Integrand size = 40, antiderivative size = 265 \[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1+e^{2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {3 b \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^2 \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\frac {3 b^3 \operatorname {PolyLog}\left (4,-e^{2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{4 c} \] Output:
1/4*(a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^4/b/c-(a+b*arccosh((-c*x+1 )^(1/2)/(c*x+1)^(1/2)))^3*ln(1+((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/ 2)/(c*x+1)^(1/2)-1)^(1/2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2)/c-3/2 *b*(a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2*polylog(2,-((-c*x+1)^(1/2 )/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)^(1/2)*((-c*x+1)^(1/2)/(c* x+1)^(1/2)+1)^(1/2))^2)/c+3/2*b^2*(a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2 )))*polylog(3,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/2) -1)^(1/2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2)/c-3/4*b^3*polylog(4,- ((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)^(1/2)*((-c* x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2)/c
\[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx \] Input:
Integrate[(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]
Output:
Integrate[(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x]
Result contains complex when optimal does not.
Time = 1.07 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.93, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {7232, 6297, 25, 3042, 26, 4201, 2620, 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.
| \(\displaystyle \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{1-c^2 x^2} \, dx\) |
| \(\Big \downarrow \) 7232 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c x+1} \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{\sqrt {1-c x}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle -\frac {\int -\frac {(1-c x)^{3/2} \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}\right )}{(c x+1)^{3/2}}d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(1-c x)^{3/2} \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}\right )}{(c x+1)^{3/2}}d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\frac {i (1-c x)^{3/2} \tan \left (\frac {i a}{b}-\frac {i \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}\right )}{(c x+1)^{3/2}}d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int \frac {(1-c x)^{3/2} \tan \left (\frac {i a}{b}-\frac {i \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}\right )}{(c x+1)^{3/2}}d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {i \left (2 i \int \frac {e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )} (1-c x)^{3/2}}{\left (1+e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) (c x+1)^{3/2}}d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {3}{2} b \int \frac {(1-c x) \log \left (1+e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{c x+1}d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {b (1-c x)^{3/2} \log \left (e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}+1\right )}{2 (c x+1)^{3/2}}\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {3}{2} b \left (\frac {b (1-c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{2 (c x+1)}-b \int \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (2,-e^{\frac {2 \left (a-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}}\right )d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )\right )-\frac {b (1-c x)^{3/2} \log \left (e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}+1\right )}{2 (c x+1)^{3/2}}\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {3}{2} b \left (\frac {b (1-c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{2 (c x+1)}-b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )d\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{\frac {2 \left (a-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}}\right )\right )\right )-\frac {b (1-c x)^{3/2} \log \left (e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}+1\right )}{2 (c x+1)^{3/2}}\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {3}{2} b \left (\frac {b (1-c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{2 (c x+1)}-b \left (-\frac {1}{4} b^2 \int \frac {\sqrt {c x+1} \operatorname {PolyLog}\left (3,-a-b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{\sqrt {1-c x}}de^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}-\frac {1}{2} b \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{\frac {2 \left (a-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}}\right )\right )\right )-\frac {b (1-c x)^{3/2} \log \left (e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}+1\right )}{2 (c x+1)^{3/2}}\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {3}{2} b \left (\frac {b (1-c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{2 (c x+1)}-b \left (-\frac {1}{4} b^2 \operatorname {PolyLog}\left (4,-a-b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{\frac {2 \left (a-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}}\right )\right )\right )-\frac {b (1-c x)^{3/2} \log \left (e^{-2 \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}+1\right )}{2 (c x+1)^{3/2}}\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
Input:
Int[(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]
Output:
((-I)*(((-1/4*I)*(1 - c*x)^2)/(1 + c*x)^2 + (2*I)*(-1/2*(b*(1 - c*x)^(3/2) *Log[1 + E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(1 + c*x)^(3/2) + ( 3*b*((b*(1 - c*x)*PolyLog[2, -E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])] )/(2*(1 + c*x)) - b*(-1/2*(b*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])* PolyLog[3, -E^((2*(a - Sqrt[1 - c*x]/Sqrt[1 + c*x]))/b)]) - (b^2*PolyLog[4 , -a - b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]]])/4)))/2)))/(b*c)
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[(((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[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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && 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]
Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.) *(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e*(g/(C*(e*f - d *g))) Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(843\) vs. \(2(381)=762\).
Time = 0.42 (sec) , antiderivative size = 844, normalized size of antiderivative = 3.18
| method | result | size |
| default | \(\frac {a^{3} \ln \left (c x +1\right )}{2 c}-\frac {a^{3} \ln \left (c x -1\right )}{2 c}-b^{3} \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{4}}{4 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3} \ln \left (1+\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}+\frac {3 \operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}-\frac {3 \,\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (3, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}+\frac {3 \operatorname {polylog}\left (4, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{4 c}\right )-3 a \,b^{2} \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}-\frac {\operatorname {polylog}\left (3, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}\right )-3 a^{2} b \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}\right )\) | \(844\) |
| parts | \(\frac {a^{3} \ln \left (c x +1\right )}{2 c}-\frac {a^{3} \ln \left (c x -1\right )}{2 c}-b^{3} \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{4}}{4 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3} \ln \left (1+\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}+\frac {3 \operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}-\frac {3 \,\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (3, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}+\frac {3 \operatorname {polylog}\left (4, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{4 c}\right )-3 a \,b^{2} \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}-\frac {\operatorname {polylog}\left (3, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}\right )-3 a^{2} b \left (-\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}+\frac {\operatorname {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c}\right )\) | \(844\) |
Input:
int((a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x,method=_R ETURNVERBOSE)
Output:
1/2*a^3/c*ln(c*x+1)-1/2*a^3/c*ln(c*x-1)-b^3*(-1/4/c*arccosh((-c*x+1)^(1/2) /(c*x+1)^(1/2))^4+1/c*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2))^3*ln(1+((-c*x+ 1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)^(1/2)*((-c*x+1)^(1 /2)/(c*x+1)^(1/2)+1)^(1/2))^2)+3/2/c*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)) ^2*polylog(2,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/2)- 1)^(1/2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2)-3/2/c*arccosh((-c*x+1) ^(1/2)/(c*x+1)^(1/2))*polylog(3,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^( 1/2)/(c*x+1)^(1/2)-1)^(1/2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2)+3/4 /c*polylog(4,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/2)- 1)^(1/2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2))-3*a*b^2*(-1/3*arccosh ((-c*x+1)^(1/2)/(c*x+1)^(1/2))^3/c+1/c*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2 ))^2*ln(1+((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)^( 1/2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2)+1/c*arccosh((-c*x+1)^(1/2) /(c*x+1)^(1/2))*polylog(2,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/( c*x+1)^(1/2)-1)^(1/2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2)-1/2/c*pol ylog(3,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)^(1/ 2)*((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2))-3*a^2*b*(-1/2*arccosh((-c*x +1)^(1/2)/(c*x+1)^(1/2))^2/c+1/c*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2))*ln( 1+((-c*x+1)^(1/2)/(c*x+1)^(1/2)+((-c*x+1)^(1/2)/(c*x+1)^(1/2)-1)^(1/2)*((- c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^(1/2))^2)+1/2/c*polylog(2,-((-c*x+1)^(1/2...
\[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \] Input:
integrate((a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x, al gorithm="fricas")
Output:
integral(-(b^3*arccosh(sqrt(-c*x + 1)/sqrt(c*x + 1))^3 + 3*a*b^2*arccosh(s qrt(-c*x + 1)/sqrt(c*x + 1))^2 + 3*a^2*b*arccosh(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a^3)/(c^2*x^2 - 1), x)
\[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=- \int \frac {a^{3}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{3} \operatorname {acosh}^{3}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a b^{2} \operatorname {acosh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a^{2} b \operatorname {acosh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \] Input:
integrate((a+b*acosh((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**3/(-c**2*x**2+1),x)
Output:
-Integral(a**3/(c**2*x**2 - 1), x) - Integral(b**3*acosh(sqrt(-c*x + 1)/sq rt(c*x + 1))**3/(c**2*x**2 - 1), x) - Integral(3*a*b**2*acosh(sqrt(-c*x + 1)/sqrt(c*x + 1))**2/(c**2*x**2 - 1), x) - Integral(3*a**2*b*acosh(sqrt(-c *x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1), x)
\[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \] Input:
integrate((a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x, al gorithm="maxima")
Output:
1/2*a^3*(log(c*x + 1)/c - log(c*x - 1)/c) + 1/2*(b^3*log(c*x + 1) - b^3*lo g(-c*x + 1))*log(sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + sqrt(-c*x + 1))^3/c + integrate(1/8*(((c*x + 1)*sqrt(- c*x + 1)*b^3 - (-c*x + 1)^(3/2)*b^3)*log(c*x + 1)^3 - 6*((c*x + 1)*sqrt(-c *x + 1)*a*b^2 - (-c*x + 1)^(3/2)*a*b^2)*log(c*x + 1)^2 - 6*((4*a*b^2 + (b^ 3*c*x + b^3)*log(c*x + 1) - (b^3*c*x + b^3)*log(-c*x + 1))*(c*x + 1)*sqrt( -c*x + 1) - (4*a*b^2 + (b^3*c*x + b^3)*log(c*x + 1) - (b^3*c*x + b^3)*log( -c*x + 1))*(-c*x + 1)^(3/2) + ((4*a*b^2 + (b^3*c*x - b^3)*log(c*x + 1) - ( b^3*c*x - b^3)*log(-c*x + 1))*(c*x + 1) + (4*a*b^2 + (b^3*c*x + b^3)*log(c *x + 1) - (b^3*c*x + b^3)*log(-c*x + 1))*(c*x - 1) - 2*((c*x + 1)*b^3 + (c *x - 1)*b^3)*log(c*x + 1))*sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt (c*x + 1) + sqrt(-c*x + 1)) - 2*((c*x + 1)*sqrt(-c*x + 1)*b^3 - (-c*x + 1) ^(3/2)*b^3)*log(c*x + 1))*log(sqrt(sqrt(c*x + 1) + sqrt(-c*x + 1))*sqrt(-s qrt(c*x + 1) + sqrt(-c*x + 1)) + sqrt(-c*x + 1))^2 + (((c*x + 1)*b^3 + (c* x - 1)*b^3)*log(c*x + 1)^3 - 6*((c*x + 1)*a*b^2 + (c*x - 1)*a*b^2)*log(c*x + 1)^2 + 12*((c*x + 1)*a^2*b + (c*x - 1)*a^2*b)*log(c*x + 1))*sqrt(sqrt(c *x + 1) + sqrt(-c*x + 1))*sqrt(-sqrt(c*x + 1) + sqrt(-c*x + 1)) + 12*((c*x + 1)*sqrt(-c*x + 1)*a^2*b - (-c*x + 1)^(3/2)*a^2*b)*log(c*x + 1) - 6*(4*( c*x + 1)*sqrt(-c*x + 1)*a^2*b - 4*(-c*x + 1)^(3/2)*a^2*b + ((c*x + 1)*sqrt (-c*x + 1)*b^3 - (-c*x + 1)^(3/2)*b^3)*log(c*x + 1)^2 + (4*(c*x + 1)*a^...
Timed out. \[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\text {Timed out} \] Input:
integrate((a+b*arccosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x, al gorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {acosh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^3}{c^2\,x^2-1} \,d x \] Input:
int(-(a + b*acosh((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^3/(c^2*x^2 - 1),x)
Output:
int(-(a + b*acosh((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^3/(c^2*x^2 - 1), x)
\[ \int \frac {\left (a+b \text {arccosh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\frac {-6 \left (\int \frac {\mathit {acosh} \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )}{c^{2} x^{2}-1}d x \right ) a^{2} b c -2 \left (\int \frac {\mathit {acosh} \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{c^{2} x^{2}-1}d x \right ) b^{3} c -6 \left (\int \frac {\mathit {acosh} \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{c^{2} x^{2}-1}d x \right ) a \,b^{2} c -\mathrm {log}\left (c^{2} x -c \right ) a^{3}+\mathrm {log}\left (c^{2} x +c \right ) a^{3}}{2 c} \] Input:
int((a+b*acosh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x)
Output:
( - 6*int(acosh(sqrt( - c*x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1),x)*a**2*b* c - 2*int(acosh(sqrt( - c*x + 1)/sqrt(c*x + 1))**3/(c**2*x**2 - 1),x)*b**3 *c - 6*int(acosh(sqrt( - c*x + 1)/sqrt(c*x + 1))**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)