Integrand size = 16, antiderivative size = 66 \[ \int e^{a+b x} \text {sech}^n(a+b x) \, dx=\frac {e^{a+b x} \left (1+e^{2 a+2 b x}\right )^n \operatorname {Hypergeometric2F1}\left (n,\frac {1+n}{2},\frac {3+n}{2},-e^{2 a+2 b x}\right ) \text {sech}^n(a+b x)}{b (1+n)} \] Output:
exp(b*x+a)*(1+exp(2*b*x+2*a))^n*hypergeom([n, 1/2+1/2*n],[3/2+1/2*n],-exp( 2*b*x+2*a))*sech(b*x+a)^n/b/(1+n)
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int e^{a+b x} \text {sech}^n(a+b x) \, dx=\frac {e^{a+b x} \left (1+e^{2 (a+b x)}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {3-n}{2},\frac {3+n}{2},-e^{2 (a+b x)}\right ) \text {sech}^n(a+b x)}{b (1+n)} \] Input:
Integrate[E^(a + b*x)*Sech[a + b*x]^n,x]
Output:
(E^(a + b*x)*(1 + E^(2*(a + b*x)))*Hypergeometric2F1[1, (3 - n)/2, (3 + n) /2, -E^(2*(a + b*x))]*Sech[a + b*x]^n)/(b*(1 + n))
Time = 0.45 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2720, 27, 7270, 278}
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.
\(\displaystyle \int e^{a+b x} \text {sech}^n(a+b x) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int 2^n \left (\frac {e^{a+b x}}{1+e^{2 a+2 b x}}\right )^nde^{a+b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2^n \int \left (\frac {e^{a+b x}}{1+e^{2 a+2 b x}}\right )^nde^{a+b x}}{b}\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle \frac {2^n \left (e^{a+b x}\right )^{-n} \left (\frac {e^{a+b x}}{e^{2 a+2 b x}+1}\right )^n \left (e^{2 a+2 b x}+1\right )^n \int \left (e^{a+b x}\right )^n \left (1+e^{2 a+2 b x}\right )^{-n}de^{a+b x}}{b}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {2^n e^{a+b x} \left (\frac {e^{a+b x}}{e^{2 a+2 b x}+1}\right )^n \left (e^{2 a+2 b x}+1\right )^n \operatorname {Hypergeometric2F1}\left (n,\frac {n+1}{2},\frac {n+3}{2},-e^{2 a+2 b x}\right )}{b (n+1)}\) |
Input:
Int[E^(a + b*x)*Sech[a + b*x]^n,x]
Output:
(2^n*E^(a + b*x)*(E^(a + b*x)/(1 + E^(2*a + 2*b*x)))^n*(1 + E^(2*a + 2*b*x ))^n*Hypergeometric2F1[n, (1 + n)/2, (3 + n)/2, -E^(2*a + 2*b*x)])/(b*(1 + n))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 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[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
\[\int {\mathrm e}^{b x +a} \operatorname {sech}\left (b x +a \right )^{n}d x\]
Input:
int(exp(b*x+a)*sech(b*x+a)^n,x)
Output:
int(exp(b*x+a)*sech(b*x+a)^n,x)
\[ \int e^{a+b x} \text {sech}^n(a+b x) \, dx=\int { \operatorname {sech}\left (b x + a\right )^{n} e^{\left (b x + a\right )} \,d x } \] Input:
integrate(exp(b*x+a)*sech(b*x+a)^n,x, algorithm="fricas")
Output:
integral(sech(b*x + a)^n*e^(b*x + a), x)
\[ \int e^{a+b x} \text {sech}^n(a+b x) \, dx=e^{a} \int e^{b x} \operatorname {sech}^{n}{\left (a + b x \right )}\, dx \] Input:
integrate(exp(b*x+a)*sech(b*x+a)**n,x)
Output:
exp(a)*Integral(exp(b*x)*sech(a + b*x)**n, x)
\[ \int e^{a+b x} \text {sech}^n(a+b x) \, dx=\int { \operatorname {sech}\left (b x + a\right )^{n} e^{\left (b x + a\right )} \,d x } \] Input:
integrate(exp(b*x+a)*sech(b*x+a)^n,x, algorithm="maxima")
Output:
integrate(sech(b*x + a)^n*e^(b*x + a), x)
\[ \int e^{a+b x} \text {sech}^n(a+b x) \, dx=\int { \operatorname {sech}\left (b x + a\right )^{n} e^{\left (b x + a\right )} \,d x } \] Input:
integrate(exp(b*x+a)*sech(b*x+a)^n,x, algorithm="giac")
Output:
integrate(sech(b*x + a)^n*e^(b*x + a), x)
Timed out. \[ \int e^{a+b x} \text {sech}^n(a+b x) \, dx=\int {\mathrm {e}}^{a+b\,x}\,{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^n \,d x \] Input:
int(exp(a + b*x)*(1/cosh(a + b*x))^n,x)
Output:
int(exp(a + b*x)*(1/cosh(a + b*x))^n, x)
\[ \int e^{a+b x} \text {sech}^n(a+b x) \, dx=e^{a} \left (\int e^{b x} \mathrm {sech}\left (b x +a \right )^{n}d x \right ) \] Input:
int(exp(b*x+a)*sech(b*x+a)^n,x)
Output:
e**a*int(e**(b*x)*sech(a + b*x)**n,x)