3.315 \(\int \frac{(c-a^2 c x^2)^3}{\cosh ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=98 \[ \frac{35 c^3 \text{Chi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac{63 c^3 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac{35 c^3 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac{7 c^3 \text{Chi}\left (7 \cosh ^{-1}(a x)\right )}{64 a}+\frac{c^3 (a x-1)^{7/2} (a x+1)^{7/2}}{a \cosh ^{-1}(a x)} \]

[Out]

(c^3*(-1 + a*x)^(7/2)*(1 + a*x)^(7/2))/(a*ArcCosh[a*x]) + (35*c^3*CoshIntegral[ArcCosh[a*x]])/(64*a) - (63*c^3
*CoshIntegral[3*ArcCosh[a*x]])/(64*a) + (35*c^3*CoshIntegral[5*ArcCosh[a*x]])/(64*a) - (7*c^3*CoshIntegral[7*A
rcCosh[a*x]])/(64*a)

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Rubi [A]  time = 0.325086, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5695, 5781, 5448, 3301} \[ \frac{35 c^3 \text{Chi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac{63 c^3 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac{35 c^3 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac{7 c^3 \text{Chi}\left (7 \cosh ^{-1}(a x)\right )}{64 a}+\frac{c^3 (a x-1)^{7/2} (a x+1)^{7/2}}{a \cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^3/ArcCosh[a*x]^2,x]

[Out]

(c^3*(-1 + a*x)^(7/2)*(1 + a*x)^(7/2))/(a*ArcCosh[a*x]) + (35*c^3*CoshIntegral[ArcCosh[a*x]])/(64*a) - (63*c^3
*CoshIntegral[3*ArcCosh[a*x]])/(64*a) + (35*c^3*CoshIntegral[5*ArcCosh[a*x]])/(64*a) - (7*c^3*CoshIntegral[7*A
rcCosh[a*x]])/(64*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{\left (c-a^2 c x^2\right )^3}{\cosh ^{-1}(a x)^2} \, dx &=\frac{c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}-\left (7 a c^3\right ) \int \frac{x (-1+a x)^{5/2} (1+a x)^{5/2}}{\cosh ^{-1}(a x)} \, dx\\ &=\frac{c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}-\frac{\left (7 c^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^6(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac{c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}-\frac{\left (7 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{5 \cosh (x)}{64 x}+\frac{9 \cosh (3 x)}{64 x}-\frac{5 \cosh (5 x)}{64 x}+\frac{\cosh (7 x)}{64 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac{c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}-\frac{\left (7 c^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (7 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}+\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}+\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}-\frac{\left (63 c^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}\\ &=\frac{c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}+\frac{35 c^3 \text{Chi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac{63 c^3 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac{35 c^3 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac{7 c^3 \text{Chi}\left (7 \cosh ^{-1}(a x)\right )}{64 a}\\ \end{align*}

Mathematica [A]  time = 0.474794, size = 128, normalized size = 1.31 \[ \frac{c^3 \left (112 \left (\text{Chi}\left (\cosh ^{-1}(a x)\right )-\text{Chi}\left (3 \cosh ^{-1}(a x)\right )\right )+56 \left (-2 \text{Chi}\left (\cosh ^{-1}(a x)\right )+\text{Chi}\left (3 \cosh ^{-1}(a x)\right )+\text{Chi}\left (5 \cosh ^{-1}(a x)\right )\right )+7 \left (5 \text{Chi}\left (\cosh ^{-1}(a x)\right )-\text{Chi}\left (3 \cosh ^{-1}(a x)\right )-3 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )-\text{Chi}\left (7 \cosh ^{-1}(a x)\right )\right )+\frac{64 (a x-1)^3 \sqrt{\frac{a x-1}{a x+1}} (a x+1)^4}{\cosh ^{-1}(a x)}\right )}{64 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a^2*c*x^2)^3/ArcCosh[a*x]^2,x]

[Out]

(c^3*((64*(-1 + a*x)^3*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)^4)/ArcCosh[a*x] + 112*(CoshIntegral[ArcCosh[a*x]]
- CoshIntegral[3*ArcCosh[a*x]]) + 56*(-2*CoshIntegral[ArcCosh[a*x]] + CoshIntegral[3*ArcCosh[a*x]] + CoshInteg
ral[5*ArcCosh[a*x]]) + 7*(5*CoshIntegral[ArcCosh[a*x]] - CoshIntegral[3*ArcCosh[a*x]] - 3*CoshIntegral[5*ArcCo
sh[a*x]] - CoshIntegral[7*ArcCosh[a*x]])))/(64*a)

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Maple [A]  time = 0.049, size = 107, normalized size = 1.1 \begin{align*}{\frac{{c}^{3}}{64\,a{\rm arccosh} \left (ax\right )} \left ( 35\,{\it Chi} \left ({\rm arccosh} \left (ax\right ) \right ){\rm arccosh} \left (ax\right )-63\,{\it Chi} \left ( 3\,{\rm arccosh} \left (ax\right ) \right ){\rm arccosh} \left (ax\right )+35\,{\it Chi} \left ( 5\,{\rm arccosh} \left (ax\right ) \right ){\rm arccosh} \left (ax\right )-7\,{\it Chi} \left ( 7\,{\rm arccosh} \left (ax\right ) \right ){\rm arccosh} \left (ax\right )-35\,\sqrt{ax-1}\sqrt{ax+1}+21\,\sinh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) -7\,\sinh \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) +\sinh \left ( 7\,{\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)^3/arccosh(a*x)^2,x)

[Out]

1/64/a*c^3*(35*Chi(arccosh(a*x))*arccosh(a*x)-63*Chi(3*arccosh(a*x))*arccosh(a*x)+35*Chi(5*arccosh(a*x))*arcco
sh(a*x)-7*Chi(7*arccosh(a*x))*arccosh(a*x)-35*(a*x-1)^(1/2)*(a*x+1)^(1/2)+21*sinh(3*arccosh(a*x))-7*sinh(5*arc
cosh(a*x))+sinh(7*arccosh(a*x)))/arccosh(a*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{9} c^{3} x^{9} - 4 \, a^{7} c^{3} x^{7} + 6 \, a^{5} c^{3} x^{5} - 4 \, a^{3} c^{3} x^{3} + a c^{3} x +{\left (a^{8} c^{3} x^{8} - 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} - 4 \, a^{2} c^{3} x^{2} + c^{3}\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{7 \, a^{10} c^{3} x^{10} - 29 \, a^{8} c^{3} x^{8} + 46 \, a^{6} c^{3} x^{6} - 34 \, a^{4} c^{3} x^{4} + 11 \, a^{2} c^{3} x^{2} +{\left (7 \, a^{8} c^{3} x^{8} - 20 \, a^{6} c^{3} x^{6} + 18 \, a^{4} c^{3} x^{4} - 4 \, a^{2} c^{3} x^{2} - c^{3}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )} - c^{3} + 7 \,{\left (2 \, a^{9} c^{3} x^{9} - 7 \, a^{7} c^{3} x^{7} + 9 \, a^{5} c^{3} x^{5} - 5 \, a^{3} c^{3} x^{3} + a c^{3} x\right )} \sqrt{a x + 1} \sqrt{a x - 1}}{{\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)^3/arccosh(a*x)^2,x, algorithm="maxima")

[Out]

(a^9*c^3*x^9 - 4*a^7*c^3*x^7 + 6*a^5*c^3*x^5 - 4*a^3*c^3*x^3 + a*c^3*x + (a^8*c^3*x^8 - 4*a^6*c^3*x^6 + 6*a^4*
c^3*x^4 - 4*a^2*c^3*x^2 + c^3)*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((7*a^10*c^3*x^10 - 29*a^8*c^3*x^8 + 46*a^6*c^3*x^6 - 34*a
^4*c^3*x^4 + 11*a^2*c^3*x^2 + (7*a^8*c^3*x^8 - 20*a^6*c^3*x^6 + 18*a^4*c^3*x^4 - 4*a^2*c^3*x^2 - c^3)*(a*x + 1
)*(a*x - 1) - c^3 + 7*(2*a^9*c^3*x^9 - 7*a^7*c^3*x^7 + 9*a^5*c^3*x^5 - 5*a^3*c^3*x^3 + a*c^3*x)*sqrt(a*x + 1)*
sqrt(a*x - 1))/((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)*sqrt(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^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}{\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)^3/arccosh(a*x)^2,x, algorithm="fricas")

[Out]

integral(-(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)/arccosh(a*x)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - c^{3} \left (\int \frac{3 a^{2} x^{2}}{\operatorname{acosh}^{2}{\left (a x \right )}}\, dx + \int - \frac{3 a^{4} x^{4}}{\operatorname{acosh}^{2}{\left (a x \right )}}\, dx + \int \frac{a^{6} x^{6}}{\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)**3/acosh(a*x)**2,x)

[Out]

-c**3*(Integral(3*a**2*x**2/acosh(a*x)**2, x) + Integral(-3*a**4*x**4/acosh(a*x)**2, x) + Integral(a**6*x**6/a
cosh(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{{\left (a^{2} c x^{2} - c\right )}^{3}}{\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)^3/arccosh(a*x)^2,x, algorithm="giac")

[Out]

integrate(-(a^2*c*x^2 - c)^3/arccosh(a*x)^2, x)