right )}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle -c \left (-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{b}+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i a}{b}+i \text {csch}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {csch}^{-1}(c x)}d\text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle -c \left (\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{b}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{b}\right )\) |
-(c*((Cosh[a/b]*CoshIntegral[a/b + ArcCsch[c*x]])/b - (Sinh[a/b]*SinhInteg ral[a/b + ArcCsch[c*x]])/b))
3.1.36.3.1 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[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[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ -(c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, A rcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G tQ[n, 0] || LtQ[m, -1])
\[\int \frac {1}{x^{2} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}d x\]
\[ \int \frac {1}{x^2 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}} \,d x } \]
\[ \int \frac {1}{x^2 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=\int \frac {1}{x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}\, dx \]
\[ \int \frac {1}{x^2 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}} \,d x } \]
\[ \int \frac {1}{x^2 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{x^2 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=\int \frac {1}{x^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )} \,d x \]