3.2.65 \(\int (c-a^2 c x^2)^2 \cosh ^{-1}(a x)^2 \, dx\) [165]

Optimal. Leaf size=195 \[ \frac {298 c^2 x}{225}-\frac {76}{675} a^2 c^2 x^3+\frac {2}{125} a^4 c^2 x^5-\frac {16 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{15 a}+\frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2 \]

[Out]

298/225*c^2*x-76/675*a^2*c^2*x^3+2/125*a^4*c^2*x^5+8/45*c^2*(a*x-1)^(3/2)*(a*x+1)^(3/2)*arccosh(a*x)/a-2/25*c^
2*(a*x-1)^(5/2)*(a*x+1)^(5/2)*arccosh(a*x)/a+8/15*c^2*x*arccosh(a*x)^2+4/15*c^2*x*(-a^2*x^2+1)*arccosh(a*x)^2+
1/5*c^2*x*(-a^2*x^2+1)^2*arccosh(a*x)^2-16/15*c^2*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

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Rubi [A]
time = 0.31, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5897, 5879, 5915, 8, 41, 200} \begin {gather*} \frac {2}{125} a^4 c^2 x^5-\frac {76}{675} a^2 c^2 x^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^2-\frac {2 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {8 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac {16 c^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{15 a}+\frac {298 c^2 x}{225} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(298*c^2*x)/225 - (76*a^2*c^2*x^3)/675 + (2*a^4*c^2*x^5)/125 - (16*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*
x])/(15*a) + (8*c^2*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/(45*a) - (2*c^2*(-1 + a*x)^(5/2)*(1 + a*x)^
(5/2)*ArcCosh[a*x])/(25*a) + (8*c^2*x*ArcCosh[a*x]^2)/15 + (4*c^2*x*(1 - a^2*x^2)*ArcCosh[a*x]^2)/15 + (c^2*x*
(1 - a^2*x^2)^2*ArcCosh[a*x]^2)/5

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5897

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

Rule 5915

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

Rubi steps

\begin {align*} \int \left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)^2 \, dx &=\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{5} (4 c) \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2 \, dx-\frac {1}{5} \left (2 a c^2\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx\\ &=-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{25} \left (2 c^2\right ) \int \left (-1+a^2 x^2\right )^2 \, dx+\frac {1}{15} \left (8 c^2\right ) \int \cosh ^{-1}(a x)^2 \, dx+\frac {1}{15} \left (8 a c^2\right ) \int x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \, dx\\ &=\frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{25} \left (2 c^2\right ) \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx-\frac {1}{45} \left (8 c^2\right ) \int \left (-1+a^2 x^2\right ) \, dx-\frac {1}{15} \left (16 a c^2\right ) \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {58 c^2 x}{225}-\frac {76}{675} a^2 c^2 x^3+\frac {2}{125} a^4 c^2 x^5-\frac {16 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{15 a}+\frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{15} \left (16 c^2\right ) \int 1 \, dx\\ &=\frac {298 c^2 x}{225}-\frac {76}{675} a^2 c^2 x^3+\frac {2}{125} a^4 c^2 x^5-\frac {16 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{15 a}+\frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 101, normalized size = 0.52 \begin {gather*} \frac {c^2 \left (4470 a x-380 a^3 x^3+54 a^5 x^5-30 \sqrt {-1+a x} \sqrt {1+a x} \left (149-38 a^2 x^2+9 a^4 x^4\right ) \cosh ^{-1}(a x)+225 a x \left (15-10 a^2 x^2+3 a^4 x^4\right ) \cosh ^{-1}(a x)^2\right )}{3375 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(4470*a*x - 380*a^3*x^3 + 54*a^5*x^5 - 30*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(149 - 38*a^2*x^2 + 9*a^4*x^4)*Arc
Cosh[a*x] + 225*a*x*(15 - 10*a^2*x^2 + 3*a^4*x^4)*ArcCosh[a*x]^2))/(3375*a)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (-a^{2} c \,x^{2}+c \right )^{2} \mathrm {arccosh}\left (a x \right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 134, normalized size = 0.69 \begin {gather*} \frac {2}{125} \, a^{4} c^{2} x^{5} - \frac {76}{675} \, a^{2} c^{2} x^{3} + \frac {298}{225} \, c^{2} x - \frac {2}{225} \, {\left (9 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 38 \, \sqrt {a^{2} x^{2} - 1} c^{2} x^{2} + \frac {149 \, \sqrt {a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right ) + \frac {1}{15} \, {\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \operatorname {arcosh}\left (a x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/125*a^4*c^2*x^5 - 76/675*a^2*c^2*x^3 + 298/225*c^2*x - 2/225*(9*sqrt(a^2*x^2 - 1)*a^2*c^2*x^4 - 38*sqrt(a^2*
x^2 - 1)*c^2*x^2 + 149*sqrt(a^2*x^2 - 1)*c^2/a^2)*a*arccosh(a*x) + 1/15*(3*a^4*c^2*x^5 - 10*a^2*c^2*x^3 + 15*c
^2*x)*arccosh(a*x)^2

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Fricas [A]
time = 0.34, size = 142, normalized size = 0.73 \begin {gather*} \frac {54 \, a^{5} c^{2} x^{5} - 380 \, a^{3} c^{2} x^{3} + 4470 \, a c^{2} x + 225 \, {\left (3 \, a^{5} c^{2} x^{5} - 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (9 \, a^{4} c^{2} x^{4} - 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{3375 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3375*(54*a^5*c^2*x^5 - 380*a^3*c^2*x^3 + 4470*a*c^2*x + 225*(3*a^5*c^2*x^5 - 10*a^3*c^2*x^3 + 15*a*c^2*x)*lo
g(a*x + sqrt(a^2*x^2 - 1))^2 - 30*(9*a^4*c^2*x^4 - 38*a^2*c^2*x^2 + 149*c^2)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(
a^2*x^2 - 1)))/a

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Sympy [A]
time = 0.52, size = 182, normalized size = 0.93 \begin {gather*} \begin {cases} \frac {a^{4} c^{2} x^{5} \operatorname {acosh}^{2}{\left (a x \right )}}{5} + \frac {2 a^{4} c^{2} x^{5}}{125} - \frac {2 a^{3} c^{2} x^{4} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{25} - \frac {2 a^{2} c^{2} x^{3} \operatorname {acosh}^{2}{\left (a x \right )}}{3} - \frac {76 a^{2} c^{2} x^{3}}{675} + \frac {76 a c^{2} x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{225} + c^{2} x \operatorname {acosh}^{2}{\left (a x \right )} + \frac {298 c^{2} x}{225} - \frac {298 c^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{225 a} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} c^{2} x}{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*c**2*x**5*acosh(a*x)**2/5 + 2*a**4*c**2*x**5/125 - 2*a**3*c**2*x**4*sqrt(a**2*x**2 - 1)*acosh(
a*x)/25 - 2*a**2*c**2*x**3*acosh(a*x)**2/3 - 76*a**2*c**2*x**3/675 + 76*a*c**2*x**2*sqrt(a**2*x**2 - 1)*acosh(
a*x)/225 + c**2*x*acosh(a*x)**2 + 298*c**2*x/225 - 298*c**2*sqrt(a**2*x**2 - 1)*acosh(a*x)/(225*a), Ne(a, 0)),
(-pi**2*c**2*x/4, True))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^2\,{\left (c-a^2\,c\,x^2\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^2*(c - a^2*c*x^2)^2,x)

[Out]

int(acosh(a*x)^2*(c - a^2*c*x^2)^2, x)

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