Integrand size = 20, antiderivative size = 246 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \]
1/5*x*arccosh(a*x)/c/(-a^2*c*x^2+c)^(5/2)+4/15*x*arccosh(a*x)/c^2/(-a^2*c* x^2+c)^(3/2)+8/15*x*arccosh(a*x)/c^3/(-a^2*c*x^2+c)^(1/2)+1/20*(a*x-1)^(1/ 2)*(a*x+1)^(1/2)/a/c^3/(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^(1/2)+2/15*(a*x-1)^(1 /2)*(a*x+1)^(1/2)/a/c^3/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2)-4/15*ln(-a^2*x^2 +1)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^3/(-a^2*c*x^2+c)^(1/2)
Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.47 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {4 a x \left (15-20 a^2 x^2+8 a^4 x^4\right ) \text {arccosh}(a x)+\sqrt {-1+a x} \sqrt {1+a x} \left (11-8 a^2 x^2-16 \left (-1+a^2 x^2\right )^2 \log \left (1-a^2 x^2\right )\right )}{60 a c^3 \left (-1+a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}} \]
(4*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*ArcCosh[a*x] + Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(11 - 8*a^2*x^2 - 16*(-1 + a^2*x^2)^2*Log[1 - a^2*x^2]))/(60*a*c^3* (-1 + a^2*x^2)^2*Sqrt[c - a^2*c*x^2])
Time = 0.64 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6316, 25, 82, 241, 6316, 82, 241, 6314, 240}
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.
| \(\displaystyle \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {4 \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}-\frac {a \sqrt {a x-1} \sqrt {a x+1} \int -\frac {x}{(1-a x)^3 (a x+1)^3}dx}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x}{(1-a x)^3 (a x+1)^3}dx}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle \frac {4 \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x}{\left (1-a^2 x^2\right )^3}dx}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {4 \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x}{(1-a x)^2 (a x+1)^2}dx}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x}{\left (1-a^2 x^2\right )^2}dx}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {x \text {arccosh}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{6 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 6314 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (\frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x}{1-a^2 x^2}dx}{c \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)}{c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{6 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {4 \left (\frac {x \text {arccosh}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)}{c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {a x-1} \sqrt {a x+1} \log \left (1-a^2 x^2\right )}{2 a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{6 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}\) |
(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*a*c^3*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^ 2]) + (x*ArcCosh[a*x])/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*((Sqrt[-1 + a*x]*S qrt[1 + a*x])/(6*a*c^2*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh[a*x ])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*((x*ArcCosh[a*x])/(c*Sqrt[c - a^2*c*x^ 2]) - (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[1 - a^2*x^2])/(2*a*c*Sqrt[c - a^2* c*x^2])))/(3*c)))/(5*c)
3.2.34.3.1 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[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcCosh[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp [b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])] Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(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}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Time = 1.37 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(-\frac {16 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-20 a^{3} x^{3}-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+15 a x +16 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-8 \sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-64 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{7} x^{7}-64 a^{8} x^{8}+248 x^{5} a^{5} \sqrt {a x -1}\, \sqrt {a x +1}+280 a^{6} x^{6}+160 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )-340 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-456 a^{4} x^{4}-380 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )+165 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +328 a^{2} x^{2}+256 \,\operatorname {arccosh}\left (a x \right )-88\right )}{60 \left (40 a^{10} x^{10}-215 a^{8} x^{8}+469 a^{6} x^{6}-517 a^{4} x^{4}+287 a^{2} x^{2}-64\right ) c^{4} a}+\frac {8 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}-1\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}\) | \(419\) |
-16/15*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^4/a/(a^2*x^2-1 )*arccosh(a*x)-1/60*(-c*(a^2*x^2-1))^(1/2)*(8*a^5*x^5-20*a^3*x^3-8*(a*x-1) ^(1/2)*(a*x+1)^(1/2)*a^4*x^4+15*a*x+16*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2) -8*(a*x-1)^(1/2)*(a*x+1)^(1/2))*(-64*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^7*x^7-6 4*a^8*x^8+248*x^5*a^5*(a*x-1)^(1/2)*(a*x+1)^(1/2)+280*a^6*x^6+160*a^4*x^4* arccosh(a*x)-340*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-456*a^4*x^4-380*a^2*x ^2*arccosh(a*x)+165*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+328*a^2*x^2+256*arccos h(a*x)-88)/(40*a^10*x^10-215*a^8*x^8+469*a^6*x^6-517*a^4*x^4+287*a^2*x^2-6 4)/c^4/a+8/15*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^4/a/(a^ 2*x^2-1)*ln((a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2-1)
\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
integral(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)/(a^8*c^4*x^8 - 4*a^6*c^4*x^6 + 6*a^4*c^4*x^4 - 4*a^2*c^4*x^2 + c^4), x)
Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.78 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {1}{60} \, a {\left (\frac {16 \, \sqrt {-\frac {1}{a^{4} c}} \log \left (x^{2} - \frac {1}{a^{2}}\right )}{c^{3}} + \frac {3}{{\left (a^{6} c^{3} x^{4} \sqrt {-\frac {1}{c}} - 2 \, a^{4} c^{3} x^{2} \sqrt {-\frac {1}{c}} + a^{2} c^{3} \sqrt {-\frac {1}{c}}\right )} c} - \frac {8}{{\left (a^{4} c^{2} x^{2} \sqrt {-\frac {1}{c}} - a^{2} c^{2} \sqrt {-\frac {1}{c}}\right )} c^{2}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {-a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \operatorname {arcosh}\left (a x\right ) \]
-1/60*a*(16*sqrt(-1/(a^4*c))*log(x^2 - 1/a^2)/c^3 + 3/((a^6*c^3*x^4*sqrt(- 1/c) - 2*a^4*c^3*x^2*sqrt(-1/c) + a^2*c^3*sqrt(-1/c))*c) - 8/((a^4*c^2*x^2 *sqrt(-1/c) - a^2*c^2*sqrt(-1/c))*c^2)) + 1/15*(8*x/(sqrt(-a^2*c*x^2 + c)* c^3) + 4*x/((-a^2*c*x^2 + c)^(3/2)*c^2) + 3*x/((-a^2*c*x^2 + c)^(5/2)*c))* arccosh(a*x)
Time = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.58 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {1}{60} \, \sqrt {-c} {\left (\frac {16 \, \log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{a c^{4}} - \frac {24 \, a^{4} x^{4} - 56 \, a^{2} x^{2} + 35}{{\left (a^{2} x^{2} - 1\right )}^{2} a c^{4}}\right )} - \frac {\sqrt {-a^{2} c x^{2} + c} {\left (4 \, {\left (\frac {2 \, a^{4} x^{2}}{c} - \frac {5 \, a^{2}}{c}\right )} x^{2} + \frac {15}{c}\right )} x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{15 \, {\left (a^{2} c x^{2} - c\right )}^{3}} \]
1/60*sqrt(-c)*(16*log(abs(a^2*x^2 - 1))/(a*c^4) - (24*a^4*x^4 - 56*a^2*x^2 + 35)/((a^2*x^2 - 1)^2*a*c^4)) - 1/15*sqrt(-a^2*c*x^2 + c)*(4*(2*a^4*x^2/ c - 5*a^2/c)*x^2 + 15/c)*x*log(a*x + sqrt(a^2*x^2 - 1))/(a^2*c*x^2 - c)^3
Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]