Integrand size = 27, antiderivative size = 203 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}-\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c^3 d \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^3+c^2*d*(a+b*arccosh(c*x))* (-c^2*d*x^2+d)^(1/2)/x-1/6*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c *x+1)^(1/2)-1/2*c^3*d*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/(c*x-1)^ (1/2)/(c*x+1)^(1/2)-4/3*b*c^3*d*ln(x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/( c*x+1)^(1/2)
Time = 0.83 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.28 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {-2 b d^2 \sqrt {\frac {-1+c x}{1+c x}} \left (1-5 c^2 x^2+4 c^4 x^4\right ) \text {arccosh}(c x)+3 b c^3 d^2 x^3 (-1+c x) \text {arccosh}(c x)^2-6 a c^3 d^{3/2} x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+d^2 \left (b c x (-1+c x)-2 a \sqrt {\frac {-1+c x}{1+c x}} \left (1-5 c^2 x^2+4 c^4 x^4\right )+8 b c^3 x^3 (-1+c x) \log (c x)\right )}{6 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \]
(-2*b*d^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 - 5*c^2*x^2 + 4*c^4*x^4)*ArcCosh[c *x] + 3*b*c^3*d^2*x^3*(-1 + c*x)*ArcCosh[c*x]^2 - 6*a*c^3*d^(3/2)*x^3*Sqrt [(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2] )/(Sqrt[d]*(-1 + c^2*x^2))] + d^2*(b*c*x*(-1 + c*x) - 2*a*Sqrt[(-1 + c*x)/ (1 + c*x)]*(1 - 5*c^2*x^2 + 4*c^4*x^4) + 8*b*c^3*x^3*(-1 + c*x)*Log[c*x])) /(6*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])
Time = 0.77 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6343, 25, 82, 244, 2009, 6339, 14, 6308}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4} \, dx\) |
| \(\Big \downarrow \) 6343 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^2}dx-\frac {b c d \sqrt {d-c^2 d x^2} \int -\frac {(1-c x) (c x+1)}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^2}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1)}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^2}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {1-c^2 x^2}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^2}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {1}{x^3}-\frac {c^2}{x}\right )dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6339 |
| \(\displaystyle c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \sqrt {d-c^2 d x^2} \int \frac {1}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle c^2 (-d) \left (\frac {c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/3*((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^3 + (b*c*d*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]) - c^2 *d*(-((Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x) + (c*Sqrt[d - c^2*d*x^ 2]*(a + b*ArcCosh[c*x])^2)/(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqrt[ d - c^2*d*x^2]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]))
3.1.75.3.1 Defintions of rubi rules used
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_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Cosh[c*x])^n/(f*(m + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*Simp[Sqrt[d + e* x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x ])^(n - 1), x], x] - Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 2)*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
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*((a + b*Arc Cosh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], 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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && G tQ[p, 0] && LtQ[m, -1]
Time = 1.12 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.33
| method | result | size |
| default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \,x^{3}}+\frac {2 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d x}+\frac {2 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+a \,c^{4} d x \sqrt {-c^{2} d \,x^{2}+d}+\frac {a \,c^{4} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-8 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-8 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{6 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\) | \(270\) |
| parts | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \,x^{3}}+\frac {2 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d x}+\frac {2 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+a \,c^{4} d x \sqrt {-c^{2} d \,x^{2}+d}+\frac {a \,c^{4} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-8 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-8 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{6 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\) | \(270\) |
-1/3*a/d/x^3*(-c^2*d*x^2+d)^(5/2)+2/3*a*c^2/d/x*(-c^2*d*x^2+d)^(5/2)+2/3*a *c^4*x*(-c^2*d*x^2+d)^(3/2)+a*c^4*d*x*(-c^2*d*x^2+d)^(1/2)+a*c^4*d^2/(c^2* d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/6*b*(-d*(c^2*x^2-1 ))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/x^3*(3*arccosh(c*x)^2*x^3*c^3-8*(c*x+ 1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-8*c^3*x^3*arccosh(c*x)+8*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)*d
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
integral(-(a*c^2*d*x^2 - a*d + (b*c^2*d*x^2 - b*d)*arccosh(c*x))*sqrt(-c^2 *d*x^2 + d)/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{4}}\, dx \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
1/3*(3*sqrt(-c^2*d*x^2 + d)*c^4*d*x + 3*c^3*d^(3/2)*arcsin(c*x) + 2*(-c^2* d*x^2 + d)^(3/2)*c^2/x - (-c^2*d*x^2 + d)^(5/2)/(d*x^3))*a + b*integrate(( -c^2*d*x^2 + d)^(3/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^4, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^4} \,d x \]