Integrand size = 25, antiderivative size = 179 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x}}{2 c^4 d^2 \sqrt {1+c x}}+\frac {b \text {arccosh}(c x)}{2 c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^4 d^2}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^4 d^2} \]
1/2*b*arccosh(c*x)/c^4/d^2+1/2*x^2*(a+b*arccosh(c*x))/c^2/d^2/(-c^2*x^2+1) -1/2*(a+b*arccosh(c*x))^2/b/c^4/d^2+(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^( 1/2)*(c*x+1)^(1/2))^2)/c^4/d^2+1/2*b*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^ (1/2))^2)/c^4/d^2-1/2*b/c^4/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*b*(c*x-1)^ (1/2)/c^4/d^2/(c*x+1)^(1/2)
Time = 0.52 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.17 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {-b \sqrt {\frac {-1+c x}{1+c x}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+\frac {b c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}-\frac {2 a}{-1+c^2 x^2}+\frac {b \text {arccosh}(c x)}{1-c x}+\frac {b \text {arccosh}(c x)}{1+c x}-2 b \text {arccosh}(c x)^2+4 b \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )+4 b \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )+2 a \log \left (1-c^2 x^2\right )+4 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )+4 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 c^4 d^2} \]
(-(b*Sqrt[(-1 + c*x)/(1 + c*x)]) + (b*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x ) + (b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) - (2*a)/(-1 + c^2*x^2) + (b*ArcCosh[c*x])/(1 - c*x) + (b*ArcCosh[c*x])/(1 + c*x) - 2*b*ArcCosh[c*x] ^2 + 4*b*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] + 4*b*ArcCosh[c*x]*Log[1 + E ^ArcCosh[c*x]] + 2*a*Log[1 - c^2*x^2] + 4*b*PolyLog[2, -E^ArcCosh[c*x]] + 4*b*PolyLog[2, E^ArcCosh[c*x]])/(4*c^4*d^2)
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {6349, 27, 100, 27, 87, 43, 6328, 3042, 26, 4199, 25, 2620, 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.
| \(\displaystyle \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6349 |
| \(\displaystyle -\frac {\int \frac {x (a+b \text {arccosh}(c x))}{d \left (1-c^2 x^2\right )}dx}{c^2 d}+\frac {b \int \frac {x^2}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 c d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2 d^2}+\frac {b \int \frac {x^2}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 c d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle -\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2 d^2}+\frac {b \left (\frac {\int \frac {c^2 x}{\sqrt {c x-1} (c x+1)^{3/2}}dx}{c^3}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2 d^2}+\frac {b \left (\frac {\int \frac {x}{\sqrt {c x-1} (c x+1)^{3/2}}dx}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2 d^2}+\frac {b \left (\frac {\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{c}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -\frac {\int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c^2 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}\) |
| \(\Big \downarrow \) 6328 |
| \(\displaystyle \frac {\int \frac {c x (a+b \text {arccosh}(c x))}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -i (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle -\frac {i \left (2 i \int -\frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (-2 i \int \frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {i \left (-2 i \left (\frac {1}{2} b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {i \left (-2 i \left (\frac {1}{4} b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {i \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {\frac {\text {arccosh}(c x)}{c^2}-\frac {\sqrt {c x-1}}{c^2 \sqrt {c x+1}}}{c}-\frac {1}{c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c d^2}\) |
(x^2*(a + b*ArcCosh[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) + (b*(-(1/(c^3*Sqrt[- 1 + c*x]*Sqrt[1 + c*x])) + (-(Sqrt[-1 + c*x]/(c^2*Sqrt[1 + c*x])) + ArcCos h[c*x]/c^2)/c))/(2*c*d^2) - (I*(((-1/2*I)*(a + b*ArcCosh[c*x])^2)/b - (2*I )*(-1/2*((a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])]) - (b*PolyLog[2, E^(2*ArcCosh[c*x])])/4)))/(c^4*d^2)
3.1.38.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/e Subst[Int[(a + b*x)^n*Coth[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 = 0.66 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{4 c x +4}+\frac {\ln \left (c x +1\right )}{2}-\frac {1}{4 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}}{c^{4}}\) | \(202\) |
| default | \(\frac {\frac {a \left (\frac {1}{4 c x +4}+\frac {\ln \left (c x +1\right )}{2}-\frac {1}{4 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}}{c^{4}}\) | \(202\) |
| parts | \(\frac {a \left (\frac {1}{4 c^{4} \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2 c^{4}}-\frac {1}{4 c^{4} \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2 c^{4}}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2} c^{4}}\) | \(213\) |
1/c^4*(a/d^2*(1/4/(c*x+1)+1/2*ln(c*x+1)-1/4/(c*x-1)+1/2*ln(c*x-1))+b/d^2*( -1/2*arccosh(c*x)^2-1/2*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-c^2*x^2+arccosh(c *x)+1)/(c^2*x^2-1)+arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+poly log(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2) *(c*x+1)^(1/2))+polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))))
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x^{3}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{3} \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
(Integral(a*x**3/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*x**3*acosh (c*x)/(c**4*x**4 - 2*c**2*x**2 + 1), x))/d**2
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
-1/8*b*(((c^2*x^2 - 1)*log(c*x + 1)^2 + 2*(c^2*x^2 - 1)*log(c*x + 1)*log(c *x - 1) + (c^2*x^2 - 1)*log(c*x - 1)^2 - 4*((c^2*x^2 - 1)*log(c*x + 1) + ( c^2*x^2 - 1)*log(c*x - 1) - 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 2) /(c^6*d^2*x^2 - c^4*d^2) - 8*integrate(1/2*((c^2*x^2 - 1)*log(c*x + 1) + ( c^2*x^2 - 1)*log(c*x - 1) - 1)/(c^8*d^2*x^5 - 2*c^6*d^2*x^3 + c^4*d^2*x + (c^7*d^2*x^4 - 2*c^5*d^2*x^2 + c^3*d^2)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x)) - 1/2*a*(1/(c^6*d^2*x^2 - c^4*d^2) - log(c^2*x^2 - 1)/(c^4*d^2 ))
Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]