Integrand size = 14, antiderivative size = 117 \[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\frac {c^3 \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{4 b}+\frac {c^3 \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b}-\frac {c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b} \]
-1/4*c^3*cosh(a/b)*Shi(a/b+arcsech(c*x))/b-1/4*c^3*cosh(3*a/b)*Shi(3*a/b+3 *arcsech(c*x))/b+1/4*c^3*Chi(a/b+arcsech(c*x))*sinh(a/b)/b+1/4*c^3*Chi(3*a /b+3*arcsech(c*x))*sinh(3*a/b)/b
Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=-\frac {c^3 \left (-\text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )-\text {Chi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )\right )}{4 b} \]
-1/4*(c^3*(-(CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b]) - CoshIntegral[3* (a/b + ArcSech[c*x])]*Sinh[(3*a)/b] + Cosh[a/b]*SinhIntegral[a/b + ArcSech [c*x]] + Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSech[c*x])]))/b
Time = 0.51 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6839, 5971, 2009}
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.
\(\displaystyle \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx\) |
\(\Big \downarrow \) 6839 |
\(\displaystyle -c^3 \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c^3 x^3 \left (a+b \text {sech}^{-1}(c x)\right )}d\text {sech}^{-1}(c x)\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle -c^3 \int \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{4 c x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {\sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 \left (a+b \text {sech}^{-1}(c x)\right )}\right )d\text {sech}^{-1}(c x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -c^3 \left (-\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b}-\frac {\sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b}+\frac {\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b}\right )\) |
-(c^3*(-1/4*(CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b])/b - (CoshIntegral [(3*a)/b + 3*ArcSech[c*x]]*Sinh[(3*a)/b])/(4*b) + (Cosh[a/b]*SinhIntegral[ a/b + ArcSech[c*x]])/(4*b) + (Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcSe ch[c*x]])/(4*b)))
3.1.56.3.1 Defintions of rubi rules used
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 = 0.86 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (\frac {3 a}{b}+3 \,\operatorname {arcsech}\left (c x \right )\right )}{8 b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\frac {a}{b}+\operatorname {arcsech}\left (c x \right )\right )}{8 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsech}\left (c x \right )-\frac {a}{b}\right )}{8 b}+\frac {{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{8 b}\right )\) | \(110\) |
default | \(c^{3} \left (-\frac {{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (\frac {3 a}{b}+3 \,\operatorname {arcsech}\left (c x \right )\right )}{8 b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\frac {a}{b}+\operatorname {arcsech}\left (c x \right )\right )}{8 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsech}\left (c x \right )-\frac {a}{b}\right )}{8 b}+\frac {{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{8 b}\right )\) | \(110\) |
c^3*(-1/8/b*exp(3*a/b)*Ei(1,3*a/b+3*arcsech(c*x))-1/8/b*exp(a/b)*Ei(1,a/b+ arcsech(c*x))+1/8/b*exp(-a/b)*Ei(1,-arcsech(c*x)-a/b)+1/8/b*exp(-3*a/b)*Ei (1,-3*arcsech(c*x)-3*a/b))
\[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}} \,d x } \]
\[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\int \frac {1}{x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}\, dx \]
\[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}} \,d x } \]
\[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx=\int \frac {1}{x^4\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )} \,d x \]