Integrand size = 25, antiderivative size = 249 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {-1+c x} (1+c x)^{3/2}}-\frac {b (-1+c x)^{3/2}}{12 c^5 d^3 (1+c x)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x (a+b \text {arccosh}(c x))}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c^5 d^3}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 c^5 d^3}-\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 c^5 d^3} \]
1/12*b*x^3/c^2/d^3/(c*x-1)^(3/2)/(c*x+1)^(3/2)-1/12*b*(c*x-1)^(3/2)/c^5/d^ 3/(c*x+1)^(3/2)+1/4*x^3*(a+b*arccosh(c*x))/c^2/d^3/(-c^2*x^2+1)^2-3/8*x*(a +b*arccosh(c*x))/c^4/d^3/(-c^2*x^2+1)+3/4*(a+b*arccosh(c*x))*arctanh(c*x+( c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d^3+3/8*b*polylog(2,-c*x-(c*x-1)^(1/2)*(c* x+1)^(1/2))/c^5/d^3-3/8*b*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d ^3+1/4*b/c^5/d^3/(c*x+1)^(3/2)/(c*x-1)^(1/2)+3/8*b/c^5/d^3/(c*x-1)^(1/2)/( c*x+1)^(1/2)
Time = 1.29 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.15 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {b (-2+c x) \sqrt {1+c x}}{(-1+c x)^{3/2}}+\frac {b \sqrt {-1+c x} (2+c x)}{(1+c x)^{3/2}}+\frac {12 a c x}{\left (-1+c^2 x^2\right )^2}+\frac {30 a c x}{-1+c^2 x^2}+\frac {3 b \text {arccosh}(c x)}{(-1+c x)^2}-\frac {3 b \text {arccosh}(c x)}{(1+c x)^2}-15 b \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )-15 b \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+\frac {9}{2} b \text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-\frac {9}{2} b \text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-9 a \log (1-c x)+9 a \log (1+c x)+18 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-18 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{48 c^5 d^3} \]
(-((b*(-2 + c*x)*Sqrt[1 + c*x])/(-1 + c*x)^(3/2)) + (b*Sqrt[-1 + c*x]*(2 + c*x))/(1 + c*x)^(3/2) + (12*a*c*x)/(-1 + c^2*x^2)^2 + (30*a*c*x)/(-1 + c^ 2*x^2) + (3*b*ArcCosh[c*x])/(-1 + c*x)^2 - (3*b*ArcCosh[c*x])/(1 + c*x)^2 - 15*b*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)) - 15*b*( Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)) + (9*b*ArcCosh[c*x]*( ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]))/2 - (9*b*ArcCosh[c*x]*(ArcCosh[ c*x] - 4*Log[1 + E^ArcCosh[c*x]]))/2 - 9*a*Log[1 - c*x] + 9*a*Log[1 + c*x] + 18*b*PolyLog[2, -E^ArcCosh[c*x]] - 18*b*PolyLog[2, E^ArcCosh[c*x]])/(48 *c^5*d^3)
Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {6349, 27, 105, 100, 27, 48, 6349, 83, 6318, 3042, 26, 4670, 2715, 2838}
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.
| \(\displaystyle \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6349 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 c^2 d}-\frac {b \int \frac {x^3}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{4 c d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}-\frac {b \int \frac {x^3}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{4 c d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}-\frac {b \left (\frac {\int \frac {x^2}{(c x-1)^{3/2} (c x+1)^{5/2}}dx}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}-\frac {b \left (\frac {\frac {\int \frac {c \sqrt {c x-1}}{(c x+1)^{5/2}}dx}{c^3}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}-\frac {b \left (\frac {\frac {\int \frac {\sqrt {c x-1}}{(c x+1)^{5/2}}dx}{c^2}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {\frac {(c x-1)^{3/2}}{3 c^3 (c x+1)^{3/2}}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}\) |
| \(\Big \downarrow \) 6349 |
| \(\displaystyle -\frac {3 \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{2 c^2}+\frac {b \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{4 c^2 d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {\frac {(c x-1)^{3/2}}{3 c^3 (c x+1)^{3/2}}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle -\frac {3 \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{2 c^2}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 c^2 d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {\frac {(c x-1)^{3/2}}{3 c^3 (c x+1)^{3/2}}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 c^2 d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {\frac {(c x-1)^{3/2}}{3 c^3 (c x+1)^{3/2}}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 c^2 d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {\frac {(c x-1)^{3/2}}{3 c^3 (c x+1)^{3/2}}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {3 \left (\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 c^2 d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {\frac {(c x-1)^{3/2}}{3 c^3 (c x+1)^{3/2}}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {3 \left (\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 c^2 d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {\frac {(c x-1)^{3/2}}{3 c^3 (c x+1)^{3/2}}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {3 \left (\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 c^2 d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {\frac {(c x-1)^{3/2}}{3 c^3 (c x+1)^{3/2}}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {3 \left (\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 c^2 d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \left (\frac {\frac {(c x-1)^{3/2}}{3 c^3 (c x+1)^{3/2}}-\frac {1}{c^3 \sqrt {c x-1} (c x+1)^{3/2}}}{c}-\frac {x^3}{3 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 c d^3}\) |
-1/4*(b*(-1/3*x^3/(c*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (-(1/(c^3*Sqrt[-1 + c*x]*(1 + c*x)^(3/2))) + (-1 + c*x)^(3/2)/(3*c^3*(1 + c*x)^(3/2)))/c))/ (c*d^3) + (x^3*(a + b*ArcCosh[c*x]))/(4*c^2*d^3*(1 - c^2*x^2)^2) - (3*(-1/ 2*b/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCosh[c*x]))/(2*c^2*( 1 - c^2*x^2)) + ((I/2)*((2*I)*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^ArcCosh[c*x]]))/c^3) )/(4*c^2*d^3)
3.1.46.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(p + 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] && LtQ[p, -1] && IGtQ[m, 1]
Time = 0.72 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {5}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (-\frac {15 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+15 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-9 c x \,\operatorname {arccosh}\left (c x \right )-13 \sqrt {c x -1}\, \sqrt {c x +1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c^{5}}\) | \(256\) |
| default | \(\frac {-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {5}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (-\frac {15 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+15 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-9 c x \,\operatorname {arccosh}\left (c x \right )-13 \sqrt {c x -1}\, \sqrt {c x +1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c^{5}}\) | \(256\) |
| parts | \(-\frac {a \left (\frac {1}{16 c^{5} \left (c x +1\right )^{2}}-\frac {5}{16 c^{5} \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16 c^{5}}-\frac {1}{16 c^{5} \left (c x -1\right )^{2}}-\frac {5}{16 c^{5} \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16 c^{5}}\right )}{d^{3}}-\frac {b \left (-\frac {15 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+15 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-9 c x \,\operatorname {arccosh}\left (c x \right )-13 \sqrt {c x -1}\, \sqrt {c x +1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3} c^{5}}\) | \(273\) |
1/c^5*(-a/d^3*(1/16/(c*x+1)^2-5/16/(c*x+1)-3/16*ln(c*x+1)-1/16/(c*x-1)^2-5 /16/(c*x-1)+3/16*ln(c*x-1))-b/d^3*(-1/24*(15*c^3*x^3*arccosh(c*x)+15*(c*x- 1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-9*c*x*arccosh(c*x)-13*(c*x-1)^(1/2)*(c*x+1) ^(1/2))/(c^4*x^4-2*c^2*x^2+1)+3/8*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x +1)^(1/2))+3/8*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/8*arccosh(c*x) *ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/8*polylog(2,-c*x-(c*x-1)^(1/2)*(c *x+1)^(1/2))))
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
integral(-(b*x^4*arccosh(c*x) + a*x^4)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^ 2*d^3*x^2 - d^3), x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a x^{4}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b x^{4} \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
-(Integral(a*x**4/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integr al(b*x**4*acosh(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x))/d**3
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
1/2048*(18432*c^5*integrate(1/32*x^5*log(c*x - 1)/(c^10*d^3*x^6 - 3*c^8*d^ 3*x^4 + 3*c^6*d^3*x^2 - c^4*d^3), x) + 80*c^4*(2*(5*c^2*x^3 - 3*x)/(c^12*d ^3*x^4 - 2*c^10*d^3*x^2 + c^8*d^3) + 3*log(c*x + 1)/(c^9*d^3) - 3*log(c*x - 1)/(c^9*d^3)) - 6144*c^4*integrate(1/32*x^4*log(c*x - 1)/(c^10*d^3*x^6 - 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 - c^4*d^3), x) + 18*(c*(2*(5*c^2*x^2 + 3*c* x - 6)/(c^12*d^3*x^3 - c^11*d^3*x^2 - c^10*d^3*x + c^9*d^3) - 5*log(c*x + 1)/(c^9*d^3) + 5*log(c*x - 1)/(c^9*d^3)) + 16*(2*c^2*x^2 - 1)*log(c*x - 1) /(c^12*d^3*x^4 - 2*c^10*d^3*x^2 + c^8*d^3))*c^3 - 48*c^2*(2*(c^2*x^3 + x)/ (c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3) - log(c*x + 1)/(c^7*d^3) + log(c* x - 1)/(c^7*d^3)) + 12288*c^2*integrate(1/32*x^2*log(c*x - 1)/(c^10*d^3*x^ 6 - 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 - c^4*d^3), x) + 9*(c*(2*(3*c^2*x^2 - 3* c*x - 2)/(c^10*d^3*x^3 - c^9*d^3*x^2 - c^8*d^3*x + c^7*d^3) - 3*log(c*x + 1)/(c^7*d^3) + 3*log(c*x - 1)/(c^7*d^3)) - 16*log(c*x - 1)/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3))*c - 32*(3*(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1 )^2 + 6*(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1)*log(c*x - 1) - 4*(10*c^3*x^ 3 - 6*c*x + 3*(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1) - 3*(c^4*x^4 - 2*c^2* x^2 + 1)*log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^9*d^3*x^ 4 - 2*c^7*d^3*x^2 + c^5*d^3) + 2048*integrate(1/16*(10*c^3*x^3 - 6*c*x + 3 *(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1) - 3*(c^4*x^4 - 2*c^2*x^2 + 1)*log( c*x - 1))/(c^11*d^3*x^7 - 3*c^9*d^3*x^5 + 3*c^7*d^3*x^3 - c^5*d^3*x + (...
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]