\(\int \text {sech}^6(a+b x) \, dx\) [6]

Optimal result

Integrand size = 8, antiderivative size = 41 \[ \int \text {sech}^6(a+b x) \, dx=\frac {\tanh (a+b x)}{b}-\frac {2 \tanh ^3(a+b x)}{3 b}+\frac {\tanh ^5(a+b x)}{5 b} \]

[Out]

tanh(b*x+a)/b-2/3*tanh(b*x+a)^3/b+1/5*tanh(b*x+a)^5/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3852} \[ \int \text {sech}^6(a+b x) \, dx=\frac {\tanh ^5(a+b x)}{5 b}-\frac {2 \tanh ^3(a+b x)}{3 b}+\frac {\tanh (a+b x)}{b} \]

[In]

Int[Sech[a + b*x]^6,x]

[Out]

Tanh[a + b*x]/b - (2*Tanh[a + b*x]^3)/(3*b) + Tanh[a + b*x]^5/(5*b)

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {i \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (a+b x)\right )}{b} \\ & = \frac {\tanh (a+b x)}{b}-\frac {2 \tanh ^3(a+b x)}{3 b}+\frac {\tanh ^5(a+b x)}{5 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \text {sech}^6(a+b x) \, dx=\frac {\tanh (a+b x)}{b}-\frac {2 \tanh ^3(a+b x)}{3 b}+\frac {\tanh ^5(a+b x)}{5 b} \]

[In]

Integrate[Sech[a + b*x]^6,x]

[Out]

Tanh[a + b*x]/b - (2*Tanh[a + b*x]^3)/(3*b) + Tanh[a + b*x]^5/(5*b)

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (b x +a \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (b x +a \right )^{2}}{15}\right ) \tanh \left (b x +a \right )}{b}\)	\(33\)
default	\(\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (b x +a \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (b x +a \right )^{2}}{15}\right ) \tanh \left (b x +a \right )}{b}\)	\(33\)
risch	\(-\frac {16 \left (10 \,{\mathrm e}^{4 b x +4 a}+5 \,{\mathrm e}^{2 b x +2 a}+1\right )}{15 b \left (1+{\mathrm e}^{2 b x +2 a}\right )^{5}}\)	\(43\)
parallelrisch	\(\frac {\frac {8 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{3}+\frac {8 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}}{3}+2 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{9}+\frac {116 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{15}+2 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )}{b \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )^{5}}\)	\(85\)

[In]

int(sech(b*x+a)^6,x,method=_RETURNVERBOSE)

[Out]

1/b*(8/15+1/5*sech(b*x+a)^4+4/15*sech(b*x+a)^2)*tanh(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (37) = 74\).

Time = 0.26 (sec) , antiderivative size = 344, normalized size of antiderivative = 8.39 \[ \int \text {sech}^6(a+b x) \, dx=-\frac {16 \, {\left (11 \, \cosh \left (b x + a\right )^{2} + 18 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 11 \, \sinh \left (b x + a\right )^{2} + 5\right )}}{15 \, {\left (b \cosh \left (b x + a\right )^{8} + 8 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + b \sinh \left (b x + a\right )^{8} + 5 \, b \cosh \left (b x + a\right )^{6} + {\left (28 \, b \cosh \left (b x + a\right )^{2} + 5 \, b\right )} \sinh \left (b x + a\right )^{6} + 2 \, {\left (28 \, b \cosh \left (b x + a\right )^{3} + 15 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 10 \, b \cosh \left (b x + a\right )^{4} + 5 \, {\left (14 \, b \cosh \left (b x + a\right )^{4} + 15 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (14 \, b \cosh \left (b x + a\right )^{5} + 25 \, b \cosh \left (b x + a\right )^{3} + 10 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 11 \, b \cosh \left (b x + a\right )^{2} + {\left (28 \, b \cosh \left (b x + a\right )^{6} + 75 \, b \cosh \left (b x + a\right )^{4} + 60 \, b \cosh \left (b x + a\right )^{2} + 11 \, b\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (4 \, b \cosh \left (b x + a\right )^{7} + 15 \, b \cosh \left (b x + a\right )^{5} + 20 \, b \cosh \left (b x + a\right )^{3} + 9 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 5 \, b\right )}} \]

[In]

integrate(sech(b*x+a)^6,x, algorithm="fricas")

[Out]

-16/15*(11*cosh(b*x + a)^2 + 18*cosh(b*x + a)*sinh(b*x + a) + 11*sinh(b*x + a)^2 + 5)/(b*cosh(b*x + a)^8 + 8*b
*cosh(b*x + a)*sinh(b*x + a)^7 + b*sinh(b*x + a)^8 + 5*b*cosh(b*x + a)^6 + (28*b*cosh(b*x + a)^2 + 5*b)*sinh(b
*x + a)^6 + 2*(28*b*cosh(b*x + a)^3 + 15*b*cosh(b*x + a))*sinh(b*x + a)^5 + 10*b*cosh(b*x + a)^4 + 5*(14*b*cos
h(b*x + a)^4 + 15*b*cosh(b*x + a)^2 + 2*b)*sinh(b*x + a)^4 + 4*(14*b*cosh(b*x + a)^5 + 25*b*cosh(b*x + a)^3 +
10*b*cosh(b*x + a))*sinh(b*x + a)^3 + 11*b*cosh(b*x + a)^2 + (28*b*cosh(b*x + a)^6 + 75*b*cosh(b*x + a)^4 + 60
*b*cosh(b*x + a)^2 + 11*b)*sinh(b*x + a)^2 + 2*(4*b*cosh(b*x + a)^7 + 15*b*cosh(b*x + a)^5 + 20*b*cosh(b*x + a
)^3 + 9*b*cosh(b*x + a))*sinh(b*x + a) + 5*b)

Sympy [F]

\[ \int \text {sech}^6(a+b x) \, dx=\int \operatorname {sech}^{6}{\left (a + b x \right )}\, dx \]

[In]

integrate(sech(b*x+a)**6,x)

[Out]

Integral(sech(a + b*x)**6, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (37) = 74\).

Time = 0.19 (sec) , antiderivative size = 205, normalized size of antiderivative = 5.00 \[ \int \text {sech}^6(a+b x) \, dx=\frac {16 \, e^{\left (-2 \, b x - 2 \, a\right )}}{3 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} + \frac {32 \, e^{\left (-4 \, b x - 4 \, a\right )}}{3 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} + \frac {16}{15 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} \]

[In]

integrate(sech(b*x+a)^6,x, algorithm="maxima")

[Out]

16/3*e^(-2*b*x - 2*a)/(b*(5*e^(-2*b*x - 2*a) + 10*e^(-4*b*x - 4*a) + 10*e^(-6*b*x - 6*a) + 5*e^(-8*b*x - 8*a)
+ e^(-10*b*x - 10*a) + 1)) + 32/3*e^(-4*b*x - 4*a)/(b*(5*e^(-2*b*x - 2*a) + 10*e^(-4*b*x - 4*a) + 10*e^(-6*b*x
- 6*a) + 5*e^(-8*b*x - 8*a) + e^(-10*b*x - 10*a) + 1)) + 16/15/(b*(5*e^(-2*b*x - 2*a) + 10*e^(-4*b*x - 4*a) +
10*e^(-6*b*x - 6*a) + 5*e^(-8*b*x - 8*a) + e^(-10*b*x - 10*a) + 1))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \text {sech}^6(a+b x) \, dx=-\frac {16 \, {\left (10 \, e^{\left (4 \, b x + 4 \, a\right )} + 5 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}{15 \, b {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{5}} \]

[In]

integrate(sech(b*x+a)^6,x, algorithm="giac")

[Out]

-16/15*(10*e^(4*b*x + 4*a) + 5*e^(2*b*x + 2*a) + 1)/(b*(e^(2*b*x + 2*a) + 1)^5)

Mupad [B] (verification not implemented)

Time = 2.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \text {sech}^6(a+b x) \, dx=-\frac {16\,\left (5\,{\mathrm {e}}^{2\,a+2\,b\,x}+10\,{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}{15\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}^5} \]

[In]

int(1/cosh(a + b*x)^6,x)

[Out]

-(16*(5*exp(2*a + 2*b*x) + 10*exp(4*a + 4*b*x) + 1))/(15*b*(exp(2*a + 2*b*x) + 1)^5)