Integrand size = 25, antiderivative size = 191 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=\frac {32 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{3 b c \left (1+e^{2 c (a+b x)}\right )^6}-\frac {192 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{5 b c \left (1+e^{2 c (a+b x)}\right )^5}+\frac {48 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^4}-\frac {64 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{3 b c \left (1+e^{2 c (a+b x)}\right )^3} \]
32/3*cosh(b*c*x+a*c)*(sech(b*c*x+a*c)^2)^(1/2)/b/c/(1+exp(2*c*(b*x+a)))^6- 192/5*cosh(b*c*x+a*c)*(sech(b*c*x+a*c)^2)^(1/2)/b/c/(1+exp(2*c*(b*x+a)))^5 +48*cosh(b*c*x+a*c)*(sech(b*c*x+a*c)^2)^(1/2)/b/c/(1+exp(2*c*(b*x+a)))^4-6 4/3*cosh(b*c*x+a*c)*(sech(b*c*x+a*c)^2)^(1/2)/b/c/(1+exp(2*c*(b*x+a)))^3
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.44 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=-\frac {16 \left (1+6 e^{2 c (a+b x)}+15 e^{4 c (a+b x)}+20 e^{6 c (a+b x)}\right ) \cosh (c (a+b x)) \sqrt {\text {sech}^2(c (a+b x))}}{15 b c \left (1+e^{2 c (a+b x)}\right )^6} \]
(-16*(1 + 6*E^(2*c*(a + b*x)) + 15*E^(4*c*(a + b*x)) + 20*E^(6*c*(a + b*x) ))*Cosh[c*(a + b*x)]*Sqrt[Sech[c*(a + b*x)]^2])/(15*b*c*(1 + E^(2*c*(a + b *x)))^6)
Time = 0.44 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.55, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {7271, 2720, 27, 243, 53, 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.
\(\displaystyle \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)} \int e^{c (a+b x)} \text {sech}^7(a c+b x c)dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)} \int \frac {128 e^{7 c (a+b x)}}{\left (1+e^{2 c (a+b x)}\right )^7}de^{c (a+b x)}}{b c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {128 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)} \int \frac {e^{7 c (a+b x)}}{\left (1+e^{2 c (a+b x)}\right )^7}de^{c (a+b x)}}{b c}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {64 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)} \int \frac {e^{3 c (a+b x)}}{\left (1+e^{2 c (a+b x)}\right )^7}de^{2 c (a+b x)}}{b c}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {64 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)} \int \left (\frac {1}{\left (1+e^{2 c (a+b x)}\right )^4}-\frac {3}{\left (1+e^{2 c (a+b x)}\right )^5}+\frac {3}{\left (1+e^{2 c (a+b x)}\right )^6}-\frac {1}{\left (1+e^{2 c (a+b x)}\right )^7}\right )de^{2 c (a+b x)}}{b c}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {64 \left (-\frac {1}{3 \left (e^{2 c (a+b x)}+1\right )^3}+\frac {3}{4 \left (e^{2 c (a+b x)}+1\right )^4}-\frac {3}{5 \left (e^{2 c (a+b x)}+1\right )^5}+\frac {1}{6 \left (e^{2 c (a+b x)}+1\right )^6}\right ) \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c}\) |
(64*(1/(6*(1 + E^(2*c*(a + b*x)))^6) - 3/(5*(1 + E^(2*c*(a + b*x)))^5) + 3 /(4*(1 + E^(2*c*(a + b*x)))^4) - 1/(3*(1 + E^(2*c*(a + b*x)))^3))*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(b*c)
3.2.51.3.1 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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 127.39 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.45
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\operatorname {sech}\left (c \left (b x +a \right )\right )\right ) \left (\frac {\tanh \left (c \left (b x +a \right )\right )^{6}}{6}+\frac {\tanh \left (c \left (b x +a \right )\right )^{5}}{5}-\frac {\tanh \left (c \left (b x +a \right )\right )^{4}}{2}-\frac {2 \tanh \left (c \left (b x +a \right )\right )^{3}}{3}+\frac {\tanh \left (c \left (b x +a \right )\right )^{2}}{2}+\tanh \left (c \left (b x +a \right )\right )\right )}{c b}\) | \(86\) |
risch | \(-\frac {16 \left (20 \,{\mathrm e}^{6 c \left (b x +a \right )}+15 \,{\mathrm e}^{4 c \left (b x +a \right )}+6 \,{\mathrm e}^{2 c \left (b x +a \right )}+1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, {\mathrm e}^{-c \left (b x +a \right )}}{15 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{5}}\) | \(91\) |
csgn(sech(c*(b*x+a)))/c/b*(1/6*tanh(c*(b*x+a))^6+1/5*tanh(c*(b*x+a))^5-1/2 *tanh(c*(b*x+a))^4-2/3*tanh(c*(b*x+a))^3+1/2*tanh(c*(b*x+a))^2+tanh(c*(b*x +a)))
Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (173) = 346\).
Time = 0.29 (sec) , antiderivative size = 589, normalized size of antiderivative = 3.08 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=-\frac {16 \, {\left (21 \, \cosh \left (b c x + a c\right )^{3} + 63 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} + 19 \, \sinh \left (b c x + a c\right )^{3} + 3 \, {\left (19 \, \cosh \left (b c x + a c\right )^{2} + 3\right )} \sinh \left (b c x + a c\right ) + 21 \, \cosh \left (b c x + a c\right )\right )}}{15 \, {\left (b c \cosh \left (b c x + a c\right )^{9} + 9 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{8} + b c \sinh \left (b c x + a c\right )^{9} + 6 \, b c \cosh \left (b c x + a c\right )^{7} + 6 \, {\left (6 \, b c \cosh \left (b c x + a c\right )^{2} + b c\right )} \sinh \left (b c x + a c\right )^{7} + 15 \, b c \cosh \left (b c x + a c\right )^{5} + 42 \, {\left (2 \, b c \cosh \left (b c x + a c\right )^{3} + b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{6} + 3 \, {\left (42 \, b c \cosh \left (b c x + a c\right )^{4} + 42 \, b c \cosh \left (b c x + a c\right )^{2} + 5 \, b c\right )} \sinh \left (b c x + a c\right )^{5} + 21 \, b c \cosh \left (b c x + a c\right )^{3} + 3 \, {\left (42 \, b c \cosh \left (b c x + a c\right )^{5} + 70 \, b c \cosh \left (b c x + a c\right )^{3} + 25 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{4} + {\left (84 \, b c \cosh \left (b c x + a c\right )^{6} + 210 \, b c \cosh \left (b c x + a c\right )^{4} + 150 \, b c \cosh \left (b c x + a c\right )^{2} + 19 \, b c\right )} \sinh \left (b c x + a c\right )^{3} + 21 \, b c \cosh \left (b c x + a c\right ) + 3 \, {\left (12 \, b c \cosh \left (b c x + a c\right )^{7} + 42 \, b c \cosh \left (b c x + a c\right )^{5} + 50 \, b c \cosh \left (b c x + a c\right )^{3} + 21 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} + 3 \, {\left (3 \, b c \cosh \left (b c x + a c\right )^{8} + 14 \, b c \cosh \left (b c x + a c\right )^{6} + 25 \, b c \cosh \left (b c x + a c\right )^{4} + 19 \, b c \cosh \left (b c x + a c\right )^{2} + 3 \, b c\right )} \sinh \left (b c x + a c\right )\right )}} \]
-16/15*(21*cosh(b*c*x + a*c)^3 + 63*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^2 + 19*sinh(b*c*x + a*c)^3 + 3*(19*cosh(b*c*x + a*c)^2 + 3)*sinh(b*c*x + a*c ) + 21*cosh(b*c*x + a*c))/(b*c*cosh(b*c*x + a*c)^9 + 9*b*c*cosh(b*c*x + a* c)*sinh(b*c*x + a*c)^8 + b*c*sinh(b*c*x + a*c)^9 + 6*b*c*cosh(b*c*x + a*c) ^7 + 6*(6*b*c*cosh(b*c*x + a*c)^2 + b*c)*sinh(b*c*x + a*c)^7 + 15*b*c*cosh (b*c*x + a*c)^5 + 42*(2*b*c*cosh(b*c*x + a*c)^3 + b*c*cosh(b*c*x + a*c))*s inh(b*c*x + a*c)^6 + 3*(42*b*c*cosh(b*c*x + a*c)^4 + 42*b*c*cosh(b*c*x + a *c)^2 + 5*b*c)*sinh(b*c*x + a*c)^5 + 21*b*c*cosh(b*c*x + a*c)^3 + 3*(42*b* c*cosh(b*c*x + a*c)^5 + 70*b*c*cosh(b*c*x + a*c)^3 + 25*b*c*cosh(b*c*x + a *c))*sinh(b*c*x + a*c)^4 + (84*b*c*cosh(b*c*x + a*c)^6 + 210*b*c*cosh(b*c* x + a*c)^4 + 150*b*c*cosh(b*c*x + a*c)^2 + 19*b*c)*sinh(b*c*x + a*c)^3 + 2 1*b*c*cosh(b*c*x + a*c) + 3*(12*b*c*cosh(b*c*x + a*c)^7 + 42*b*c*cosh(b*c* x + a*c)^5 + 50*b*c*cosh(b*c*x + a*c)^3 + 21*b*c*cosh(b*c*x + a*c))*sinh(b *c*x + a*c)^2 + 3*(3*b*c*cosh(b*c*x + a*c)^8 + 14*b*c*cosh(b*c*x + a*c)^6 + 25*b*c*cosh(b*c*x + a*c)^4 + 19*b*c*cosh(b*c*x + a*c)^2 + 3*b*c)*sinh(b* c*x + a*c))
Timed out. \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (173) = 346\).
Time = 0.21 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.02 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=-\frac {64 \, e^{\left (6 \, b c x + 6 \, a c\right )}}{3 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {16 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {32 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{5 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {16}{15 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
-64/3*e^(6*b*c*x + 6*a*c)/(b*c*(e^(12*b*c*x + 12*a*c) + 6*e^(10*b*c*x + 10 *a*c) + 15*e^(8*b*c*x + 8*a*c) + 20*e^(6*b*c*x + 6*a*c) + 15*e^(4*b*c*x + 4*a*c) + 6*e^(2*b*c*x + 2*a*c) + 1)) - 16*e^(4*b*c*x + 4*a*c)/(b*c*(e^(12* b*c*x + 12*a*c) + 6*e^(10*b*c*x + 10*a*c) + 15*e^(8*b*c*x + 8*a*c) + 20*e^ (6*b*c*x + 6*a*c) + 15*e^(4*b*c*x + 4*a*c) + 6*e^(2*b*c*x + 2*a*c) + 1)) - 32/5*e^(2*b*c*x + 2*a*c)/(b*c*(e^(12*b*c*x + 12*a*c) + 6*e^(10*b*c*x + 10 *a*c) + 15*e^(8*b*c*x + 8*a*c) + 20*e^(6*b*c*x + 6*a*c) + 15*e^(4*b*c*x + 4*a*c) + 6*e^(2*b*c*x + 2*a*c) + 1)) - 16/15/(b*c*(e^(12*b*c*x + 12*a*c) + 6*e^(10*b*c*x + 10*a*c) + 15*e^(8*b*c*x + 8*a*c) + 20*e^(6*b*c*x + 6*a*c) + 15*e^(4*b*c*x + 4*a*c) + 6*e^(2*b*c*x + 2*a*c) + 1))
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.34 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=-\frac {16 \, {\left (20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}}{15 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{6}} \]
-16/15*(20*e^(6*b*c*x + 6*a*c) + 15*e^(4*b*c*x + 4*a*c) + 6*e^(2*b*c*x + 2 *a*c) + 1)/(b*c*(e^(2*b*c*x + 2*a*c) + 1)^6)
Time = 0.15 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.12 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=\frac {24\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left (2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}+1\right )}{b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^4}-\frac {32\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left (2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}+1\right )}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^3}-\frac {96\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left (2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}+1\right )}{5\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^5}+\frac {16\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left (2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}+1\right )}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^6} \]
(24*(1/(exp(a*c + b*c*x)/2 + exp(- a*c - b*c*x)/2)^2)^(1/2)*(2*exp(2*a*c + 2*b*c*x) + exp(4*a*c + 4*b*c*x) + 1))/(b*c*(exp(a*c + b*c*x) + exp(3*a*c + 3*b*c*x))*(exp(2*a*c + 2*b*c*x) + 1)^4) - (32*(1/(exp(a*c + b*c*x)/2 + e xp(- a*c - b*c*x)/2)^2)^(1/2)*(2*exp(2*a*c + 2*b*c*x) + exp(4*a*c + 4*b*c* x) + 1))/(3*b*c*(exp(a*c + b*c*x) + exp(3*a*c + 3*b*c*x))*(exp(2*a*c + 2*b *c*x) + 1)^3) - (96*(1/(exp(a*c + b*c*x)/2 + exp(- a*c - b*c*x)/2)^2)^(1/2 )*(2*exp(2*a*c + 2*b*c*x) + exp(4*a*c + 4*b*c*x) + 1))/(5*b*c*(exp(a*c + b *c*x) + exp(3*a*c + 3*b*c*x))*(exp(2*a*c + 2*b*c*x) + 1)^5) + (16*(1/(exp( a*c + b*c*x)/2 + exp(- a*c - b*c*x)/2)^2)^(1/2)*(2*exp(2*a*c + 2*b*c*x) + exp(4*a*c + 4*b*c*x) + 1))/(3*b*c*(exp(a*c + b*c*x) + exp(3*a*c + 3*b*c*x) )*(exp(2*a*c + 2*b*c*x) + 1)^6)