3.346 \(\int \frac{x}{\sqrt{1-c^2 x^2} (a+b \cosh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=130 \[ -\frac{\sqrt{c x-1} \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c^2 \sqrt{1-c x}}+\frac{\sqrt{c x-1} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c^2 \sqrt{1-c x}}-\frac{x \sqrt{c x-1}}{b c \sqrt{1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.43128, antiderivative size = 169, normalized size of antiderivative = 1.3, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5798, 5775, 5658, 3303, 3298, 3301} \[ -\frac{\sqrt{c x-1} \sqrt{c x+1} \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c^2 \sqrt{1-c^2 x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c^2 \sqrt{1-c^2 x^2}}-\frac{x \sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2),x]

[Out]

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

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 5658

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Dist[(b*c)^(-1), Subst[Int[x^n*Sinh[a/b - x/b], x]
, x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

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

Mathematica [A]  time = 0.217522, size = 107, normalized size = 0.82 \[ \frac{\sqrt{1-c^2 x^2} \left (\sinh \left (\frac{a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-\cosh \left (\frac{a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+b c x\right )}{b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.168, size = 283, normalized size = 2.2 \begin{align*}{\frac{1}{2\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ){b}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{cx-1}\sqrt{cx+1} \left ({\rm arccosh} \left (cx\right ){{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) b+\sqrt{cx+1}\sqrt{cx-1}b+{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) a+xbc \right ) }-{\frac{1}{2\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) b \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) }-{\frac{1}{2\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ){b}^{2}} \left ( \sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) \sqrt{-{c}^{2}{x}^{2}+1}{\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ){{\rm e}^{-{\frac{b{\rm arccosh} \left (cx\right )-a}{b}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c^{3} x^{4} - c x^{2} +{\left (c^{2} x^{3} - x\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{{\left ({\left (c x + 1\right )} \sqrt{c x - 1} b^{2} c^{2} x +{\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) +{\left ({\left (c x + 1\right )} \sqrt{c x - 1} a b c^{2} x +{\left (a b c^{3} x^{2} - a b c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}} + \int \frac{c^{5} x^{5} +{\left (c x + 1\right )}{\left (c x - 1\right )} c^{3} x^{3} - 3 \, c^{3} x^{3} +{\left (2 \, c^{4} x^{4} - 3 \, c^{2} x^{2} + 1\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 2 \, c x}{{\left ({\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} b^{2} c^{3} x^{2} + 2 \,{\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{4} - 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) +{\left ({\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} a b c^{3} x^{2} + 2 \,{\left (a b c^{4} x^{3} - a b c^{2} x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (a b c^{5} x^{4} - 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*x^4 - c*x^2 + (c^2*x^3 - x)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x + (b^2*c^3*
x^2 - b^2*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)*sqrt(c*x - 1)*a
*b*c^2*x + (a*b*c^3*x^2 - a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)) + integrate((c^5*x^5 + (c*x + 1)*(c*x - 1)*c^3
*x^3 - 3*c^3*x^3 + (2*c^4*x^4 - 3*c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(c*x - 1) + 2*c*x)/(((c*x + 1)^(3/2)*(c*x - 1
)*b^2*c^3*x^2 + 2*(b^2*c^4*x^3 - b^2*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 - 2*b^2*c^3*x^2 + b^2*c)*sq
rt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)^(3/2)*(c*x - 1)*a*b*c^3*x^2 +
2*(a*b*c^4*x^3 - a*b*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^5*x^4 - 2*a*b*c^3*x^2 + a*b*c)*sqrt(c*x + 1))*sqr
t(-c*x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-c^2*x^2 + 1)*x/(a^2*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arccosh(c*x)^2 - a^2 + 2*(a*b*c^2*x^2 - a*b)
*arccosh(c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(sqrt(-(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(sqrt(-c^2*x^2 + 1)*(b*arccosh(c*x) + a)^2), x)