\(\int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx\) [151]

Optimal result

Integrand size = 25, antiderivative size = 191 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=\frac {32 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{3 b c \left (1+e^{2 c (a+b x)}\right )^6}-\frac {192 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{5 b c \left (1+e^{2 c (a+b x)}\right )^5}+\frac {48 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^4}-\frac {64 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{3 b c \left (1+e^{2 c (a+b x)}\right )^3} \]

[Out]

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
-64/3*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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6852, 2320, 12, 272, 45} \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=-\frac {64 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{3 b c \left (e^{2 c (a+b x)}+1\right )^3}+\frac {48 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (e^{2 c (a+b x)}+1\right )^4}-\frac {192 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{5 b c \left (e^{2 c (a+b x)}+1\right )^5}+\frac {32 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{3 b c \left (e^{2 c (a+b x)}+1\right )^6} \]

[In]

Int[E^(c*(a + b*x))*(Sech[a*c + b*c*x]^2)^(7/2),x]

[Out]

(32*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(3*b*c*(1 + E^(2*c*(a + b*x)))^6) - (192*Cosh[a*c + b*c*x]*Sq
rt[Sech[a*c + b*c*x]^2])/(5*b*c*(1 + E^(2*c*(a + b*x)))^5) + (48*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/
(b*c*(1 + E^(2*c*(a + b*x)))^4) - (64*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(3*b*c*(1 + E^(2*c*(a + b*x
)))^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

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] && NeQ[b*c - a*d, 0] && 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])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6852

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[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]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rubi steps \begin{align*} \text {integral}& = \left (\cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \int e^{c (a+b x)} \text {sech}^7(a c+b c x) \, dx \\ & = \frac {\left (\cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {128 x^7}{\left (1+x^2\right )^7} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {\left (128 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {x^7}{\left (1+x^2\right )^7} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {\left (64 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {x^3}{(1+x)^7} \, dx,x,e^{2 c (a+b x)}\right )}{b c} \\ & = \frac {\left (64 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \text {Subst}\left (\int \left (-\frac {1}{(1+x)^7}+\frac {3}{(1+x)^6}-\frac {3}{(1+x)^5}+\frac {1}{(1+x)^4}\right ) \, dx,x,e^{2 c (a+b x)}\right )}{b c} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.44 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=-\frac {16 \left (1+6 e^{2 c (a+b x)}+15 e^{4 c (a+b x)}+20 e^{6 c (a+b x)}\right ) \cosh (c (a+b x)) \sqrt {\text {sech}^2(c (a+b x))}}{15 b c \left (1+e^{2 c (a+b x)}\right )^6} \]

[In]

Integrate[E^(c*(a + b*x))*(Sech[a*c + b*c*x]^2)^(7/2),x]

[Out]

(-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)

Maple [C] (warning: unable to verify)

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	\(-\frac {16 \left (20 \,{\mathrm e}^{6 c \left (b x +a \right )}+15 \,{\mathrm e}^{4 c \left (b x +a \right )}+6 \,{\mathrm e}^{2 c \left (b x +a \right )}+1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, {\mathrm e}^{-c \left (b x +a \right )}}{15 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{5}}\)	\(91\)

[In]

int(exp(c*(b*x+a))*(sech(b*c*x+a*c)^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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)))

Fricas [B] (verification not implemented)

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 )}} \]

[In]

integrate(exp(c*(b*x+a))*(sech(b*c*x+a*c)^2)^(7/2),x, algorithm="fricas")

[Out]

-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*cos
h(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
))*sinh(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 + 21*b*c*cosh(b*c*x + a*c) + 3*(12*b*c*cosh(b*c*x + a*c)^7 + 42*b*c*c
osh(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*cos
h(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))

Sympy [F(-1)]

Timed out. \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=\text {Timed out} \]

[In]

integrate(exp(c*(b*x+a))*(sech(b*c*x+a*c)**2)**(7/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

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 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=-\frac {64 \, e^{\left (6 \, b c x + 6 \, a c\right )}}{3 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {16 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {32 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{5 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {16}{15 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]

[In]

integrate(exp(c*(b*x+a))*(sech(b*c*x+a*c)^2)^(7/2),x, algorithm="maxima")

[Out]

-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^(1
2*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))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.34 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=-\frac {16 \, {\left (20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}}{15 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{6}} \]

[In]

integrate(exp(c*(b*x+a))*(sech(b*c*x+a*c)^2)^(7/2),x, algorithm="giac")

[Out]

-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)

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.12 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{7/2} \, dx=\frac {24\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left (2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}+1\right )}{b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^4}-\frac {32\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left (2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}+1\right )}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^3}-\frac {96\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left (2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}+1\right )}{5\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^5}+\frac {16\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left (2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}+1\right )}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^6} \]

[In]

int(exp(c*(a + b*x))*(1/cosh(a*c + b*c*x)^2)^(7/2),x)

[Out]

(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 +
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)^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))*(e
xp(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
)