3.54 \(\int \frac{1}{x^2 (a+b \text{sech}^{-1}(c x))} \, dx\)

Optimal. Leaf size=46 \[ \frac{c \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b}-\frac{c \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b} \]

[Out]

(c*CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b])/b - (c*Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]])/b

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Rubi [A]  time = 0.110401, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6285, 3303, 3298, 3301} \[ \frac{c \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b}-\frac{c \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*ArcSech[c*x])),x]

[Out]

(c*CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b])/b - (c*Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]])/b

Rule 6285

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[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]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b \text{sech}^{-1}(c x)\right )} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=-\left (\left (c \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\text{sech}^{-1}(c x)\right )\right )+\left (c \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=\frac{c \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{b}-\frac{c \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0747712, size = 43, normalized size = 0.93 \[ \frac{c \left (\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )-\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*ArcSech[c*x])),x]

[Out]

(c*(CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]]))/b

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Maple [A]  time = 0.211, size = 54, normalized size = 1.2 \begin{align*} c \left ( -{\frac{1}{2\,b}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\frac{a}{b}}+{\rm arcsech} \left (cx\right ) \right ) }+{\frac{1}{2\,b}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arcsech} \left (cx\right )-{\frac{a}{b}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b*arcsech(c*x)),x)

[Out]

c*(-1/2/b*exp(a/b)*Ei(1,a/b+arcsech(c*x))+1/2/b*exp(-a/b)*Ei(1,-arcsech(c*x)-a/b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*arcsech(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((b*arcsech(c*x) + a)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b x^{2} \operatorname{arsech}\left (c x\right ) + a x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*arcsech(c*x)),x, algorithm="fricas")

[Out]

integral(1/(b*x^2*arcsech(c*x) + a*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (a + b \operatorname{asech}{\left (c x \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+b*asech(c*x)),x)

[Out]

Integral(1/(x**2*(a + b*asech(c*x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*arcsech(c*x)),x, algorithm="giac")

[Out]

integrate(1/((b*arcsech(c*x) + a)*x^2), x)