3.134 \(\int \frac{\cosh ^{-1}(a x)}{(c-a^2 c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=246 \[ \frac{2 \sqrt{a x-1} \sqrt{a x+1}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}-\frac{4 \sqrt{a x-1} \sqrt{a x+1} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{x \cosh ^{-1}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]

[Out]

(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*a*c^3*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x
])/(15*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh[a*x])/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*ArcCosh[
a*x])/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcCosh[a*x])/(15*c^3*Sqrt[c - a^2*c*x^2]) - (4*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]*Log[1 - a^2*x^2])/(15*a*c^3*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.339178, antiderivative size = 276, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5713, 5691, 5688, 260, 261} \[ \frac{2 \sqrt{a x-1} \sqrt{a x+1}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}-\frac{4 \sqrt{a x-1} \sqrt{a x+1} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^3 (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)}{5 c^3 (1-a x)^2 (a x+1)^2 \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]/(c - a^2*c*x^2)^(7/2),x]

[Out]

(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*a*c^3*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x
])/(15*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) + (8*x*ArcCosh[a*x])/(15*c^3*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh
[a*x])/(5*c^3*(1 - a*x)^2*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]) + (4*x*ArcCosh[a*x])/(15*c^3*(1 - a*x)*(1 + a*x)*Sq
rt[c - a^2*c*x^2]) - (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[1 - a^2*x^2])/(15*a*c^3*Sqrt[c - a^2*c*x^2])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((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[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5691

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol]
:> -Simp[(x*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*d1*d2*(p + 1)), x] + (Dist[(2*
p + 3)/(2*d1*d2*(p + 1)), Int[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b
*c*n*(-(d1*d2))^(p + 1/2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x])/(2*(p + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[x*(-1
+ c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1,
c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[p + 1/2]

Rule 5688

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(((d1_) + (e1_.)*(x_))^(3/2)*((d2_) + (e2_.)*(x_))^(3/2)), x_Sym
bol] :> Simp[(x*(a + b*ArcCosh[c*x])^n)/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Dist[(b*c*n*Sqrt[1 + c*x
]*Sqrt[-1 + c*x])/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(x*(a + b*ArcCosh[c*x])^(n - 1))/(1 - c^2*x^2),
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\cosh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{(-1+a x)^{7/2} (1+a x)^{7/2}} \, dx}{c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\left (-1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{-1+a x} \sqrt{1+a x}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}-\frac{\left (8 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (4 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\left (-1+a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{-1+a x} \sqrt{1+a x}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\left (8 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{1-a^2 x^2} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{-1+a x} \sqrt{1+a x}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0882938, size = 116, normalized size = 0.47 \[ \frac{\sqrt{a x-1} \sqrt{a x+1} \left (-8 a^2 x^2-16 \left (a^2 x^2-1\right )^2 \log \left (1-a^2 x^2\right )+11\right )+4 a x \left (8 a^4 x^4-20 a^2 x^2+15\right ) \cosh ^{-1}(a x)}{60 a c^3 \left (a^2 x^2-1\right )^2 \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a*x]/(c - a^2*c*x^2)^(7/2),x]

[Out]

(4*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*ArcCosh[a*x] + Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(11 - 8*a^2*x^2 - 16*(-1 + a^
2*x^2)^2*Log[1 - a^2*x^2]))/(60*a*c^3*(-1 + a^2*x^2)^2*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.229, size = 419, normalized size = 1.7 \begin{align*} -{\frac{16\,{\rm arccosh} \left (ax\right )}{15\,a{c}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{ax-1}\sqrt{ax+1}}-{\frac{1}{ \left ( 2400\,{a}^{10}{x}^{10}-12900\,{x}^{8}{a}^{8}+28140\,{x}^{6}{a}^{6}-31020\,{x}^{4}{a}^{4}+17220\,{a}^{2}{x}^{2}-3840 \right ) a{c}^{4}}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 8\,{x}^{5}{a}^{5}-20\,{x}^{3}{a}^{3}-8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{4}{a}^{4}+15\,ax+16\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-8\,\sqrt{ax-1}\sqrt{ax+1} \right ) \left ( -64\,\sqrt{ax+1}\sqrt{ax-1}{x}^{7}{a}^{7}-64\,{x}^{8}{a}^{8}+248\,\sqrt{ax+1}\sqrt{ax-1}{x}^{5}{a}^{5}+280\,{x}^{6}{a}^{6}+160\,{\rm arccosh} \left (ax\right ){x}^{4}{a}^{4}-340\,{a}^{3}{x}^{3}\sqrt{ax-1}\sqrt{ax+1}-456\,{x}^{4}{a}^{4}-380\,{a}^{2}{x}^{2}{\rm arccosh} \left (ax\right )+165\,\sqrt{ax+1}\sqrt{ax-1}ax+328\,{a}^{2}{x}^{2}+256\,{\rm arccosh} \left (ax\right )-88 \right ) }+{\frac{8}{15\,a{c}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{ax-1}\sqrt{ax+1}\ln \left ( \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) ^{2}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)/(-a^2*c*x^2+c)^(7/2),x)

[Out]

-16/15*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^4/a/(a^2*x^2-1)*arccosh(a*x)-1/60*(-c*(a^2*x^2-1))
^(1/2)*(8*x^5*a^5-20*x^3*a^3-8*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^4*a^4+15*a*x+16*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a
^2-8*(a*x-1)^(1/2)*(a*x+1)^(1/2))*(-64*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^7*a^7-64*x^8*a^8+248*(a*x+1)^(1/2)*(a*x-1
)^(1/2)*x^5*a^5+280*x^6*a^6+160*arccosh(a*x)*x^4*a^4-340*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-456*x^4*a^4-380*a
^2*x^2*arccosh(a*x)+165*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+328*a^2*x^2+256*arccosh(a*x)-88)/(40*a^10*x^10-215*a^8
*x^8+469*a^6*x^6-517*a^4*x^4+287*a^2*x^2-64)/a/c^4+8/15*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^4
/a/(a^2*x^2-1)*ln((a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2-1)

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Maxima [A]  time = 1.27007, size = 258, normalized size = 1.05 \begin{align*} -\frac{1}{60} \, a{\left (\frac{16 \, \sqrt{-\frac{1}{a^{4} c}} \log \left (x^{2} - \frac{1}{a^{2}}\right )}{c^{3}} + \frac{3}{{\left (a^{6} c^{3} x^{4} \sqrt{-\frac{1}{c}} - 2 \, a^{4} c^{3} x^{2} \sqrt{-\frac{1}{c}} + a^{2} c^{3} \sqrt{-\frac{1}{c}}\right )} c} - \frac{8}{{\left (a^{4} c^{2} x^{2} \sqrt{-\frac{1}{c}} - a^{2} c^{2} \sqrt{-\frac{1}{c}}\right )} c^{2}}\right )} + \frac{1}{15} \,{\left (\frac{8 \, x}{\sqrt{-a^{2} c x^{2} + c} c^{3}} + \frac{4 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} c^{2}} + \frac{3 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} c}\right )} \operatorname{arcosh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

-1/60*a*(16*sqrt(-1/(a^4*c))*log(x^2 - 1/a^2)/c^3 + 3/((a^6*c^3*x^4*sqrt(-1/c) - 2*a^4*c^3*x^2*sqrt(-1/c) + a^
2*c^3*sqrt(-1/c))*c) - 8/((a^4*c^2*x^2*sqrt(-1/c) - a^2*c^2*sqrt(-1/c))*c^2)) + 1/15*(8*x/(sqrt(-a^2*c*x^2 + c
)*c^3) + 4*x/((-a^2*c*x^2 + c)^(3/2)*c^2) + 3*x/((-a^2*c*x^2 + c)^(5/2)*c))*arccosh(a*x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )}{a^{8} c^{4} x^{8} - 4 \, a^{6} c^{4} x^{6} + 6 \, a^{4} c^{4} x^{4} - 4 \, a^{2} c^{4} x^{2} + c^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)/(a^8*c^4*x^8 - 4*a^6*c^4*x^6 + 6*a^4*c^4*x^4 - 4*a^2*c^4*x^2 + c^4)
, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)/(-a**2*c*x**2+c)**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.45557, size = 192, normalized size = 0.78 \begin{align*} \frac{1}{60} \, \sqrt{-c}{\left (\frac{16 \, \log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{a c^{4}} - \frac{24 \, a^{4} x^{4} - 56 \, a^{2} x^{2} + 35}{{\left (a^{2} x^{2} - 1\right )}^{2} a c^{4}}\right )} - \frac{\sqrt{-a^{2} c x^{2} + c}{\left (4 \,{\left (\frac{2 \, a^{4} x^{2}}{c} - \frac{5 \, a^{2}}{c}\right )} x^{2} + \frac{15}{c}\right )} x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{15 \,{\left (a^{2} c x^{2} - c\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

1/60*sqrt(-c)*(16*log(abs(a^2*x^2 - 1))/(a*c^4) - (24*a^4*x^4 - 56*a^2*x^2 + 35)/((a^2*x^2 - 1)^2*a*c^4)) - 1/
15*sqrt(-a^2*c*x^2 + c)*(4*(2*a^4*x^2/c - 5*a^2/c)*x^2 + 15/c)*x*log(a*x + sqrt(a^2*x^2 - 1))/(a^2*c*x^2 - c)^
3