Integrand size = 21, antiderivative size = 98 \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {3 b d x \sqrt {-1+c x} \sqrt {1+c x}}{32 c}+\frac {b d x (-1+c x)^{3/2} (1+c x)^{3/2}}{16 c}+\frac {3 b d \text {arccosh}(c x)}{32 c^2}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2} \] Output:
-3/32*b*d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c+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
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02 \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {d \left (c x \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (5-2 c^2 x^2\right )+8 a c x \left (-2+c^2 x^2\right )\right )+8 b c^2 x^2 \left (-2+c^2 x^2\right ) \text {arccosh}(c x)+10 b \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{32 c^2} \] Input:
Integrate[x*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]
Output:
-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[Sqr t[(-1 + c*x)/(1 + c*x)]]))/c^2
Time = 0.25 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6329, 40, 40, 43}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx\) |
\(\Big \downarrow \) 6329 |
\(\displaystyle \frac {b d \int (c x-1)^{3/2} (c x+1)^{3/2}dx}{4 c}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\) |
\(\Big \downarrow \) 40 |
\(\displaystyle \frac {b d \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \int \sqrt {c x-1} \sqrt {c x+1}dx\right )}{4 c}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\) |
\(\Big \downarrow \) 40 |
\(\displaystyle \frac {b d \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )\right )}{4 c}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\) |
\(\Big \downarrow \) 43 |
\(\displaystyle \frac {b d \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\) |
Input:
Int[x*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]
Output:
-1/4*(d*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/c^2 + (b*d*((x*(-1 + c*x)^(3 /2)*(1 + c*x)^(3/2))/4 - (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/2 - ArcCosh[ c*x]/(2*c)))/4))/(4*c)
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] + Simp[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]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
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] - Simp[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]
Time = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.39
method | result | size |
derivativedivides | \(\frac {-\frac {d a \left (c^{2} x^{2}-1\right )^{2}}{4}-d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}+\frac {\operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-5 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\) | \(136\) |
default | \(\frac {-\frac {d a \left (c^{2} x^{2}-1\right )^{2}}{4}-d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}+\frac {\operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-5 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\) | \(136\) |
orering | \(\frac {\left (14 c^{4} x^{4}-33 c^{2} x^{2}+10\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{32 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\left (2 c^{2} x^{2}-5\right ) \left (\left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-2 x^{2} c^{2} d \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+\frac {x \left (-c^{2} d \,x^{2}+d \right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{32 c^{2}}\) | \(137\) |
parts | \(-\frac {d a \left (c^{2} x^{2}-1\right )^{2}}{4 c^{2}}-\frac {d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}+\frac {\operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-5 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\) | \(138\) |
Input:
int(x*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
Output:
1/c^2*(-1/4*d*a*(c^2*x^2-1)^2-d*b*(1/4*arccosh(c*x)*c^4*x^4-1/2*c^2*x^2*ar ccosh(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)))
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {8 \, a c^{4} d x^{4} - 16 \, a c^{2} d x^{2} + {\left (8 \, b c^{4} d x^{4} - 16 \, b c^{2} d x^{2} + 5 \, b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{3} d x^{3} - 5 \, b c d x\right )} \sqrt {c^{2} x^{2} - 1}}{32 \, c^{2}} \] Input:
integrate(x*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="fricas")
Output:
-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
\[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=- d \left (\int \left (- a x\right )\, dx + \int a c^{2} x^{3}\, dx + \int \left (- b x \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{2} x^{3} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:
integrate(x*(-c**2*d*x**2+d)*(a+b*acosh(c*x)),x)
Output:
-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))
Time = 0.03 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.65 \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{4} \, a c^{2} d x^{4} - \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b c^{2} d + \frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d \] Input:
integrate(x*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="maxima")
Output:
-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
Exception generated. \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \] Input:
int(x*(a + b*acosh(c*x))*(d - c^2*d*x^2),x)
Output:
int(x*(a + b*acosh(c*x))*(d - c^2*d*x^2), x)
Time = 0.18 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05 \[ \int x \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {d \left (-8 \mathit {acosh} \left (c x \right ) b \,c^{4} x^{4}+16 \mathit {acosh} \left (c x \right ) b \,c^{2} x^{2}+2 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} x^{3}-5 \sqrt {c^{2} x^{2}-1}\, b c x -5 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b -8 a \,c^{4} x^{4}+16 a \,c^{2} x^{2}\right )}{32 c^{2}} \] Input:
int(x*(-c^2*d*x^2+d)*(a+b*acosh(c*x)),x)
Output:
(d*( - 8*acosh(c*x)*b*c**4*x**4 + 16*acosh(c*x)*b*c**2*x**2 + 2*sqrt(c**2* x**2 - 1)*b*c**3*x**3 - 5*sqrt(c**2*x**2 - 1)*b*c*x - 5*log(sqrt(c**2*x**2 - 1) + c*x)*b - 8*a*c**4*x**4 + 16*a*c**2*x**2))/(32*c**2)