3.27 \(\int \frac{1}{(a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=85 \[ -\frac{\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}+\frac{\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{\sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]

[Out]

-(Sqrt[1 + c^2*x^2]/(b*c*(a + b*ArcSinh[c*x]))) - (CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(b^2*c) + (
Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c)

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Rubi [A]  time = 0.179424, antiderivative size = 81, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5655, 5779, 3303, 3298, 3301} \[ -\frac{\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}+\frac{\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}-\frac{\sqrt{c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])^(-2),x]

[Out]

-(Sqrt[1 + c^2*x^2]/(b*c*(a + b*ArcSinh[c*x]))) - (CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b])/(b^2*c) + (Cosh
[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b^2*c)

Rule 5655

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1
))/(b*c*(n + 1)), x] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSinh[c*x])^(n + 1))/Sqrt[1 + c^2*x^2], x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5779

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Sinh[x]^m*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

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

Mathematica [A]  time = 0.198078, size = 71, normalized size = 0.84 \[ \frac{-\frac{b \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}-\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )+\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSinh[c*x])^(-2),x]

[Out]

(-((b*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])) - CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + Cosh[a/b]*SinhIn
tegral[a/b + ArcSinh[c*x]])/(b^2*c)

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Maple [A]  time = 0., size = 118, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{1}{2\,b \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ( cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{1}{2\,{b}^{2}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\it Arcsinh} \left ( cx \right ) +{\frac{a}{b}} \right ) }-{\frac{1}{2\,b \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{1}{2\,{b}^{2}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\it Arcsinh} \left ( cx \right ) -{\frac{a}{b}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c^{3} x^{3} + c x +{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{a b c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} a b c^{2} x + a b c +{\left (b^{2} c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )} + \int \frac{c^{4} x^{4} + 2 \, c^{2} x^{2} +{\left (c^{2} x^{2} + 1\right )}{\left (c^{2} x^{2} - 1\right )} +{\left (2 \, c^{3} x^{3} + c x\right )} \sqrt{c^{2} x^{2} + 1} + 1}{a b c^{4} x^{4} +{\left (c^{2} x^{2} + 1\right )} a b c^{2} x^{2} + 2 \, a b c^{2} x^{2} + a b +{\left (b^{2} c^{4} x^{4} +{\left (c^{2} x^{2} + 1\right )} b^{2} c^{2} x^{2} + 2 \, b^{2} c^{2} x^{2} + b^{2} + 2 \,{\left (b^{2} c^{3} x^{3} + b^{2} c x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \,{\left (a b c^{3} x^{3} + a b c x\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2))/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*c^2*x + a*b*c + (b^2*c^3*x^2 + sqr
t(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate((c^4*x^4 + 2*c^2*x^2 + (c^2*x^2 +
1)*(c^2*x^2 - 1) + (2*c^3*x^3 + c*x)*sqrt(c^2*x^2 + 1) + 1)/(a*b*c^4*x^4 + (c^2*x^2 + 1)*a*b*c^2*x^2 + 2*a*b*c
^2*x^2 + a*b + (b^2*c^4*x^4 + (c^2*x^2 + 1)*b^2*c^2*x^2 + 2*b^2*c^2*x^2 + b^2 + 2*(b^2*c^3*x^3 + b^2*c*x)*sqrt
(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^3*x^3 + a*b*c*x)*sqrt(c^2*x^2 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asinh(c*x))**(-2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsinh(c*x) + a)^(-2), x)