Integrand size = 13, antiderivative size = 69 \[ \int \text {sech}^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {16 e^{4 a} x \left (c x^n\right )^{4 b} \operatorname {Hypergeometric2F1}\left (4,\frac {1}{2} \left (4+\frac {1}{b n}\right ),\frac {1}{2} \left (6+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+4 b n} \]
16*exp(4*a)*x*(c*x^n)^(4*b)*hypergeom([4, 2+1/2/b/n],[3+1/2/b/n],-exp(2*a) *(c*x^n)^(2*b))/(4*b*n+1)
Leaf count is larger than twice the leaf count of optimal. \(750\) vs. \(2(69)=138\).
Time = 14.12 (sec) , antiderivative size = 750, normalized size of antiderivative = 10.87 \[ \int \text {sech}^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {\left (-1+4 b^2 n^2\right ) x \text {sech}\left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \text {sech}\left (a+b n \log (x)+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \sinh (b n \log (x))}{6 b^3 n^3}+\frac {x \text {sech}\left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \text {sech}^3\left (a+b n \log (x)+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \sinh (b n \log (x))}{3 b n}+\frac {x \text {sech}\left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \text {sech}^2\left (a+b n \log (x)+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \left (\cosh \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+2 b n \sinh \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{6 b^2 n^2}+\frac {e^{-\frac {a+b \left (-n \log (x)+\log \left (c x^n\right )\right )}{b n}} \text {sech}\left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \left (e^{\left (2+\frac {1}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )} \cosh \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{2 b n},2+\frac {1}{2 b n},-e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )-e^{\frac {a}{b n}+\frac {-n \log (x)+\log \left (c x^n\right )}{n}} (1+2 b n) x \left (\cosh \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 b n},1+\frac {1}{2 b n},-e^{2 \left (a+b n \log (x)+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}\right )+\sinh \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{6 b^3 n^3 (1+2 b n)}-\frac {2 e^{-\frac {a+b \left (-n \log (x)+\log \left (c x^n\right )\right )}{b n}} \text {sech}\left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \left (e^{\left (2+\frac {1}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )} \cosh \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{2 b n},2+\frac {1}{2 b n},-e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )-e^{\frac {a}{b n}+\frac {-n \log (x)+\log \left (c x^n\right )}{n}} (1+2 b n) x \left (\cosh \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 b n},1+\frac {1}{2 b n},-e^{2 \left (a+b n \log (x)+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}\right )+\sinh \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{3 b n (1+2 b n)} \]
((-1 + 4*b^2*n^2)*x*Sech[a + b*(-(n*Log[x]) + Log[c*x^n])]*Sech[a + b*n*Lo g[x] + b*(-(n*Log[x]) + Log[c*x^n])]*Sinh[b*n*Log[x]])/(6*b^3*n^3) + (x*Se ch[a + b*(-(n*Log[x]) + Log[c*x^n])]*Sech[a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])]^3*Sinh[b*n*Log[x]])/(3*b*n) + (x*Sech[a + b*(-(n*Log[x]) + Log[c*x^n])]*Sech[a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])]^2*(Cosh[a + b*(-(n*Log[x]) + Log[c*x^n])] + 2*b*n*Sinh[a + b*(-(n*Log[x]) + Log[c*x ^n])]))/(6*b^2*n^2) + (Sech[a + b*(-(n*Log[x]) + Log[c*x^n])]*(E^((2 + 1/( b*n))*(a + b*Log[c*x^n]))*Cosh[a + b*(-(n*Log[x]) + Log[c*x^n])]*Hypergeom etric2F1[1, 1 + 1/(2*b*n), 2 + 1/(2*b*n), -E^(2*(a + b*Log[c*x^n]))] - E^( a/(b*n) + (-(n*Log[x]) + Log[c*x^n])/n)*(1 + 2*b*n)*x*(Cosh[a + b*(-(n*Log [x]) + Log[c*x^n])]*Hypergeometric2F1[1, 1/(2*b*n), 1 + 1/(2*b*n), -E^(2*( a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])))] + Sinh[a + b*(-(n*Log[x]) + Log[c*x^n])])))/(6*b^3*E^((a + b*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*n^3 *(1 + 2*b*n)) - (2*Sech[a + b*(-(n*Log[x]) + Log[c*x^n])]*(E^((2 + 1/(b*n) )*(a + b*Log[c*x^n]))*Cosh[a + b*(-(n*Log[x]) + Log[c*x^n])]*Hypergeometri c2F1[1, 1 + 1/(2*b*n), 2 + 1/(2*b*n), -E^(2*(a + b*Log[c*x^n]))] - E^(a/(b *n) + (-(n*Log[x]) + Log[c*x^n])/n)*(1 + 2*b*n)*x*(Cosh[a + b*(-(n*Log[x]) + Log[c*x^n])]*Hypergeometric2F1[1, 1/(2*b*n), 1 + 1/(2*b*n), -E^(2*(a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])))] + Sinh[a + b*(-(n*Log[x]) + L og[c*x^n])])))/(3*b*E^((a + b*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*n*(1 +...
Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6079, 6081, 795, 888}
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.
| \(\displaystyle \int \text {sech}^4\left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6079 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \text {sech}^4\left (a+b \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 6081 |
| \(\displaystyle \frac {16 e^{-4 a} x \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{-4 b+\frac {1}{n}-1}}{\left (e^{-2 a} \left (c x^n\right )^{-2 b}+1\right )^4}d\left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle \frac {16 e^{-4 a} x \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{4 b+\frac {1}{n}-1}}{\left (\left (c x^n\right )^{2 b}+e^{-2 a}\right )^4}d\left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {16 e^{4 a} x \left (c x^n\right )^{4 b} \operatorname {Hypergeometric2F1}\left (4,\frac {1}{2} \left (4+\frac {1}{b n}\right ),\frac {1}{2} \left (6+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )}{4 b n+1}\) |
(16*E^(4*a)*x*(c*x^n)^(4*b)*Hypergeometric2F1[4, (4 + 1/(b*n))/2, (6 + 1/( b*n))/2, -(E^(2*a)*(c*x^n)^(2*b))])/(1 + 4*b*n)
3.2.85.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[Sech[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> S imp[x/(n*(c*x^n)^(1/n)) Subst[Int[x^(1/n - 1)*Sech[d*(a + b*Log[x])]^p, x ], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1] )
Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Simp[2^p/E^(a*d*p) Int[(e*x)^m*(1/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b *d)))^p)), x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
\[\int {\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}d x\]
\[ \int \text {sech}^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{4} \,d x } \]
\[ \int \text {sech}^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \operatorname {sech}^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
\[ \int \text {sech}^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{4} \,d x } \]
16*(4*b^2*n^2 - 1)*integrate(1/48/(b^3*c^(2*b)*n^3*e^(2*b*log(x^n) + 2*a) + b^3*n^3), x) + 1/3*((2*b*c^(4*b)*n + c^(4*b))*x*e^(4*b*log(x^n) + 4*a) - 2*(6*b^2*c^(2*b)*n^2 - b*c^(2*b)*n - c^(2*b))*x*e^(2*b*log(x^n) + 2*a) - (4*b^2*n^2 - 1)*x)/(b^3*c^(6*b)*n^3*e^(6*b*log(x^n) + 6*a) + 3*b^3*c^(4*b) *n^3*e^(4*b*log(x^n) + 4*a) + 3*b^3*c^(2*b)*n^3*e^(2*b*log(x^n) + 2*a) + b ^3*n^3)
\[ \int \text {sech}^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{4} \,d x } \]
Timed out. \[ \int \text {sech}^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {1}{{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^4} \,d x \]