\(\int (d-c^2 d x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [14]

Optimal result

Integrand size = 22, antiderivative size = 143 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {8 b d^2 \sqrt {-1+c x} \sqrt {1+c x}}{15 c}+\frac {4 b d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{45 c}-\frac {b d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{25 c}+d^2 x (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x)) \]

[Out]

4/45*b*d^2*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c-1/25*b*d^2*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c+d^2*x*(a+b*arccosh(c*x))-2/3
*c^2*d^2*x^3*(a+b*arccosh(c*x))+1/5*c^4*d^2*x^5*(a+b*arccosh(c*x))-8/15*b*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 177, 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, 534, 1261, 712} \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+d^2 x (a+b \text {arccosh}(c x))+\frac {b d^2 \left (1-c^2 x^2\right )^3}{25 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b d^2 \left (1-c^2 x^2\right )^2}{45 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^2 \left (1-c^2 x^2\right )}{15 c \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(8*b*d^2*(1 - c^2*x^2))/(15*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d^2*(1 - c^2*x^2)^2)/(45*c*Sqrt[-1 + c*x]*S
qrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3)/(25*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^2*x*(a + b*ArcCosh[c*x]) - (2*
c^2*d^2*x^3*(a + b*ArcCosh[c*x]))/3 + (c^4*d^2*x^5*(a + b*ArcCosh[c*x]))/5

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 534

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*
a2 + b1*b2*x^n)^FracPart[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

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^2 x (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-(b c) \int \frac {d^2 x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = d^2 x (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {1}{15} \left (b c d^2\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = d^2 x (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{15 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = d^2 x (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{30 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = d^2 x (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {-1+c^2 x}}-4 \sqrt {-1+c^2 x}+3 \left (-1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{30 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {8 b d^2 \left (1-c^2 x^2\right )}{15 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b d^2 \left (1-c^2 x^2\right )^2}{45 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3}{25 c \sqrt {-1+c x} \sqrt {1+c x}}+d^2 x (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (-149+38 c^2 x^2-9 c^4 x^4\right )+15 a c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+15 b c x \left (15-10 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)\right )}{225 c} \]

[In]

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

[Out]

(d^2*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-149 + 38*c^2*x^2 - 9*c^4*x^4) + 15*a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4)
+ 15*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4)*ArcCosh[c*x]))/(225*c)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69

method	result	size
parts	\(d^{2} a \left (\frac {1}{5} c^{4} x^{5}-\frac {2}{3} x^{3} c^{2}+x \right )+\frac {d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\)	\(99\)
derivativedivides	\(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\)	\(102\)
default	\(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\)	\(102\)

[In]

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

[Out]

d^2*a*(1/5*c^4*x^5-2/3*x^3*c^2+x)+d^2*b/c*(1/5*arccosh(c*x)*c^5*x^5-2/3*c^3*x^3*arccosh(c*x)+c*x*arccosh(c*x)-
1/225*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(9*c^4*x^4-38*c^2*x^2+149))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.93 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {45 \, a c^{5} d^{2} x^{5} - 150 \, a c^{3} d^{2} x^{3} + 225 \, a c d^{2} x + 15 \, {\left (3 \, b c^{5} d^{2} x^{5} - 10 \, b c^{3} d^{2} x^{3} + 15 \, b c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (9 \, b c^{4} d^{2} x^{4} - 38 \, b c^{2} d^{2} x^{2} + 149 \, b d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, c} \]

[In]

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

[Out]

1/225*(45*a*c^5*d^2*x^5 - 150*a*c^3*d^2*x^3 + 225*a*c*d^2*x + 15*(3*b*c^5*d^2*x^5 - 10*b*c^3*d^2*x^3 + 15*b*c*
d^2*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (9*b*c^4*d^2*x^4 - 38*b*c^2*d^2*x^2 + 149*b*d^2)*sqrt(c^2*x^2 - 1))/c

Sympy [F]

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

[In]

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

[Out]

d**2*(Integral(a, x) + Integral(b*acosh(c*x), x) + Integral(-2*a*c**2*x**2, x) + Integral(a*c**4*x**4, x) + In
tegral(-2*b*c**2*x**2*acosh(c*x), x) + Integral(b*c**4*x**4*acosh(c*x), x))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.36 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{5} \, a c^{4} d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{4} d^{2} - \frac {2}{3} \, a c^{2} d^{2} x^{3} - \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{2} d^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{2}}{c} \]

[In]

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

[Out]

1/5*a*c^4*d^2*x^5 + 1/75*(15*x^5*arccosh(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^2 - 2/3*a*c^2*d^2*x^3 - 2/9*(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^2 + a*d^2*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^2/c

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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