3.317 \(\int \frac{c-a^2 c x^2}{\cosh ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=58 \[ \frac{3 c \text{Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac{3 c \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}+\frac{c (a x-1)^{3/2} (a x+1)^{3/2}}{a \cosh ^{-1}(a x)} \]

[Out]

(c*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2))/(a*ArcCosh[a*x]) + (3*c*CoshIntegral[ArcCosh[a*x]])/(4*a) - (3*c*CoshInte
gral[3*ArcCosh[a*x]])/(4*a)

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Rubi [A]  time = 0.235204, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5695, 5781, 5448, 3301} \[ \frac{3 c \text{Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac{3 c \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}+\frac{c (a x-1)^{3/2} (a x+1)^{3/2}}{a \cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)/ArcCosh[a*x]^2,x]

[Out]

(c*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2))/(a*ArcCosh[a*x]) + (3*c*CoshIntegral[ArcCosh[a*x]])/(4*a) - (3*c*CoshInte
gral[3*ArcCosh[a*x]])/(4*a)

Rule 5695

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

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

Rule 5448

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]

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{c-a^2 c x^2}{\cosh ^{-1}(a x)^2} \, dx &=\frac{c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-(3 a c) \int \frac{x \sqrt{-1+a x} \sqrt{1+a x}}{\cosh ^{-1}(a x)} \, dx\\ &=\frac{c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac{c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-\frac{(3 c) \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 x}+\frac{\cosh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac{c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}\\ &=\frac{c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}+\frac{3 c \text{Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac{3 c \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}\\ \end{align*}

Mathematica [B]  time = 0.897519, size = 140, normalized size = 2.41 \[ \frac{c \sqrt{a x-1} \left (\left (4 \sqrt{a x-1}-\sqrt{\frac{a x-1}{a x+1}} \sqrt{a x+1}\right ) \cosh ^{-1}(a x) \text{Chi}\left (\cosh ^{-1}(a x)\right )+\sqrt{a x+1} \left (4 (a x-1)^2 (a x+1)-3 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x) \text{Chi}\left (3 \cosh ^{-1}(a x)\right )\right )\right ) \text{csch}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )}{8 a \cosh ^{-1}(a x)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a^2*c*x^2)/ArcCosh[a*x]^2,x]

[Out]

(c*Sqrt[-1 + a*x]*((4*Sqrt[-1 + a*x] - Sqrt[(-1 + a*x)/(1 + a*x)]*Sqrt[1 + a*x])*ArcCosh[a*x]*CoshIntegral[Arc
Cosh[a*x]] + Sqrt[1 + a*x]*(4*(-1 + a*x)^2*(1 + a*x) - 3*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*CoshIntegral[
3*ArcCosh[a*x]]))*Csch[ArcCosh[a*x]/2]^2)/(8*a*ArcCosh[a*x])

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Maple [A]  time = 0.036, size = 61, normalized size = 1.1 \begin{align*}{\frac{c}{4\,a{\rm arccosh} \left (ax\right )} \left ( 3\,{\it Chi} \left ({\rm arccosh} \left (ax\right ) \right ){\rm arccosh} \left (ax\right )-3\,{\it Chi} \left ( 3\,{\rm arccosh} \left (ax\right ) \right ){\rm arccosh} \left (ax\right )-3\,\sqrt{ax-1}\sqrt{ax+1}+\sinh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a*c*(3*Chi(arccosh(a*x))*arccosh(a*x)-3*Chi(3*arccosh(a*x))*arccosh(a*x)-3*(a*x-1)^(1/2)*(a*x+1)^(1/2)+sin
h(3*arccosh(a*x)))/arccosh(a*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{5} c x^{5} - 2 \, a^{3} c x^{3} + a c x +{\left (a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c\right )} \sqrt{a x + 1} \sqrt{a x - 1}}{{\left (a^{3} x^{2} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )} - \int \frac{3 \, a^{6} c x^{6} - 7 \, a^{4} c x^{4} + 5 \, a^{2} c x^{2} +{\left (3 \, a^{4} c x^{4} - 2 \, a^{2} c x^{2} - c\right )}{\left (a x + 1\right )}{\left (a x - 1\right )} + 3 \,{\left (2 \, a^{5} c x^{5} - 3 \, a^{3} c x^{3} + a c x\right )} \sqrt{a x + 1} \sqrt{a x - 1} - c}{{\left (a^{4} x^{4} +{\left (a x + 1\right )}{\left (a x - 1\right )} a^{2} x^{2} - 2 \, a^{2} x^{2} + 2 \,{\left (a^{3} x^{3} - a x\right )} \sqrt{a x + 1} \sqrt{a x - 1} + 1\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^5*c*x^5 - 2*a^3*c*x^3 + a*c*x + (a^4*c*x^4 - 2*a^2*c*x^2 + c)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^3*x^2 + sqrt
(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) - integrate((3*a^6*c*x^6 - 7*a^4*c*
x^4 + 5*a^2*c*x^2 + (3*a^4*c*x^4 - 2*a^2*c*x^2 - c)*(a*x + 1)*(a*x - 1) + 3*(2*a^5*c*x^5 - 3*a^3*c*x^3 + a*c*x
)*sqrt(a*x + 1)*sqrt(a*x - 1) - c)/((a^4*x^4 + (a*x + 1)*(a*x - 1)*a^2*x^2 - 2*a^2*x^2 + 2*(a^3*x^3 - a*x)*sqr
t(a*x + 1)*sqrt(a*x - 1) + 1)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^2*c*x^2 - c)/arccosh(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(a^2*c*x^2 - c)/arccosh(a*x)^2, x)