Integrand size = 25, antiderivative size = 190 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {61}{25} b c d^3 \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{25} b c^3 d^3 x^2 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{25} b c^5 d^3 x^4 \sqrt {-1+c x} \sqrt {1+c x}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+b c d^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \] Output:
61/25*b*c*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)-7/25*b*c^3*d^3*x^2*(c*x-1)^(1/2) *(c*x+1)^(1/2)+1/25*b*c^5*d^3*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)-d^3*(a+b*arc cosh(c*x))/x-3*c^2*d^3*x*(a+b*arccosh(c*x))+c^4*d^3*x^3*(a+b*arccosh(c*x)) -1/5*c^6*d^3*x^5*(a+b*arccosh(c*x))+b*c*d^3*arctan((c*x-1)^(1/2)*(c*x+1)^( 1/2))
Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{25} d^3 \left (-\frac {25 a}{x}-75 a c^2 x+25 a c^4 x^3-5 a c^6 x^5+b c \sqrt {-1+c x} \sqrt {1+c x} \left (61-7 c^2 x^2+c^4 x^4\right )-\frac {5 b \left (5+15 c^2 x^2-5 c^4 x^4+c^6 x^6\right ) \text {arccosh}(c x)}{x}-25 b c \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right ) \] Input:
Integrate[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^2,x]
Output:
(d^3*((-25*a)/x - 75*a*c^2*x + 25*a*c^4*x^3 - 5*a*c^6*x^5 + b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(61 - 7*c^2*x^2 + c^4*x^4) - (5*b*(5 + 15*c^2*x^2 - 5*c ^4*x^4 + c^6*x^6)*ArcCosh[c*x])/x - 25*b*c*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])]))/25
Time = 0.71 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6336, 27, 2113, 2331, 2123, 2009}
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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx\) |
\(\Big \downarrow \) 6336 |
\(\displaystyle -b c \int -\frac {d^3 \left (c^6 x^6-5 c^4 x^4+15 c^2 x^2+5\right )}{5 x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} b c d^3 \int \frac {c^6 x^6-5 c^4 x^4+15 c^2 x^2+5}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}\) |
\(\Big \downarrow \) 2113 |
\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {c^6 x^6-5 c^4 x^4+15 c^2 x^2+5}{x \sqrt {c^2 x^2-1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}\) |
\(\Big \downarrow \) 2331 |
\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {c^6 x^6-5 c^4 x^4+15 c^2 x^2+5}{x^2 \sqrt {c^2 x^2-1}}dx^2}{10 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \int \left (\left (c^2 x^2-1\right )^{3/2} c^2-3 \sqrt {c^2 x^2-1} c^2+\frac {11 c^2}{\sqrt {c^2 x^2-1}}+\frac {5}{x^2 \sqrt {c^2 x^2-1}}\right )dx^2}{10 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+\frac {b c d^3 \sqrt {c^2 x^2-1} \left (10 \arctan \left (\sqrt {c^2 x^2-1}\right )+\frac {2}{5} \left (c^2 x^2-1\right )^{5/2}-2 \left (c^2 x^2-1\right )^{3/2}+22 \sqrt {c^2 x^2-1}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^2,x]
Output:
-((d^3*(a + b*ArcCosh[c*x]))/x) - 3*c^2*d^3*x*(a + b*ArcCosh[c*x]) + c^4*d ^3*x^3*(a + b*ArcCosh[c*x]) - (c^6*d^3*x^5*(a + b*ArcCosh[c*x]))/5 + (b*c* d^3*Sqrt[-1 + c^2*x^2]*(22*Sqrt[-1 + c^2*x^2] - 2*(-1 + c^2*x^2)^(3/2) + ( 2*(-1 + c^2*x^2)^(5/2))/5 + 10*ArcTan[Sqrt[-1 + c^2*x^2]]))/(10*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. )*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ m]/(a*c + b*d*x^2)^FracPart[m]) Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a *d, 0] && EqQ[m, n] && !IntegerQ[m]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
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]}, Simp [(a + b*ArcCosh[c*x]) u, x] - Simp[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] && E qQ[c^2*d + e, 0] && IGtQ[p, 0]
Time = 0.17 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.94
method | result | size |
parts | \(-d^{3} a \left (\frac {c^{6} x^{5}}{5}-c^{4} x^{3}+3 c^{2} x +\frac {1}{x}\right )-d^{3} b c \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+7 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+25 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-61 \sqrt {c^{2} x^{2}-1}\right )}{25 \sqrt {c^{2} x^{2}-1}}\right )\) | \(178\) |
derivativedivides | \(c \left (-d^{3} a \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+7 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+25 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-61 \sqrt {c^{2} x^{2}-1}\right )}{25 \sqrt {c^{2} x^{2}-1}}\right )\right )\) | \(181\) |
default | \(c \left (-d^{3} a \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+7 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+25 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-61 \sqrt {c^{2} x^{2}-1}\right )}{25 \sqrt {c^{2} x^{2}-1}}\right )\right )\) | \(181\) |
Input:
int((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^2,x,method=_RETURNVERBOSE)
Output:
-d^3*a*(1/5*c^6*x^5-c^4*x^3+3*c^2*x+1/x)-d^3*b*c*(1/5*arccosh(c*x)*c^5*x^5 -c^3*x^3*arccosh(c*x)+3*c*x*arccosh(c*x)+arccosh(c*x)/c/x+1/25*(c*x-1)^(1/ 2)*(c*x+1)^(1/2)*(-c^4*x^4*(c^2*x^2-1)^(1/2)+7*c^2*x^2*(c^2*x^2-1)^(1/2)+2 5*arctan(1/(c^2*x^2-1)^(1/2))-61*(c^2*x^2-1)^(1/2))/(c^2*x^2-1)^(1/2))
Time = 0.14 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.31 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {5 \, a c^{6} d^{3} x^{6} - 25 \, a c^{4} d^{3} x^{4} + 75 \, a c^{2} d^{3} x^{2} - 50 \, b c d^{3} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 5 \, {\left (b c^{6} - 5 \, b c^{4} + 15 \, b c^{2} + 5 \, b\right )} d^{3} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 25 \, a d^{3} + 5 \, {\left (b c^{6} d^{3} x^{6} - 5 \, b c^{4} d^{3} x^{4} + 15 \, b c^{2} d^{3} x^{2} - {\left (b c^{6} - 5 \, b c^{4} + 15 \, b c^{2} + 5 \, b\right )} d^{3} x + 5 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{5} d^{3} x^{5} - 7 \, b c^{3} d^{3} x^{3} + 61 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} - 1}}{25 \, x} \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^2,x, algorithm="fricas")
Output:
-1/25*(5*a*c^6*d^3*x^6 - 25*a*c^4*d^3*x^4 + 75*a*c^2*d^3*x^2 - 50*b*c*d^3* x*arctan(-c*x + sqrt(c^2*x^2 - 1)) - 5*(b*c^6 - 5*b*c^4 + 15*b*c^2 + 5*b)* d^3*x*log(-c*x + sqrt(c^2*x^2 - 1)) + 25*a*d^3 + 5*(b*c^6*d^3*x^6 - 5*b*c^ 4*d^3*x^4 + 15*b*c^2*d^3*x^2 - (b*c^6 - 5*b*c^4 + 15*b*c^2 + 5*b)*d^3*x + 5*b*d^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^5*d^3*x^5 - 7*b*c^3*d^3*x^3 + 61*b*c*d^3*x)*sqrt(c^2*x^2 - 1))/x
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=- d^{3} \left (\int 3 a c^{2}\, dx + \int \left (- \frac {a}{x^{2}}\right )\, dx + \int \left (- 3 a c^{4} x^{2}\right )\, dx + \int a c^{6} x^{4}\, dx + \int 3 b c^{2} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{2}}\right )\, dx + \int \left (- 3 b c^{4} x^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{4} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)**3*(a+b*acosh(c*x))/x**2,x)
Output:
-d**3*(Integral(3*a*c**2, x) + Integral(-a/x**2, x) + Integral(-3*a*c**4*x **2, x) + Integral(a*c**6*x**4, x) + Integral(3*b*c**2*acosh(c*x), x) + In tegral(-b*acosh(c*x)/x**2, x) + Integral(-3*b*c**4*x**2*acosh(c*x), x) + I ntegral(b*c**6*x**4*acosh(c*x), x))
Time = 0.11 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {1}{5} \, a c^{6} d^{3} x^{5} - \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{6} d^{3} + a c^{4} d^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{4} d^{3} - 3 \, a c^{2} d^{3} x - 3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c d^{3} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{3} - \frac {a d^{3}}{x} \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^2,x, algorithm="maxima")
Output:
-1/5*a*c^6*d^3*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^6*d^3 + a*c^4*d^3*x^3 + 1/3*(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^4*d^3 - 3*a*c^2*d^3*x - 3*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*c*d^3 - (c*arcsin(1/(c*abs(x))) + arccosh(c*x)/x)* b*d^3 - a*d^3/x
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^2,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. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x^2} \,d x \] Input:
int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^3)/x^2,x)
Output:
int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^3)/x^2, x)
Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {d^{3} \left (-5 \mathit {acosh} \left (c x \right ) b \,c^{6} x^{6}+25 \mathit {acosh} \left (c x \right ) b \,c^{4} x^{4}-75 \mathit {acosh} \left (c x \right ) b \,c^{2} x^{2}-25 \mathit {acosh} \left (c x \right ) b -50 \mathit {atan} \left (\sqrt {c^{2} x^{2}-1}+c x \right ) b c x +\sqrt {c^{2} x^{2}-1}\, b \,c^{5} x^{5}-7 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} x^{3}-14 \sqrt {c^{2} x^{2}-1}\, b c x +75 \sqrt {c x +1}\, \sqrt {c x -1}\, b c x -5 a \,c^{6} x^{6}+25 a \,c^{4} x^{4}-75 a \,c^{2} x^{2}-25 a \right )}{25 x} \] Input:
int((-c^2*d*x^2+d)^3*(a+b*acosh(c*x))/x^2,x)
Output:
(d**3*( - 5*acosh(c*x)*b*c**6*x**6 + 25*acosh(c*x)*b*c**4*x**4 - 75*acosh( c*x)*b*c**2*x**2 - 25*acosh(c*x)*b - 50*atan(sqrt(c**2*x**2 - 1) + c*x)*b* c*x + sqrt(c**2*x**2 - 1)*b*c**5*x**5 - 7*sqrt(c**2*x**2 - 1)*b*c**3*x**3 - 14*sqrt(c**2*x**2 - 1)*b*c*x + 75*sqrt(c*x + 1)*sqrt(c*x - 1)*b*c*x - 5* a*c**6*x**6 + 25*a*c**4*x**4 - 75*a*c**2*x**2 - 25*a))/(25*x)