\(\int \frac {\cosh ^4(x)}{a+a \text {sech}(x)} \, dx\) [68]

Optimal result

Integrand size = 13, antiderivative size = 67 \[ \int \frac {\cosh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {15 x}{8 a}-\frac {4 \sinh (x)}{a}+\frac {15 \cosh (x) \sinh (x)}{8 a}+\frac {5 \cosh ^3(x) \sinh (x)}{4 a}-\frac {\cosh ^3(x) \sinh (x)}{a+a \text {sech}(x)}-\frac {4 \sinh ^3(x)}{3 a} \]

[Out]

15/8*x/a-4*sinh(x)/a+15/8*cosh(x)*sinh(x)/a+5/4*cosh(x)^3*sinh(x)/a-cosh(x)^3*sinh(x)/(a+a*sech(x))-4/3*sinh(x
)^3/a

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3904, 3872, 2715, 8, 2713} \[ \int \frac {\cosh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {15 x}{8 a}-\frac {4 \sinh ^3(x)}{3 a}-\frac {4 \sinh (x)}{a}+\frac {5 \sinh (x) \cosh ^3(x)}{4 a}+\frac {15 \sinh (x) \cosh (x)}{8 a}-\frac {\sinh (x) \cosh ^3(x)}{a \text {sech}(x)+a} \]

[In]

Int[Cosh[x]^4/(a + a*Sech[x]),x]

[Out]

(15*x)/(8*a) - (4*Sinh[x])/a + (15*Cosh[x]*Sinh[x])/(8*a) + (5*Cosh[x]^3*Sinh[x])/(4*a) - (Cosh[x]^3*Sinh[x])/
(a + a*Sech[x]) - (4*Sinh[x]^3)/(3*a)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3904

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[Cot[e + f*x
]*((d*Csc[e + f*x])^n/(f*(a + b*Csc[e + f*x]))), x] - Dist[1/a^2, Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[
e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh ^3(x) \sinh (x)}{a+a \text {sech}(x)}-\frac {\int \cosh ^4(x) (-5 a+4 a \text {sech}(x)) \, dx}{a^2} \\ & = -\frac {\cosh ^3(x) \sinh (x)}{a+a \text {sech}(x)}-\frac {4 \int \cosh ^3(x) \, dx}{a}+\frac {5 \int \cosh ^4(x) \, dx}{a} \\ & = \frac {5 \cosh ^3(x) \sinh (x)}{4 a}-\frac {\cosh ^3(x) \sinh (x)}{a+a \text {sech}(x)}-\frac {(4 i) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (x)\right )}{a}+\frac {15 \int \cosh ^2(x) \, dx}{4 a} \\ & = -\frac {4 \sinh (x)}{a}+\frac {15 \cosh (x) \sinh (x)}{8 a}+\frac {5 \cosh ^3(x) \sinh (x)}{4 a}-\frac {\cosh ^3(x) \sinh (x)}{a+a \text {sech}(x)}-\frac {4 \sinh ^3(x)}{3 a}+\frac {15 \int 1 \, dx}{8 a} \\ & = \frac {15 x}{8 a}-\frac {4 \sinh (x)}{a}+\frac {15 \cosh (x) \sinh (x)}{8 a}+\frac {5 \cosh ^3(x) \sinh (x)}{4 a}-\frac {\cosh ^3(x) \sinh (x)}{a+a \text {sech}(x)}-\frac {4 \sinh ^3(x)}{3 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \frac {\cosh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\text {sech}\left (\frac {x}{2}\right ) \left (360 x \cosh \left (\frac {x}{2}\right )-360 \sinh \left (\frac {x}{2}\right )-120 \sinh \left (\frac {3 x}{2}\right )+40 \sinh \left (\frac {5 x}{2}\right )-5 \sinh \left (\frac {7 x}{2}\right )+3 \sinh \left (\frac {9 x}{2}\right )\right )}{192 a} \]

[In]

Integrate[Cosh[x]^4/(a + a*Sech[x]),x]

[Out]

(Sech[x/2]*(360*x*Cosh[x/2] - 360*Sinh[x/2] - 120*Sinh[(3*x)/2] + 40*Sinh[(5*x)/2] - 5*Sinh[(7*x)/2] + 3*Sinh[
(9*x)/2]))/(192*a)

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64

method	result	size
parallelrisch	\(-\frac {\operatorname {csch}\left (x \right ) \left (-360 x \sinh \left (x \right )+240 \cosh \left (x \right )+8 \cosh \left (4 x \right )+160 \cosh \left (2 x \right )-3 \cosh \left (5 x \right )-45 \cosh \left (3 x \right )-360\right )}{192 a}\)	\(43\)
risch	\(\frac {3 \,{\mathrm e}^{5 x}-5 \,{\mathrm e}^{4 x}+40 \,{\mathrm e}^{3 x}-120 \,{\mathrm e}^{2 x}+552+120 \,{\mathrm e}^{-x}-40 \,{\mathrm e}^{-2 x}+5 \,{\mathrm e}^{-3 x}+360 x \,{\mathrm e}^{x}-168 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{-4 x}+360 x}{192 \left ({\mathrm e}^{x}+1\right ) a}\)	\(74\)
default	\(\frac {-\tanh \left (\frac {x}{2}\right )-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {5}{6 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {15}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {25}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {15 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {5}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {15}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {25}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {15 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}}{a}\)	\(110\)

[In]

int(cosh(x)^4/(a+a*sech(x)),x,method=_RETURNVERBOSE)

[Out]

-1/192*csch(x)*(-360*x*sinh(x)+240*cosh(x)+8*cosh(4*x)+160*cosh(2*x)-3*cosh(5*x)-45*cosh(3*x)-360)/a

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (59) = 118\).

Time = 0.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.07 \[ \int \frac {\cosh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {3 \, \cosh \left (x\right )^{5} + {\left (15 \, \cosh \left (x\right ) - 8\right )} \sinh \left (x\right )^{4} + 3 \, \sinh \left (x\right )^{5} - 8 \, \cosh \left (x\right )^{4} + {\left (30 \, \cosh \left (x\right )^{2} - 8 \, \cosh \left (x\right ) + 35\right )} \sinh \left (x\right )^{3} + 45 \, \cosh \left (x\right )^{3} + {\left (30 \, \cosh \left (x\right )^{3} - 48 \, \cosh \left (x\right )^{2} + 135 \, \cosh \left (x\right ) - 160\right )} \sinh \left (x\right )^{2} + 24 \, {\left (15 \, x - 2\right )} \cosh \left (x\right ) - 160 \, \cosh \left (x\right )^{2} + {\left (15 \, \cosh \left (x\right )^{4} - 8 \, \cosh \left (x\right )^{3} + 105 \, \cosh \left (x\right )^{2} + 360 \, x - 160 \, \cosh \left (x\right ) - 288\right )} \sinh \left (x\right ) + 360 \, x + 552}{192 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + a\right )}} \]

[In]

integrate(cosh(x)^4/(a+a*sech(x)),x, algorithm="fricas")

[Out]

1/192*(3*cosh(x)^5 + (15*cosh(x) - 8)*sinh(x)^4 + 3*sinh(x)^5 - 8*cosh(x)^4 + (30*cosh(x)^2 - 8*cosh(x) + 35)*
sinh(x)^3 + 45*cosh(x)^3 + (30*cosh(x)^3 - 48*cosh(x)^2 + 135*cosh(x) - 160)*sinh(x)^2 + 24*(15*x - 2)*cosh(x)
- 160*cosh(x)^2 + (15*cosh(x)^4 - 8*cosh(x)^3 + 105*cosh(x)^2 + 360*x - 160*cosh(x) - 288)*sinh(x) + 360*x +
552)/(a*cosh(x) + a*sinh(x) + a)

Sympy [F]

\[ \int \frac {\cosh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\cosh ^{4}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(cosh(x)**4/(a+a*sech(x)),x)

[Out]

Integral(cosh(x)**4/(sech(x) + 1), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.19 \[ \int \frac {\cosh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {15 \, x}{8 \, a} + \frac {168 \, e^{\left (-x\right )} - 48 \, e^{\left (-2 \, x\right )} + 8 \, e^{\left (-3 \, x\right )} - 3 \, e^{\left (-4 \, x\right )}}{192 \, a} - \frac {5 \, e^{\left (-x\right )} - 40 \, e^{\left (-2 \, x\right )} + 120 \, e^{\left (-3 \, x\right )} + 552 \, e^{\left (-4 \, x\right )} - 3}{192 \, {\left (a e^{\left (-4 \, x\right )} + a e^{\left (-5 \, x\right )}\right )}} \]

[In]

integrate(cosh(x)^4/(a+a*sech(x)),x, algorithm="maxima")

[Out]

15/8*x/a + 1/192*(168*e^(-x) - 48*e^(-2*x) + 8*e^(-3*x) - 3*e^(-4*x))/a - 1/192*(5*e^(-x) - 40*e^(-2*x) + 120*
e^(-3*x) + 552*e^(-4*x) - 3)/(a*e^(-4*x) + a*e^(-5*x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.28 \[ \int \frac {\cosh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {15 \, x}{8 \, a} + \frac {{\left (552 \, e^{\left (4 \, x\right )} + 120 \, e^{\left (3 \, x\right )} - 40 \, e^{\left (2 \, x\right )} + 5 \, e^{x} - 3\right )} e^{\left (-4 \, x\right )}}{192 \, a {\left (e^{x} + 1\right )}} + \frac {3 \, a^{3} e^{\left (4 \, x\right )} - 8 \, a^{3} e^{\left (3 \, x\right )} + 48 \, a^{3} e^{\left (2 \, x\right )} - 168 \, a^{3} e^{x}}{192 \, a^{4}} \]

[In]

integrate(cosh(x)^4/(a+a*sech(x)),x, algorithm="giac")

[Out]

15/8*x/a + 1/192*(552*e^(4*x) + 120*e^(3*x) - 40*e^(2*x) + 5*e^x - 3)*e^(-4*x)/(a*(e^x + 1)) + 1/192*(3*a^3*e^
(4*x) - 8*a^3*e^(3*x) + 48*a^3*e^(2*x) - 168*a^3*e^x)/a^4

Mupad [B] (verification not implemented)

Time = 2.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int \frac {\cosh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {7\,{\mathrm {e}}^{-x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{4\,a}+\frac {{\mathrm {e}}^{2\,x}}{4\,a}+\frac {{\mathrm {e}}^{-3\,x}}{24\,a}-\frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a}+\frac {{\mathrm {e}}^{4\,x}}{64\,a}+\frac {15\,x}{8\,a}+\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {7\,{\mathrm {e}}^x}{8\,a} \]

[In]

int(cosh(x)^4/(a + a/cosh(x)),x)

[Out]

(7*exp(-x))/(8*a) - exp(-2*x)/(4*a) + exp(2*x)/(4*a) + exp(-3*x)/(24*a) - exp(3*x)/(24*a) - exp(-4*x)/(64*a) +
exp(4*x)/(64*a) + (15*x)/(8*a) + 2/(a*(exp(x) + 1)) - (7*exp(x))/(8*a)