\(\int (d-c^2 d x^2)^3 (a+b \text {arccosh}(c x)) \, dx\) [23]

Optimal result

Integrand size = 22, antiderivative size = 191 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {16 b d^3 \sqrt {-1+c x} \sqrt {1+c x}}{35 c}+\frac {8 b d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}{105 c}-\frac {6 b d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{175 c}+\frac {b d^3 (-1+c x)^{7/2} (1+c x)^{7/2}}{49 c}+d^3 x (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x)) \]

[Out]

8/105*b*d^3*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c-6/175*b*d^3*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c+1/49*b*d^3*(c*x-1)^(7/2)*(
c*x+1)^(7/2)/c+d^3*x*(a+b*arccosh(c*x))-c^2*d^3*x^3*(a+b*arccosh(c*x))+3/5*c^4*d^3*x^5*(a+b*arccosh(c*x))-1/7*
c^6*d^3*x^7*(a+b*arccosh(c*x))-16/35*b*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {200, 5894, 12, 1624, 1813, 1864} \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+d^3 x (a+b \text {arccosh}(c x))+\frac {b d^3 \left (1-c^2 x^2\right )^4}{49 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {6 b d^3 \left (1-c^2 x^2\right )^3}{175 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^3 \left (1-c^2 x^2\right )^2}{105 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {16 b d^3 \left (1-c^2 x^2\right )}{35 c \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]

[Out]

(16*b*d^3*(1 - c^2*x^2))/(35*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^3*(1 - c^2*x^2)^2)/(105*c*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) + (6*b*d^3*(1 - c^2*x^2)^3)/(175*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*(1 - c^2*x^2)^4)/(49
*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^3*x*(a + b*ArcCosh[c*x]) - c^2*d^3*x^3*(a + b*ArcCosh[c*x]) + (3*c^4*d^3*
x^5*(a + b*ArcCosh[c*x]))/5 - (c^6*d^3*x^7*(a + b*ArcCosh[c*x]))/7

Rule 12

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

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 1624

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[(a
+ b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]), Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

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]

Rubi steps \begin{align*} \text {integral}& = d^3 x (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))-(b c) \int \frac {d^3 x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = d^3 x (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {1}{35} \left (b c d^3\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = d^3 x (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {-1+c^2 x^2}} \, dx}{35 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = d^3 x (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{70 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = d^3 x (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {16}{\sqrt {-1+c^2 x}}-8 \sqrt {-1+c^2 x}+6 \left (-1+c^2 x\right )^{3/2}-5 \left (-1+c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{70 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {16 b d^3 \left (1-c^2 x^2\right )}{35 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {8 b d^3 \left (1-c^2 x^2\right )^2}{105 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {6 b d^3 \left (1-c^2 x^2\right )^3}{175 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^3 \left (1-c^2 x^2\right )^4}{49 c \sqrt {-1+c x} \sqrt {1+c x}}+d^3 x (a+b \text {arccosh}(c x))-c^2 d^3 x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {1}{7} c^6 d^3 x^7 (a+b \text {arccosh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.64 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {d^3 \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (2161-757 c^2 x^2+351 c^4 x^4-75 c^6 x^6\right )+105 a c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+105 b c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right ) \text {arccosh}(c x)\right )}{3675 c} \]

[In]

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

[Out]

-1/3675*(d^3*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2161 - 757*c^2*x^2 + 351*c^4*x^4 - 75*c^6*x^6) + 105*a*c*x*(-35
+ 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 105*b*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcCosh[c*x]))/
c

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.68

method	result	size
parts	\(-d^{3} a \left (\frac {1}{7} c^{6} x^{7}-\frac {3}{5} c^{4} x^{5}+x^{3} c^{2}-x \right )-\frac {d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right )}{3675}\right )}{c}\)	\(130\)
derivativedivides	\(\frac {-d^{3} a \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right )}{3675}\right )}{c}\)	\(132\)
default	\(\frac {-d^{3} a \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right )}{3675}\right )}{c}\)	\(132\)

[In]

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

[Out]

-d^3*a*(1/7*c^6*x^7-3/5*c^4*x^5+x^3*c^2-x)-d^3*b/c*(1/7*arccosh(c*x)*c^7*x^7-3/5*arccosh(c*x)*c^5*x^5+c^3*x^3*
arccosh(c*x)-c*x*arccosh(c*x)-1/3675*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*x^6-351*c^4*x^4+757*c^2*x^2-2161))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.88 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {525 \, a c^{7} d^{3} x^{7} - 2205 \, a c^{5} d^{3} x^{5} + 3675 \, a c^{3} d^{3} x^{3} - 3675 \, a c d^{3} x + 105 \, {\left (5 \, b c^{7} d^{3} x^{7} - 21 \, b c^{5} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{3} - 35 \, b c d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{6} d^{3} x^{6} - 351 \, b c^{4} d^{3} x^{4} + 757 \, b c^{2} d^{3} x^{2} - 2161 \, b d^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c} \]

[In]

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

[Out]

-1/3675*(525*a*c^7*d^3*x^7 - 2205*a*c^5*d^3*x^5 + 3675*a*c^3*d^3*x^3 - 3675*a*c*d^3*x + 105*(5*b*c^7*d^3*x^7 -
21*b*c^5*d^3*x^5 + 35*b*c^3*d^3*x^3 - 35*b*c*d^3*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (75*b*c^6*d^3*x^6 - 351*b*
c^4*d^3*x^4 + 757*b*c^2*d^3*x^2 - 2161*b*d^3)*sqrt(c^2*x^2 - 1))/c

Sympy [F]

\[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=- d^{3} \left (\int \left (- a\right )\, dx + \int \left (- b \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int 3 a c^{2} x^{2}\, dx + \int \left (- 3 a c^{4} x^{4}\right )\, dx + \int a c^{6} x^{6}\, dx + \int 3 b c^{2} x^{2} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 3 b c^{4} x^{4} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{6} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]

[In]

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

[Out]

-d**3*(Integral(-a, x) + Integral(-b*acosh(c*x), x) + Integral(3*a*c**2*x**2, x) + Integral(-3*a*c**4*x**4, x)
+ Integral(a*c**6*x**6, x) + Integral(3*b*c**2*x**2*acosh(c*x), x) + Integral(-3*b*c**4*x**4*acosh(c*x), x) +
Integral(b*c**6*x**6*acosh(c*x), x))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.58 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{7} \, a c^{6} d^{3} x^{7} + \frac {3}{5} \, a c^{4} d^{3} x^{5} - \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{4} d^{3} - a c^{2} d^{3} x^{3} - \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{2} d^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{3}}{c} \]

[In]

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

[Out]

-1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 - 1/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^2*x^2 - 1)*x^6/c^2 + 6*sqrt(c^
2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)*x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*c^6*d^3 + 1/25*(15*x^5*arcco
sh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*c^4*d^3 -
a*c^2*d^3*x^3 - 1/3*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*c^2*d^3
+ a*d^3*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^3/c

Giac [F(-2)]

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

[In]

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

[In]

int((a + b*acosh(c*x))*(d - c^2*d*x^2)^3,x)

[Out]

int((a + b*acosh(c*x))*(d - c^2*d*x^2)^3, x)