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

Optimal result

Integrand size = 21, antiderivative size = 98 \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {3 b d x \sqrt {-1+c x} \sqrt {1+c x}}{32 c}+\frac {b d x (-1+c x)^{3/2} (1+c x)^{3/2}}{16 c}+\frac {3 b d \text {arccosh}(c x)}{32 c^2}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2} \]

[Out]

1/16*b*d*x*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c+3/32*b*d*arccosh(c*x)/c^2-1/4*d*(-c^2*x^2+1)^2*(a+b*arccosh(c*x))/c^2
-3/32*b*d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5914, 38, 54} \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}+\frac {3 b d \text {arccosh}(c x)}{32 c^2}+\frac {b d x (c x-1)^{3/2} (c x+1)^{3/2}}{16 c}-\frac {3 b d x \sqrt {c x-1} \sqrt {c x+1}}{32 c} \]

[In]

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

[Out]

(-3*b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c) + (b*d*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(16*c) + (3*b*d*ArcC
osh[c*x])/(32*c^2) - (d*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/(4*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 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}+\frac {(b d) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{4 c} \\ & = \frac {b d x (-1+c x)^{3/2} (1+c x)^{3/2}}{16 c}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}-\frac {(3 b d) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{16 c} \\ & = -\frac {3 b d x \sqrt {-1+c x} \sqrt {1+c x}}{32 c}+\frac {b d x (-1+c x)^{3/2} (1+c x)^{3/2}}{16 c}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}+\frac {(3 b d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c} \\ & = -\frac {3 b d x \sqrt {-1+c x} \sqrt {1+c x}}{32 c}+\frac {b d x (-1+c x)^{3/2} (1+c x)^{3/2}}{16 c}+\frac {3 b d \text {arccosh}(c x)}{32 c^2}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02 \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {d \left (c x \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (5-2 c^2 x^2\right )+8 a c x \left (-2+c^2 x^2\right )\right )+8 b c^2 x^2 \left (-2+c^2 x^2\right ) \text {arccosh}(c x)+10 b \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{32 c^2} \]

[In]

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

[Out]

-1/32*(d*(c*x*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(5 - 2*c^2*x^2) + 8*a*c*x*(-2 + c^2*x^2)) + 8*b*c^2*x^2*(-2 + c^
2*x^2)*ArcCosh[c*x] + 10*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/c^2

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(\frac {-\frac {d a \left (c^{2} x^{2}-1\right )^{2}}{4}-d b \left (\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}+\frac {\operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-5 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\)	\(136\)
default	\(\frac {-\frac {d a \left (c^{2} x^{2}-1\right )^{2}}{4}-d b \left (\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}+\frac {\operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-5 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\)	\(136\)
parts	\(-\frac {d a \left (c^{2} x^{2}-1\right )^{2}}{4 c^{2}}-\frac {d b \left (\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}+\frac {\operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-5 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\)	\(138\)

[In]

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

[Out]

1/c^2*(-1/4*d*a*(c^2*x^2-1)^2-d*b*(1/4*c^4*x^4*arccosh(c*x)-1/2*c^2*x^2*arccosh(c*x)+1/4*arccosh(c*x)-1/32*(c*
x-1)^(1/2)*(c*x+1)^(1/2)*(2*(c^2*x^2-1)^(1/2)*c^3*x^3-5*c*x*(c^2*x^2-1)^(1/2)+3*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 = 98, normalized size of antiderivative = 1.00 \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {8 \, a c^{4} d x^{4} - 16 \, a c^{2} d x^{2} + {\left (8 \, b c^{4} d x^{4} - 16 \, b c^{2} d x^{2} + 5 \, b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{3} d x^{3} - 5 \, b c d x\right )} \sqrt {c^{2} x^{2} - 1}}{32 \, c^{2}} \]

[In]

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

[Out]

-1/32*(8*a*c^4*d*x^4 - 16*a*c^2*d*x^2 + (8*b*c^4*d*x^4 - 16*b*c^2*d*x^2 + 5*b*d)*log(c*x + sqrt(c^2*x^2 - 1))
- (2*b*c^3*d*x^3 - 5*b*c*d*x)*sqrt(c^2*x^2 - 1))/c^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.65 \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{4} \, a c^{2} d x^{4} - \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b c^{2} d + \frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d \]

[In]

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

[Out]

-1/4*a*c^2*d*x^4 - 1/32*(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 + 1/2*a*d*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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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