Integrand size = 27, antiderivative size = 119 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 d x^3}-\frac {b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:
-1/6*b*c*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*(-c^2*d* x^2+d)^(3/2)*(a+b*arccosh(c*x))/d/x^3-1/3*b*c^3*(-c^2*d*x^2+d)^(1/2)*ln(x) /(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\frac {(-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}-\frac {1}{3} b c \left (\frac {1}{2 x^2}+c^2 \log (x)\right )\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \] Input:
Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^4,x]
Output:
(Sqrt[d - c^2*d*x^2]*(((-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x ]))/(3*x^3) - (b*c*(1/(2*x^2) + c^2*Log[x]))/3))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
Time = 0.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.74, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6332, 25, 82, 244, 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.
\(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^4} \, dx\) |
\(\Big \downarrow \) 6332 |
\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {(1-c x) (c x+1)}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 d x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1)}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 d x^3}\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {1-c^2 x^2}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 d x^3}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \left (\frac {1}{x^3}-\frac {c^2}{x}\right )dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 d x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 d x^3}\) |
Input:
Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^4,x]
Output:
-1/3*((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(d*x^3) + (b*c*Sqrt[d - c^2*d*x^2]*(-1/2*1/x^2 - c^2*Log[x]))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(f*x)^(m + 1)*(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, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3 , 0] && NeQ[m, -1]
Time = 0.35 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.29
method | result | size |
default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 d \,x^{3}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}+2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-2 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-c x \right )}{6 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\) | \(153\) |
parts | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 d \,x^{3}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}+2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-2 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-c x \right )}{6 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\) | \(153\) |
Input:
int((-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*a/d/x^3*(-c^2*d*x^2+d)^(3/2)+1/6*b*(-d*(c^2*x^2-1))^(1/2)*(2*arccosh( c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^2*x^2+2*c^3*x^3*arccosh(c*x)-2*ln(1+(c* x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^3*c^3-2*arccosh(c*x)*(c*x-1)^(1/2)*(c* x+1)^(1/2)-c*x)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/x^3
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (99) = 198\).
Time = 0.14 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.89 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\left [\frac {2 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} - 1} + 2 \, {\left (a c^{4} x^{4} - 2 \, a c^{2} x^{2} + a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} x^{5} - x^{3}\right )}}, -\frac {2 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} - 1\right )} \sqrt {d}}{c^{2} d x^{4} + {\left (c^{2} - 1\right )} d x^{2} - d}\right ) - 2 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} - 1} - 2 \, {\left (a c^{4} x^{4} - 2 \, a c^{2} x^{2} + a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} x^{5} - x^{3}\right )}}\right ] \] Input:
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/x^4,x, algorithm="fricas ")
Output:
[1/6*(2*(b*c^4*x^4 - 2*b*c^2*x^2 + b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt( c^2*x^2 - 1)) + (b*c^5*x^5 - b*c^3*x^3)*sqrt(-d)*log((c^2*d*x^6 + c^2*d*x^ 2 - d*x^4 + sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^4 - 1)*sqrt(-d) - d) /(c^2*x^4 - x^2)) + sqrt(-c^2*d*x^2 + d)*(b*c*x^3 - b*c*x)*sqrt(c^2*x^2 - 1) + 2*(a*c^4*x^4 - 2*a*c^2*x^2 + a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^5 - x^3) , -1/6*(2*(b*c^5*x^5 - b*c^3*x^3)*sqrt(d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt (c^2*x^2 - 1)*(x^2 - 1)*sqrt(d)/(c^2*d*x^4 + (c^2 - 1)*d*x^2 - d)) - 2*(b* c^4*x^4 - 2*b*c^2*x^2 + b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1 )) - sqrt(-c^2*d*x^2 + d)*(b*c*x^3 - b*c*x)*sqrt(c^2*x^2 - 1) - 2*(a*c^4*x ^4 - 2*a*c^2*x^2 + a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^5 - x^3)]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{4}}\, dx \] Input:
integrate((-c**2*d*x**2+d)**(1/2)*(a+b*acosh(c*x))/x**4,x)
Output:
Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))/x**4, x)
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {{\left (c^{4} d^{2} \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right ) + i \, \left (-1\right )^{-2 \, c^{2} d x^{2} + 2 \, d} c^{2} d^{\frac {3}{2}} \log \left (-2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) + \frac {\sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} d}{x^{2}}\right )} b c}{6 \, d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b \operatorname {arcosh}\left (c x\right )}{3 \, d x^{3}} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, d x^{3}} \] Input:
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/x^4,x, algorithm="maxima ")
Output:
1/6*(c^4*d^2*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2) + I*(-1)^(-2*c^2*d*x^2 + 2* d)*c^2*d^(3/2)*log(-2*c^2*d + 2*d/x^2) + sqrt(-c^4*d*x^4 + 2*c^2*d*x^2 - d )*d/x^2)*b*c/d - 1/3*(-c^2*d*x^2 + d)^(3/2)*b*arccosh(c*x)/(d*x^3) - 1/3*( -c^2*d*x^2 + d)^(3/2)*a/(d*x^3)
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/x^4,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. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^4} \,d x \] Input:
int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(1/2))/x^4,x)
Output:
int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(1/2))/x^4, x)
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {\sqrt {d}\, \left (\sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, a +3 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{4}}d x \right ) b \,x^{3}\right )}{3 x^{3}} \] Input:
int((-c^2*d*x^2+d)^(1/2)*(a+b*acosh(c*x))/x^4,x)
Output:
(sqrt(d)*(sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - sqrt( - c**2*x**2 + 1)*a + 3*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**4,x)*b*x**3))/(3*x**3)