Integrand size = 26, antiderivative size = 473 \[ \int \frac {x^{11} \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}+\frac {7 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{90 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {13 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{150 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}+\frac {3 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{7/2}}{70 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{9/2}}{90 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}+\frac {4 b \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{15 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x} \]
1/3*(-c^4*x^4+1)^(3/2)*(a+b*arcsech(c*x))/c^12-1/10*(-c^4*x^4+1)^(5/2)*(a+ b*arcsech(c*x))/c^12+7/90*b*(c^2*x^2+1)^(3/2)*(-c^2*x^2+1)^(1/2)/c^13/x/(- 1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)-13/150*b*(c^2*x^2+1)^(5/2)*(-c^2*x^2+1)^(1/ 2)/c^13/x/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)+3/70*b*(c^2*x^2+1)^(7/2)*(-c^2* x^2+1)^(1/2)/c^13/x/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)-1/90*b*(c^2*x^2+1)^(9 /2)*(-c^2*x^2+1)^(1/2)/c^13/x/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)+4/15*b*arct anh((c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/c^13/x/(-1+1/c/x)^(1/2)/(1+1/c/x )^(1/2)-4/15*b*(-c^2*x^2+1)^(1/2)*(c^2*x^2+1)^(1/2)/c^13/x/(-1+1/c/x)^(1/2 )/(1+1/c/x)^(1/2)-1/2*(a+b*arcsech(c*x))*(-c^4*x^4+1)^(1/2)/c^12
Time = 0.63 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.45 \[ \int \frac {x^{11} \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\frac {-105 a \sqrt {1-c^4 x^4} \left (8+4 c^4 x^4+3 c^8 x^8\right )+\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^4 x^4} \left (768+36 c^2 x^2+78 c^4 x^4+5 c^6 x^6+35 c^8 x^8\right )}{-1+c x}-105 b \sqrt {1-c^4 x^4} \left (8+4 c^4 x^4+3 c^8 x^8\right ) \text {sech}^{-1}(c x)+840 b \log (x (1-c x))-840 b \log \left (1-c x-\sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^4 x^4}\right )}{3150 c^{12}} \]
(-105*a*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8) + (b*Sqrt[(1 - c*x)/ (1 + c*x)]*Sqrt[1 - c^4*x^4]*(768 + 36*c^2*x^2 + 78*c^4*x^4 + 5*c^6*x^6 + 35*c^8*x^8))/(-1 + c*x) - 105*b*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x ^8)*ArcSech[c*x] + 840*b*Log[x*(1 - c*x)] - 840*b*Log[1 - c*x - Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^4*x^4]])/(3150*c^12)
Time = 0.72 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.51, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6863, 27, 1388, 2331, 2123, 2009}
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.
| \(\displaystyle \int \frac {x^{11} \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx\) |
| \(\Big \downarrow \) 6863 |
| \(\displaystyle \frac {b \sqrt {1-c^2 x^2} \int -\frac {\sqrt {1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right )}{30 c^{12} x \sqrt {1-c^2 x^2}}dx}{c x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int \frac {\sqrt {1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right )}{x \sqrt {1-c^2 x^2}}dx}{30 c^{13} x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int \frac {\sqrt {c^2 x^2+1} \left (3 c^8 x^8+4 c^4 x^4+8\right )}{x}dx}{30 c^{13} x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int \frac {\sqrt {c^2 x^2+1} \left (3 c^8 x^8+4 c^4 x^4+8\right )}{x^2}dx^2}{60 c^{13} x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int \left (3 c^2 \left (c^2 x^2+1\right )^{7/2}-9 c^2 \left (c^2 x^2+1\right )^{5/2}+13 c^2 \left (c^2 x^2+1\right )^{3/2}-7 c^2 \sqrt {c^2 x^2+1}+\frac {8 \sqrt {c^2 x^2+1}}{x^2}\right )dx^2}{60 c^{13} x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}-\frac {b \sqrt {1-c^2 x^2} \left (-16 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{3} \left (c^2 x^2+1\right )^{9/2}-\frac {18}{7} \left (c^2 x^2+1\right )^{7/2}+\frac {26}{5} \left (c^2 x^2+1\right )^{5/2}-\frac {14}{3} \left (c^2 x^2+1\right )^{3/2}+16 \sqrt {c^2 x^2+1}\right )}{60 c^{13} x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}\) |
-1/2*(Sqrt[1 - c^4*x^4]*(a + b*ArcSech[c*x]))/c^12 + ((1 - c^4*x^4)^(3/2)* (a + b*ArcSech[c*x]))/(3*c^12) - ((1 - c^4*x^4)^(5/2)*(a + b*ArcSech[c*x]) )/(10*c^12) - (b*Sqrt[1 - c^2*x^2]*(16*Sqrt[1 + c^2*x^2] - (14*(1 + c^2*x^ 2)^(3/2))/3 + (26*(1 + c^2*x^2)^(5/2))/5 - (18*(1 + c^2*x^2)^(7/2))/7 + (2 *(1 + c^2*x^2)^(9/2))/3 - 16*ArcTanh[Sqrt[1 + c^2*x^2]]))/(60*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)
3.2.86.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHid e[u, x]}, Simp[(a + b*ArcSech[c*x]) v, x] + Simp[b*(Sqrt[1 - c^2*x^2]/(c* x*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])) Int[SimplifyIntegrand[v/(x*Sqrt[ 1 - c^2*x^2]), x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c} , x]
\[\int \frac {x^{11} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}d x\]
Time = 0.28 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.83 \[ \int \frac {x^{11} \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {105 \, {\left (3 \, b c^{10} x^{10} - 3 \, b c^{8} x^{8} + 4 \, b c^{6} x^{6} - 4 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} - 8 \, b\right )} \sqrt {-c^{4} x^{4} + 1} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (35 \, b c^{9} x^{9} + 5 \, b c^{7} x^{7} + 78 \, b c^{5} x^{5} + 36 \, b c^{3} x^{3} + 768 \, b c x\right )} \sqrt {-c^{4} x^{4} + 1} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 420 \, {\left (b c^{2} x^{2} - b\right )} \log \left (\frac {c^{2} x^{2} + \sqrt {-c^{4} x^{4} + 1} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) - 420 \, {\left (b c^{2} x^{2} - b\right )} \log \left (-\frac {c^{2} x^{2} - \sqrt {-c^{4} x^{4} + 1} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) + 105 \, {\left (3 \, a c^{10} x^{10} - 3 \, a c^{8} x^{8} + 4 \, a c^{6} x^{6} - 4 \, a c^{4} x^{4} + 8 \, a c^{2} x^{2} - 8 \, a\right )} \sqrt {-c^{4} x^{4} + 1}}{3150 \, {\left (c^{14} x^{2} - c^{12}\right )}} \]
-1/3150*(105*(3*b*c^10*x^10 - 3*b*c^8*x^8 + 4*b*c^6*x^6 - 4*b*c^4*x^4 + 8* b*c^2*x^2 - 8*b)*sqrt(-c^4*x^4 + 1)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2) ) + 1)/(c*x)) - (35*b*c^9*x^9 + 5*b*c^7*x^7 + 78*b*c^5*x^5 + 36*b*c^3*x^3 + 768*b*c*x)*sqrt(-c^4*x^4 + 1)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 420*(b*c^ 2*x^2 - b)*log((c^2*x^2 + sqrt(-c^4*x^4 + 1)*c*x*sqrt(-(c^2*x^2 - 1)/(c^2* x^2)) - 1)/(c^2*x^2 - 1)) - 420*(b*c^2*x^2 - b)*log(-(c^2*x^2 - sqrt(-c^4* x^4 + 1)*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c^2*x^2 - 1)) + 105*(3*a *c^10*x^10 - 3*a*c^8*x^8 + 4*a*c^6*x^6 - 4*a*c^4*x^4 + 8*a*c^2*x^2 - 8*a)* sqrt(-c^4*x^4 + 1))/(c^14*x^2 - c^12)
Timed out. \[ \int \frac {x^{11} \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {x^{11} \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{11}}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \]
-1/30*a*(3*(-c^4*x^4 + 1)^(5/2)/c^12 - 10*(-c^4*x^4 + 1)^(3/2)/c^12 + 15*s qrt(-c^4*x^4 + 1)/c^12) + 1/30*b*((3*c^12*x^12 + c^8*x^8 + 4*c^4*x^4 - 8)* log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)/(sqrt(c^2*x^2 + 1)*sqrt(c*x + 1)*sqr t(-c*x + 1)*c^12) - 30*integrate(1/30*(30*c^10*x^21*log(c) + 60*c^10*x^21* log(sqrt(x)) + (60*c^10*x^21*log(sqrt(x)) + (3*c^10*x^10*(10*log(c) + 1) + 3*c^8*x^8 + 4*c^6*x^6 + 4*c^4*x^4 + 8*c^2*x^2 + 8)*x^11)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)))/((c^10*x^10*e^(log(c*x + 1) + log(-c*x + 1)) + c ^10*x^10*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)))*sqrt(c^2*x^2 + 1)), x))
Exception generated. \[ \int \frac {x^{11} \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^{11} \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int \frac {x^{11}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \]