\(\int x (d-c^2 d x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [13]

Optimal result

Integrand size = 23, antiderivative size = 136 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {5 b d^2 x \sqrt {-1+c x} \sqrt {1+c x}}{96 c}+\frac {5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac {b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}+\frac {5 b d^2 \text {arccosh}(c x)}{96 c^2}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2} \]

[Out]

5/144*b*d^2*x*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c-1/36*b*d^2*x*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c+5/96*b*d^2*arccosh(c*x)
/c^2-1/6*d^2*(-c^2*x^2+1)^3*(a+b*arccosh(c*x))/c^2-5/96*b*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5914, 38, 54} \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2}+\frac {5 b d^2 \text {arccosh}(c x)}{96 c^2}-\frac {b d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{36 c}+\frac {5 b d^2 x (c x-1)^{3/2} (c x+1)^{3/2}}{144 c}-\frac {5 b d^2 x \sqrt {c x-1} \sqrt {c x+1}}{96 c} \]

[In]

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

[Out]

(-5*b*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c) + (5*b*d^2*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(144*c) - (b*d
^2*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(36*c) + (5*b*d^2*ArcCosh[c*x])/(96*c^2) - (d^2*(1 - c^2*x^2)^3*(a + b*
ArcCosh[c*x]))/(6*c^2)

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 5914

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

Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2}-\frac {\left (b d^2\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx}{6 c} \\ & = -\frac {b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2}+\frac {\left (5 b d^2\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{36 c} \\ & = \frac {5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac {b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2}-\frac {\left (5 b d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{48 c} \\ & = -\frac {5 b d^2 x \sqrt {-1+c x} \sqrt {1+c x}}{96 c}+\frac {5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac {b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2}+\frac {\left (5 b d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{96 c} \\ & = -\frac {5 b d^2 x \sqrt {-1+c x} \sqrt {1+c x}}{96 c}+\frac {5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac {b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}+\frac {5 b d^2 \text {arccosh}(c x)}{96 c^2}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{6 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.93 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \left (c x \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (-33+26 c^2 x^2-8 c^4 x^4\right )+48 a c x \left (3-3 c^2 x^2+c^4 x^4\right )\right )+48 b c^2 x^2 \left (3-3 c^2 x^2+c^4 x^4\right ) \text {arccosh}(c x)-66 b \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{288 c^2} \]

[In]

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

[Out]

(d^2*(c*x*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-33 + 26*c^2*x^2 - 8*c^4*x^4) + 48*a*c*x*(3 - 3*c^2*x^2 + c^4*x^4))
+ 48*b*c^2*x^2*(3 - 3*c^2*x^2 + c^4*x^4)*ArcCosh[c*x] - 66*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(288*c^2)

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {\frac {d^{2} a \left (c^{2} x^{2}-1\right )^{3}}{6}+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{2}+\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}-\frac {\operatorname {arccosh}\left (c x \right )}{6}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-8 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+26 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-33 c x \sqrt {c^{2} x^{2}-1}+15 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{288 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\)	\(170\)
default	\(\frac {\frac {d^{2} a \left (c^{2} x^{2}-1\right )^{3}}{6}+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{2}+\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}-\frac {\operatorname {arccosh}\left (c x \right )}{6}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-8 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+26 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-33 c x \sqrt {c^{2} x^{2}-1}+15 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{288 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\)	\(170\)
parts	\(\frac {d^{2} a \left (c^{2} x^{2}-1\right )^{3}}{6 c^{2}}+\frac {d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{2}+\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}-\frac {\operatorname {arccosh}\left (c x \right )}{6}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-8 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+26 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-33 c x \sqrt {c^{2} x^{2}-1}+15 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{288 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\)	\(172\)

[In]

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

[Out]

1/c^2*(1/6*d^2*a*(c^2*x^2-1)^3+d^2*b*(1/6*arccosh(c*x)*c^6*x^6-1/2*c^4*x^4*arccosh(c*x)+1/2*c^2*x^2*arccosh(c*
x)-1/6*arccosh(c*x)+1/288*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-8*c^5*x^5*(c^2*x^2-1)^(1/2)+26*(c^2*x^2-1)^(1/2)*c^3*x
^3-33*c*x*(c^2*x^2-1)^(1/2)+15*ln(c*x+(c^2*x^2-1)^(1/2)))/(c^2*x^2-1)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.10 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {48 \, a c^{6} d^{2} x^{6} - 144 \, a c^{4} d^{2} x^{4} + 144 \, a c^{2} d^{2} x^{2} + 3 \, {\left (16 \, b c^{6} d^{2} x^{6} - 48 \, b c^{4} d^{2} x^{4} + 48 \, b c^{2} d^{2} x^{2} - 11 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (8 \, b c^{5} d^{2} x^{5} - 26 \, b c^{3} d^{2} x^{3} + 33 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{288 \, c^{2}} \]

[In]

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

[Out]

1/288*(48*a*c^6*d^2*x^6 - 144*a*c^4*d^2*x^4 + 144*a*c^2*d^2*x^2 + 3*(16*b*c^6*d^2*x^6 - 48*b*c^4*d^2*x^4 + 48*
b*c^2*d^2*x^2 - 11*b*d^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (8*b*c^5*d^2*x^5 - 26*b*c^3*d^2*x^3 + 33*b*c*d^2*x)*s
qrt(c^2*x^2 - 1))/c^2

Sympy [F]

\[ \int x \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=d^{2} \left (\int a x\, dx + \int \left (- 2 a c^{2} x^{3}\right )\, dx + \int a c^{4} x^{5}\, dx + \int b x \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 2 b c^{2} x^{3} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{5} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]

[In]

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

[Out]

d**2*(Integral(a*x, x) + Integral(-2*a*c**2*x**3, x) + Integral(a*c**4*x**5, x) + Integral(b*x*acosh(c*x), x)
+ Integral(-2*b*c**2*x**3*acosh(c*x), x) + Integral(b*c**4*x**5*acosh(c*x), x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (113) = 226\).

Time = 0.25 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.11 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{6} \, a c^{4} d^{2} x^{6} - \frac {1}{2} \, a c^{2} d^{2} x^{4} + \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b c^{4} d^{2} - \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d^{2} \]

[In]

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

[Out]

1/6*a*c^4*d^2*x^6 - 1/2*a*c^2*d^2*x^4 + 1/288*(48*x^6*arccosh(c*x) - (8*sqrt(c^2*x^2 - 1)*x^5/c^2 + 10*sqrt(c^
2*x^2 - 1)*x^3/c^4 + 15*sqrt(c^2*x^2 - 1)*x/c^6 + 15*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^7)*c)*b*c^4*d^2 -
1/16*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x + 2*sqrt(c
^2*x^2 - 1)*c)/c^5)*c)*b*c^2*d^2 + 1/2*a*d^2*x^2 + 1/4*(2*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(
2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3))*b*d^2

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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