\(\int (c-a^2 c x^2) \text {arccosh}(a x)^3 \, dx\) [242]

Optimal result

Integrand size = 18, antiderivative size = 175 \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^3 \, dx=-\frac {122 c \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {2}{27} a c x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {14}{3} c x \text {arccosh}(a x)-\frac {2}{9} a^2 c x^3 \text {arccosh}(a x)-\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{3 a}+\frac {2}{3} c x \text {arccosh}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3 \]

[Out]

14/3*c*x*arccosh(a*x)-2/9*a^2*c*x^3*arccosh(a*x)+1/3*c*(a*x-1)^(3/2)*(a*x+1)^(3/2)*arccosh(a*x)^2/a+2/3*c*x*ar
ccosh(a*x)^3+1/3*c*x*(-a^2*x^2+1)*arccosh(a*x)^3-122/27*c*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a+2/27*a*c*x^2*(a*x-1)^(
1/2)*(a*x+1)^(1/2)-2*c*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5897, 5879, 5915, 75, 5889, 5894, 12, 471} \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^3 \, dx=-\frac {2}{9} a^2 c x^3 \text {arccosh}(a x)+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {2}{3} c x \text {arccosh}(a x)^3+\frac {14}{3} c x \text {arccosh}(a x)+\frac {c (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)^2}{3 a}-\frac {2 c \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{a}+\frac {2}{27} a c x^2 \sqrt {a x-1} \sqrt {a x+1}-\frac {122 c \sqrt {a x-1} \sqrt {a x+1}}{27 a} \]

[In]

Int[(c - a^2*c*x^2)*ArcCosh[a*x]^3,x]

[Out]

(-122*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) + (2*a*c*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/27 + (14*c*x*ArcCosh[a
*x])/3 - (2*a^2*c*x^3*ArcCosh[a*x])/9 - (2*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/a + (c*(-1 + a*x)^(3
/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^2)/(3*a) + (2*c*x*ArcCosh[a*x]^3)/3 + (c*x*(1 - a^2*x^2)*ArcCosh[a*x]^3)/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 75

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

Rule 471

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(
m + n*(p + 1) + 1))), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 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 5889

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2
*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5894

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 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*} \text {integral}& = \frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{3} (2 c) \int \text {arccosh}(a x)^3 \, dx+(a c) \int x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \, dx \\ & = \frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{3 a}+\frac {2}{3} c x \text {arccosh}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3-\frac {1}{3} (2 c) \int (-1+a x) (1+a x) \text {arccosh}(a x) \, dx-(2 a c) \int \frac {x \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{3 a}+\frac {2}{3} c x \text {arccosh}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3-\frac {1}{3} (2 c) \int \left (-1+a^2 x^2\right ) \text {arccosh}(a x) \, dx+(4 c) \int \text {arccosh}(a x) \, dx \\ & = \frac {14}{3} c x \text {arccosh}(a x)-\frac {2}{9} a^2 c x^3 \text {arccosh}(a x)-\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{3 a}+\frac {2}{3} c x \text {arccosh}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{3} (2 a c) \int \frac {x \left (-3+a^2 x^2\right )}{3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx-(4 a c) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {4 c \sqrt {-1+a x} \sqrt {1+a x}}{a}+\frac {14}{3} c x \text {arccosh}(a x)-\frac {2}{9} a^2 c x^3 \text {arccosh}(a x)-\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{3 a}+\frac {2}{3} c x \text {arccosh}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{9} (2 a c) \int \frac {x \left (-3+a^2 x^2\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {4 c \sqrt {-1+a x} \sqrt {1+a x}}{a}+\frac {2}{27} a c x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {14}{3} c x \text {arccosh}(a x)-\frac {2}{9} a^2 c x^3 \text {arccosh}(a x)-\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{3 a}+\frac {2}{3} c x \text {arccosh}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3-\frac {1}{27} (14 a c) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {122 c \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {2}{27} a c x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {14}{3} c x \text {arccosh}(a x)-\frac {2}{9} a^2 c x^3 \text {arccosh}(a x)-\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{3 a}+\frac {2}{3} c x \text {arccosh}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.62 \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^3 \, dx=\frac {c \left (2 \sqrt {-1+a x} \sqrt {1+a x} \left (-61+a^2 x^2\right )-6 a x \left (-21+a^2 x^2\right ) \text {arccosh}(a x)+9 \sqrt {-1+a x} \sqrt {1+a x} \left (-7+a^2 x^2\right ) \text {arccosh}(a x)^2-9 a x \left (-3+a^2 x^2\right ) \text {arccosh}(a x)^3\right )}{27 a} \]

[In]

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

[Out]

(c*(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-61 + a^2*x^2) - 6*a*x*(-21 + a^2*x^2)*ArcCosh[a*x] + 9*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]*(-7 + a^2*x^2)*ArcCosh[a*x]^2 - 9*a*x*(-3 + a^2*x^2)*ArcCosh[a*x]^3))/(27*a)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(-\frac {c \left (9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}-9 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-27 a x \operatorname {arccosh}\left (a x \right )^{3}+63 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+6 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-126 a x \,\operatorname {arccosh}\left (a x \right )+122 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{27 a}\)	\(140\)
default	\(-\frac {c \left (9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}-9 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-27 a x \operatorname {arccosh}\left (a x \right )^{3}+63 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+6 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-126 a x \,\operatorname {arccosh}\left (a x \right )+122 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{27 a}\)	\(140\)

[In]

int((-a^2*c*x^2+c)*arccosh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/27/a*c*(9*a^3*x^3*arccosh(a*x)^3-9*a^2*x^2*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-27*a*x*arccosh(a*x)^3
+63*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+6*a^3*x^3*arccosh(a*x)-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-12
6*a*x*arccosh(a*x)+122*(a*x-1)^(1/2)*(a*x+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.80 \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^3 \, dx=-\frac {9 \, {\left (a^{3} c x^{3} - 3 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 9 \, {\left (a^{2} c x^{2} - 7 \, c\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 6 \, {\left (a^{3} c x^{3} - 21 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (a^{2} c x^{2} - 61 \, c\right )} \sqrt {a^{2} x^{2} - 1}}{27 \, a} \]

[In]

integrate((-a^2*c*x^2+c)*arccosh(a*x)^3,x, algorithm="fricas")

[Out]

-1/27*(9*(a^3*c*x^3 - 3*a*c*x)*log(a*x + sqrt(a^2*x^2 - 1))^3 - 9*(a^2*c*x^2 - 7*c)*sqrt(a^2*x^2 - 1)*log(a*x
+ sqrt(a^2*x^2 - 1))^2 + 6*(a^3*c*x^3 - 21*a*c*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 2*(a^2*c*x^2 - 61*c)*sqrt(a^2
*x^2 - 1))/a

Sympy [F]

\[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^3 \, dx=- c \left (\int a^{2} x^{2} \operatorname {acosh}^{3}{\left (a x \right )}\, dx + \int \left (- \operatorname {acosh}^{3}{\left (a x \right )}\right )\, dx\right ) \]

[In]

integrate((-a**2*c*x**2+c)*acosh(a*x)**3,x)

[Out]

-c*(Integral(a**2*x**2*acosh(a*x)**3, x) + Integral(-acosh(a*x)**3, x))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.71 \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^3 \, dx=\frac {1}{3} \, {\left (\sqrt {a^{2} x^{2} - 1} c x^{2} - \frac {7 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right )^{2} - \frac {1}{3} \, {\left (a^{2} c x^{3} - 3 \, c x\right )} \operatorname {arcosh}\left (a x\right )^{3} + \frac {2}{27} \, {\left (\sqrt {a^{2} x^{2} - 1} c x^{2} - \frac {3 \, {\left (a^{2} c x^{3} - 21 \, c x\right )} \operatorname {arcosh}\left (a x\right )}{a} - \frac {61 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}}\right )} a \]

[In]

integrate((-a^2*c*x^2+c)*arccosh(a*x)^3,x, algorithm="maxima")

[Out]

1/3*(sqrt(a^2*x^2 - 1)*c*x^2 - 7*sqrt(a^2*x^2 - 1)*c/a^2)*a*arccosh(a*x)^2 - 1/3*(a^2*c*x^3 - 3*c*x)*arccosh(a
*x)^3 + 2/27*(sqrt(a^2*x^2 - 1)*c*x^2 - 3*(a^2*c*x^3 - 21*c*x)*arccosh(a*x)/a - 61*sqrt(a^2*x^2 - 1)*c/a^2)*a

Giac [F(-2)]

Exception generated. \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^3 \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^3\,\left (c-a^2\,c\,x^2\right ) \,d x \]

[In]

int(acosh(a*x)^3*(c - a^2*c*x^2),x)

[Out]

int(acosh(a*x)^3*(c - a^2*c*x^2), x)