Integrand size = 24, antiderivative size = 178 \[ \int \frac {(a+b \text {arccosh}(c x))^3}{d-c^2 d x^2} \, dx=\frac {2 (a+b \text {arccosh}(c x))^3 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d}+\frac {3 b (a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c d}-\frac {3 b (a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c d}-\frac {6 b^2 (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )}{c d}+\frac {6 b^2 (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )}{c d}+\frac {6 b^3 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(c x)}\right )}{c d}-\frac {6 b^3 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(c x)}\right )}{c d} \] Output:
2*(a+b*arccosh(c*x))^3*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d+3*b*(a +b*arccosh(c*x))^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d-3*b*(a+ b*arccosh(c*x))^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d-6*b^2*(a+ b*arccosh(c*x))*polylog(3,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d+6*b^2*(a+b *arccosh(c*x))*polylog(3,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d+6*b^3*polylo g(4,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d-6*b^3*polylog(4,c*x+(c*x-1)^(1/2 )*(c*x+1)^(1/2))/c/d
Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b \text {arccosh}(c x))^3}{d-c^2 d x^2} \, dx=\frac {-(a+b \text {arccosh}(c x))^3 \log \left (1-e^{\text {arccosh}(c x)}\right )+(a+b \text {arccosh}(c x))^3 \log \left (1+e^{\text {arccosh}(c x)}\right )+3 b (a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-3 b (a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )-6 b^2 (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )+6 b^2 (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )+6 b^3 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(c x)}\right )-6 b^3 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(c x)}\right )}{c d} \] Input:
Integrate[(a + b*ArcCosh[c*x])^3/(d - c^2*d*x^2),x]
Output:
(-((a + b*ArcCosh[c*x])^3*Log[1 - E^ArcCosh[c*x]]) + (a + b*ArcCosh[c*x])^ 3*Log[1 + E^ArcCosh[c*x]] + 3*b*(a + b*ArcCosh[c*x])^2*PolyLog[2, -E^ArcCo sh[c*x]] - 3*b*(a + b*ArcCosh[c*x])^2*PolyLog[2, E^ArcCosh[c*x]] - 6*b^2*( a + b*ArcCosh[c*x])*PolyLog[3, -E^ArcCosh[c*x]] + 6*b^2*(a + b*ArcCosh[c*x ])*PolyLog[3, E^ArcCosh[c*x]] + 6*b^3*PolyLog[4, -E^ArcCosh[c*x]] - 6*b^3* PolyLog[4, E^ArcCosh[c*x]])/(c*d)
Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6318, 3042, 26, 4670, 3011, 7163, 2720, 7143}
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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^3}{d-c^2 d x^2} \, dx\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle -\frac {\int \frac {(a+b \text {arccosh}(c x))^3}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int i (a+b \text {arccosh}(c x))^3 \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int (a+b \text {arccosh}(c x))^3 \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c d}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {i \left (3 i b \int (a+b \text {arccosh}(c x))^2 \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-3 i b \int (a+b \text {arccosh}(c x))^2 \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{c d}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{c d}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \int \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)\right )-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \int \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)\right )-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{c d}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{c d}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \operatorname {PolyLog}\left (4,e^{\text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )\right )}{c d}\) |
Input:
Int[(a + b*ArcCosh[c*x])^3/(d - c^2*d*x^2),x]
Output:
((-I)*((2*I)*(a + b*ArcCosh[c*x])^3*ArcTanh[E^ArcCosh[c*x]] - (3*I)*b*(-(( a + b*ArcCosh[c*x])^2*PolyLog[2, -E^ArcCosh[c*x]]) + 2*b*((a + b*ArcCosh[c *x])*PolyLog[3, -E^ArcCosh[c*x]] - b*PolyLog[4, -E^ArcCosh[c*x]])) + (3*I) *b*(-((a + b*ArcCosh[c*x])^2*PolyLog[2, E^ArcCosh[c*x]]) + 2*b*((a + b*Arc Cosh[c*x])*PolyLog[3, E^ArcCosh[c*x]] - b*PolyLog[4, E^ArcCosh[c*x]]))))/( c*d)
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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
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[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 590, normalized size of antiderivative = 3.31
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{3} \operatorname {arctanh}\left (c x \right )}{d}-\frac {b^{3} \left (\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-6 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+6 \operatorname {polylog}\left (4, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+6 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-6 \operatorname {polylog}\left (4, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {3 a \,b^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {3 a^{2} b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d}}{c}\) | \(590\) |
| default | \(\frac {\frac {a^{3} \operatorname {arctanh}\left (c x \right )}{d}-\frac {b^{3} \left (\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-6 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+6 \operatorname {polylog}\left (4, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+6 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-6 \operatorname {polylog}\left (4, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {3 a \,b^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {3 a^{2} b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d}}{c}\) | \(590\) |
| parts | \(\frac {a^{3} \ln \left (c x +1\right )}{2 d c}-\frac {a^{3} \ln \left (c x -1\right )}{2 d c}-\frac {b^{3} \left (\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-6 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+6 \operatorname {polylog}\left (4, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+6 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-6 \operatorname {polylog}\left (4, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d c}-\frac {3 a \,b^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d c}-\frac {3 a^{2} b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d c}\) | \(618\) |
Input:
int((a+b*arccosh(c*x))^3/(-c^2*d*x^2+d),x,method=_RETURNVERBOSE)
Output:
1/c*(a^3/d*arctanh(c*x)-b^3/d*(arccosh(c*x)^3*ln(1-c*x-(c*x-1)^(1/2)*(c*x+ 1)^(1/2))+3*arccosh(c*x)^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-6*ar ccosh(c*x)*polylog(3,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+6*polylog(4,c*x+(c*x -1)^(1/2)*(c*x+1)^(1/2))-arccosh(c*x)^3*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/ 2))-3*arccosh(c*x)^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+6*arccosh (c*x)*polylog(3,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-6*polylog(4,-c*x-(c*x-1) ^(1/2)*(c*x+1)^(1/2)))-3*a*b^2/d*(arccosh(c*x)^2*ln(1-c*x-(c*x-1)^(1/2)*(c *x+1)^(1/2))+2*arccosh(c*x)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-2*p olylog(3,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-arccosh(c*x)^2*ln(1+c*x+(c*x-1)^ (1/2)*(c*x+1)^(1/2))-2*arccosh(c*x)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^( 1/2))+2*polylog(3,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3*a^2*b/d*(-arctanh(c *x)*arccosh(c*x)-2*I*(arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-arct anh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+dilog(1+I*(c*x+1)/(-c^2*x^2+1) ^(1/2))-dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)*(1/2*c*x +1/2)^(1/2)*(1/2*c*x-1/2)^(1/2)/(c^2*x^2-1)))
\[ \int \frac {(a+b \text {arccosh}(c x))^3}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3}}{c^{2} d x^{2} - d} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^3/(-c^2*d*x^2+d),x, algorithm="fricas")
Output:
integral(-(b^3*arccosh(c*x)^3 + 3*a*b^2*arccosh(c*x)^2 + 3*a^2*b*arccosh(c *x) + a^3)/(c^2*d*x^2 - d), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^3}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a^{3}}{c^{2} x^{2} - 1}\, dx + \int \frac {b^{3} \operatorname {acosh}^{3}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx + \int \frac {3 a b^{2} \operatorname {acosh}^{2}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx + \int \frac {3 a^{2} b \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \] Input:
integrate((a+b*acosh(c*x))**3/(-c**2*d*x**2+d),x)
Output:
-(Integral(a**3/(c**2*x**2 - 1), x) + Integral(b**3*acosh(c*x)**3/(c**2*x* *2 - 1), x) + Integral(3*a*b**2*acosh(c*x)**2/(c**2*x**2 - 1), x) + Integr al(3*a**2*b*acosh(c*x)/(c**2*x**2 - 1), x))/d
\[ \int \frac {(a+b \text {arccosh}(c x))^3}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3}}{c^{2} d x^{2} - d} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^3/(-c^2*d*x^2+d),x, algorithm="maxima")
Output:
1/2*a^3*(log(c*x + 1)/(c*d) - log(c*x - 1)/(c*d)) + 1/2*(b^3*log(c*x + 1) - b^3*log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^3/(c*d) - integ rate(3/2*((2*a*b^2*c*x + (b^3*c*x*log(c*x + 1) - b^3*c*x*log(c*x - 1) + 2* a*b^2)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^3*c^2*x^2 - b^3)*log(c*x + 1) - (b ^3*c^2*x^2 - b^3)*log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 2*(a^2*b*c*x + sqrt(c*x + 1)*sqrt(c*x - 1)*a^2*b)*log(c*x + sqrt(c*x + 1) *sqrt(c*x - 1)))/(c^3*d*x^3 - c*d*x + (c^2*d*x^2 - d)*sqrt(c*x + 1)*sqrt(c *x - 1)), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^3}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3}}{c^{2} d x^{2} - d} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^3/(-c^2*d*x^2+d),x, algorithm="giac")
Output:
integrate(-(b*arccosh(c*x) + a)^3/(c^2*d*x^2 - d), x)
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^3}{d-c^2 d x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^3}{d-c^2\,d\,x^2} \,d x \] Input:
int((a + b*acosh(c*x))^3/(d - c^2*d*x^2),x)
Output:
int((a + b*acosh(c*x))^3/(d - c^2*d*x^2), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^3}{d-c^2 d x^2} \, dx=\frac {-6 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{2} x^{2}-1}d x \right ) a^{2} b c -2 \left (\int \frac {\mathit {acosh} \left (c x \right )^{3}}{c^{2} x^{2}-1}d x \right ) b^{3} c -6 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{c^{2} x^{2}-1}d x \right ) a \,b^{2} c -\mathrm {log}\left (c^{2} x -c \right ) a^{3}+\mathrm {log}\left (c^{2} x +c \right ) a^{3}}{2 c d} \] Input:
int((a+b*acosh(c*x))^3/(-c^2*d*x^2+d),x)
Output:
( - 6*int(acosh(c*x)/(c**2*x**2 - 1),x)*a**2*b*c - 2*int(acosh(c*x)**3/(c* *2*x**2 - 1),x)*b**3*c - 6*int(acosh(c*x)**2/(c**2*x**2 - 1),x)*a*b**2*c - log(c**2*x - c)*a**3 + log(c**2*x + c)*a**3)/(2*c*d)