3.2.66 \(\int (c-a^2 c x^2) \cosh ^{-1}(a x)^2 \, dx\) [166]

Optimal. Leaf size=112 \[ \frac {14 c x}{9}-\frac {2}{27} a^2 c x^3-\frac {4 c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 a}+\frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{9 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2 \]

[Out]

14/9*c*x-2/27*a^2*c*x^3+2/9*c*(a*x-1)^(3/2)*(a*x+1)^(3/2)*arccosh(a*x)/a+2/3*c*x*arccosh(a*x)^2+1/3*c*x*(-a^2*
x^2+1)*arccosh(a*x)^2-4/3*c*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

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Rubi [A]
time = 0.18, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5897, 5879, 5915, 8, 41} \begin {gather*} -\frac {2}{27} a^2 c x^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {2}{3} c x \cosh ^{-1}(a x)^2+\frac {2 c (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{9 a}-\frac {4 c \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{3 a}+\frac {14 c x}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)*ArcCosh[a*x]^2,x]

[Out]

(14*c*x)/9 - (2*a^2*c*x^3)/27 - (4*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(3*a) + (2*c*(-1 + a*x)^(3/2)*
(1 + a*x)^(3/2)*ArcCosh[a*x])/(9*a) + (2*c*x*ArcCosh[a*x]^2)/3 + (c*x*(1 - a^2*x^2)*ArcCosh[a*x]^2)/3

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 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 ) \cosh ^{-1}(a x)^2 \, dx &=\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{3} (2 c) \int \cosh ^{-1}(a x)^2 \, dx+\frac {1}{3} (2 a c) \int x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \, dx\\ &=\frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{9 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2-\frac {1}{9} (2 c) \int \left (-1+a^2 x^2\right ) \, dx-\frac {1}{3} (4 a c) \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {2 c x}{9}-\frac {2}{27} a^2 c x^3-\frac {4 c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 a}+\frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{9 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{3} (4 c) \int 1 \, dx\\ &=\frac {14 c x}{9}-\frac {2}{27} a^2 c x^3-\frac {4 c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 a}+\frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{9 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 73, normalized size = 0.65 \begin {gather*} \frac {c \left (42 a x-2 a^3 x^3+6 \sqrt {-1+a x} \sqrt {1+a x} \left (-7+a^2 x^2\right ) \cosh ^{-1}(a x)-9 a x \left (-3+a^2 x^2\right ) \cosh ^{-1}(a x)^2\right )}{27 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(42*a*x - 2*a^3*x^3 + 6*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-7 + a^2*x^2)*ArcCosh[a*x] - 9*a*x*(-3 + a^2*x^2)*Arc
Cosh[a*x]^2))/(27*a)

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Maple [A]
time = 1.18, size = 90, normalized size = 0.80 \[-\frac {c \left (9 \mathrm {arccosh}\left (a x \right )^{2} x^{3} a^{3}-6 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, x^{2} a^{2}+2 a^{3} x^{3}-27 \mathrm {arccosh}\left (a x \right )^{2} a x +42 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-42 a x \right )}{27 a}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27/a*c*(9*arccosh(a*x)^2*x^3*a^3-6*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*x^2*a^2+2*a^3*x^3-27*arccosh(a*
x)^2*a*x+42*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)-42*a*x)

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Maxima [A]
time = 0.25, size = 76, normalized size = 0.68 \begin {gather*} -\frac {2}{27} \, a^{2} c x^{3} + \frac {2}{9} \, {\left (\sqrt {a^{2} x^{2} - 1} c x^{2} - \frac {7 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right ) - \frac {1}{3} \, {\left (a^{2} c x^{3} - 3 \, c x\right )} \operatorname {arcosh}\left (a x\right )^{2} + \frac {14}{9} \, c x \end {gather*}

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]

-2/27*a^2*c*x^3 + 2/9*(sqrt(a^2*x^2 - 1)*c*x^2 - 7*sqrt(a^2*x^2 - 1)*c/a^2)*a*arccosh(a*x) - 1/3*(a^2*c*x^3 -
3*c*x)*arccosh(a*x)^2 + 14/9*c*x

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Fricas [A]
time = 0.36, size = 95, normalized size = 0.85 \begin {gather*} -\frac {2 \, a^{3} c x^{3} - 42 \, a c x + 9 \, {\left (a^{3} c x^{3} - 3 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (a^{2} c x^{2} - 7 \, c\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{27 \, a} \end {gather*}

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]

-1/27*(2*a^3*c*x^3 - 42*a*c*x + 9*(a^3*c*x^3 - 3*a*c*x)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 6*(a^2*c*x^2 - 7*c)*s
qrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)))/a

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Sympy [A]
time = 0.20, size = 105, normalized size = 0.94 \begin {gather*} \begin {cases} - \frac {a^{2} c x^{3} \operatorname {acosh}^{2}{\left (a x \right )}}{3} - \frac {2 a^{2} c x^{3}}{27} + \frac {2 a c x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{9} + c x \operatorname {acosh}^{2}{\left (a x \right )} + \frac {14 c x}{9} - \frac {14 c \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{9 a} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} c 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)*acosh(a*x)**2,x)

[Out]

Piecewise((-a**2*c*x**3*acosh(a*x)**2/3 - 2*a**2*c*x**3/27 + 2*a*c*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)/9 + c*x
*acosh(a*x)**2 + 14*c*x/9 - 14*c*sqrt(a**2*x**2 - 1)*acosh(a*x)/(9*a), Ne(a, 0)), (-pi**2*c*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)*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 ) \,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),x)

[Out]

int(acosh(a*x)^2*(c - a^2*c*x^2), x)

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