\(\int x^2 (d-c^2 d x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [12]

Optimal result

Integrand size = 25, antiderivative size = 177 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {8 b d^2 \sqrt {-1+c x} \sqrt {1+c x}}{105 c^3}+\frac {4 b d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{315 c^3}-\frac {b d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{175 c^3}-\frac {b d^2 (-1+c x)^{7/2} (1+c x)^{7/2}}{49 c^3}+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x)) \]

[Out]

4/315*b*d^2*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c^3-1/175*b*d^2*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c^3-1/49*b*d^2*(c*x-1)^(7/
2)*(c*x+1)^(7/2)/c^3+1/3*d^2*x^3*(a+b*arccosh(c*x))-2/5*c^2*d^2*x^5*(a+b*arccosh(c*x))+1/7*c^4*d^2*x^7*(a+b*ar
ccosh(c*x))-8/105*b*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {276, 5921, 12, 534, 1265, 785} \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))-\frac {b d^2 \left (1-c^2 x^2\right )^4}{49 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 \left (1-c^2 x^2\right )^3}{175 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b d^2 \left (1-c^2 x^2\right )^2}{315 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^2 \left (1-c^2 x^2\right )}{105 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(8*b*d^2*(1 - c^2*x^2))/(105*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d^2*(1 - c^2*x^2)^2)/(315*c^3*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3)/(175*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^2*(1 - c^2*x^2)^4)
/(49*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*x^3*(a + b*ArcCosh[c*x]))/3 - (2*c^2*d^2*x^5*(a + b*ArcCosh[c*x]
))/5 + (c^4*d^2*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 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && 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 785

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

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(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] &&
IntegerQ[(m - 1)/2]

Rule 5921

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(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, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-(b c) \int \frac {d^2 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )}{105 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-\frac {1}{105} \left (b c d^2\right ) \int \frac {x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{105 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x))-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x \left (35-42 c^2 x+15 c^4 x^2\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{210 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} c^4 d^2 x^7 (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}{c^2 \sqrt {-1+c^2 x}}-\frac {4 \sqrt {-1+c^2 x}}{c^2}+\frac {3 \left (-1+c^2 x\right )^{3/2}}{c^2}+\frac {15 \left (-1+c^2 x\right )^{5/2}}{c^2}\right ) \, dx,x,x^2\right )}{210 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {8 b d^2 \left (1-c^2 x^2\right )}{105 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b d^2 \left (1-c^2 x^2\right )^2}{315 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3}{175 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^2 \left (1-c^2 x^2\right )^4}{49 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arccosh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.66 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \left (105 a c^3 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )-b \sqrt {-1+c x} \sqrt {1+c x} \left (818+409 c^2 x^2-612 c^4 x^4+225 c^6 x^6\right )+105 b c^3 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right ) \text {arccosh}(c x)\right )}{11025 c^3} \]

[In]

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

[Out]

(d^2*(105*a*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(818 + 409*c^2*x^2 - 612*c
^4*x^4 + 225*c^6*x^6) + 105*b*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4)*ArcCosh[c*x]))/(11025*c^3)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.66

method	result	size
parts	\(d^{2} a \left (\frac {1}{7} c^{4} x^{7}-\frac {2}{5} c^{2} x^{5}+\frac {1}{3} x^{3}\right )+\frac {d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 c^{6} x^{6}-612 c^{4} x^{4}+409 c^{2} x^{2}+818\right )}{11025}\right )}{c^{3}}\)	\(116\)
derivativedivides	\(\frac {d^{2} a \left (\frac {1}{7} c^{7} x^{7}-\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 c^{6} x^{6}-612 c^{4} x^{4}+409 c^{2} x^{2}+818\right )}{11025}\right )}{c^{3}}\)	\(120\)
default	\(\frac {d^{2} a \left (\frac {1}{7} c^{7} x^{7}-\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 c^{6} x^{6}-612 c^{4} x^{4}+409 c^{2} x^{2}+818\right )}{11025}\right )}{c^{3}}\)	\(120\)

[In]

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

[Out]

d^2*a*(1/7*c^4*x^7-2/5*c^2*x^5+1/3*x^3)+d^2*b/c^3*(1/7*arccosh(c*x)*c^7*x^7-2/5*arccosh(c*x)*c^5*x^5+1/3*c^3*x
^3*arccosh(c*x)-1/11025*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(225*c^6*x^6-612*c^4*x^4+409*c^2*x^2+818))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.86 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1575 \, a c^{7} d^{2} x^{7} - 4410 \, a c^{5} d^{2} x^{5} + 3675 \, a c^{3} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} d^{2} x^{7} - 42 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (225 \, b c^{6} d^{2} x^{6} - 612 \, b c^{4} d^{2} x^{4} + 409 \, b c^{2} d^{2} x^{2} + 818 \, b d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{11025 \, c^{3}} \]

[In]

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

[Out]

1/11025*(1575*a*c^7*d^2*x^7 - 4410*a*c^5*d^2*x^5 + 3675*a*c^3*d^2*x^3 + 105*(15*b*c^7*d^2*x^7 - 42*b*c^5*d^2*x
^5 + 35*b*c^3*d^2*x^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (225*b*c^6*d^2*x^6 - 612*b*c^4*d^2*x^4 + 409*b*c^2*d^2*x
^2 + 818*b*d^2)*sqrt(c^2*x^2 - 1))/c^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.47 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{7} \, a c^{4} d^{2} x^{7} - \frac {2}{5} \, a c^{2} d^{2} x^{5} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{4} d^{2} - \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{2} \]

[In]

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

[Out]

1/7*a*c^4*d^2*x^7 - 2/5*a*c^2*d^2*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^4*d^2 - 2/75*(15*x^5*arccos
h(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^2*d^2 +
1/3*a*d^2*x^3 + 1/9*(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*d^2

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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