\(\int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx\) [69]

Optimal result

Integrand size = 13, antiderivative size = 54 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 x}{2 a}+\frac {4 \sinh (x)}{a}-\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh ^2(x) \sinh (x)}{a+a \text {sech}(x)}+\frac {4 \sinh ^3(x)}{3 a} \]

[Out]

-3/2*x/a+4*sinh(x)/a-3/2*cosh(x)*sinh(x)/a-cosh(x)^2*sinh(x)/(a+a*sech(x))+4/3*sinh(x)^3/a

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3904, 3872, 2713, 2715, 8} \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 x}{2 a}+\frac {4 \sinh ^3(x)}{3 a}+\frac {4 \sinh (x)}{a}-\frac {3 \sinh (x) \cosh (x)}{2 a}-\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a} \]

[In]

Int[Cosh[x]^3/(a + a*Sech[x]),x]

[Out]

(-3*x)/(2*a) + (4*Sinh[x])/a - (3*Cosh[x]*Sinh[x])/(2*a) - (Cosh[x]^2*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 ^2(x) \sinh (x)}{a+a \text {sech}(x)}-\frac {\int \cosh ^3(x) (-4 a+3 a \text {sech}(x)) \, dx}{a^2} \\ & = -\frac {\cosh ^2(x) \sinh (x)}{a+a \text {sech}(x)}-\frac {3 \int \cosh ^2(x) \, dx}{a}+\frac {4 \int \cosh ^3(x) \, dx}{a} \\ & = -\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh ^2(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 {3 \int 1 \, dx}{2 a} \\ & = -\frac {3 x}{2 a}+\frac {4 \sinh (x)}{a}-\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh ^2(x) \sinh (x)}{a+a \text {sech}(x)}+\frac {4 \sinh ^3(x)}{3 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\text {sech}\left (\frac {x}{2}\right ) \left (-36 x \cosh \left (\frac {x}{2}\right )+45 \sinh \left (\frac {x}{2}\right )+18 \sinh \left (\frac {3 x}{2}\right )-2 \sinh \left (\frac {5 x}{2}\right )+\sinh \left (\frac {7 x}{2}\right )\right )}{24 a} \]

[In]

Integrate[Cosh[x]^3/(a + a*Sech[x]),x]

[Out]

(Sech[x/2]*(-36*x*Cosh[x/2] + 45*Sinh[x/2] + 18*Sinh[(3*x)/2] - 2*Sinh[(5*x)/2] + Sinh[(7*x)/2]))/(24*a)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.65

method	result	size
parallelrisch	\(\frac {\operatorname {csch}\left (x \right ) \left (-36 x \sinh \left (x \right )-3 \cosh \left (3 x \right )+27 \cosh \left (x \right )+\cosh \left (4 x \right )+20 \cosh \left (2 x \right )-45\right )}{24 a}\)	\(35\)
risch	\(\frac {{\mathrm e}^{4 x}-2 \,{\mathrm e}^{3 x}+18 \,{\mathrm e}^{2 x}-69-18 \,{\mathrm e}^{-x}+2 \,{\mathrm e}^{-2 x}-36 x \,{\mathrm e}^{x}+21 \,{\mathrm e}^{x}-{\mathrm e}^{-3 x}-36 x}{24 \left ({\mathrm e}^{x}+1\right ) a}\)	\(60\)
default	\(\frac {\tanh \left (\frac {x}{2}\right )-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {5}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {5}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}}{a}\)	\(86\)

[In]

int(cosh(x)^3/(a+a*sech(x)),x,method=_RETURNVERBOSE)

[Out]

1/24*csch(x)*(-36*x*sinh(x)-3*cosh(3*x)+27*cosh(x)+cosh(4*x)+20*cosh(2*x)-45)/a

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (48) = 96\).

Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.85 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\cosh \left (x\right )^{4} + {\left (4 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{3} + {\left (6 \, \cosh \left (x\right )^{2} - 9 \, \cosh \left (x\right ) + 20\right )} \sinh \left (x\right )^{2} - 3 \, {\left (12 \, x - 1\right )} \cosh \left (x\right ) + 20 \, \cosh \left (x\right )^{2} + {\left (4 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} - 36 \, x + 32 \, \cosh \left (x\right ) + 39\right )} \sinh \left (x\right ) - 36 \, x - 69}{24 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + a\right )}} \]

[In]

integrate(cosh(x)^3/(a+a*sech(x)),x, algorithm="fricas")

[Out]

1/24*(cosh(x)^4 + (4*cosh(x) - 1)*sinh(x)^3 + sinh(x)^4 - 3*cosh(x)^3 + (6*cosh(x)^2 - 9*cosh(x) + 20)*sinh(x)
^2 - 3*(12*x - 1)*cosh(x) + 20*cosh(x)^2 + (4*cosh(x)^3 - 3*cosh(x)^2 - 36*x + 32*cosh(x) + 39)*sinh(x) - 36*x
- 69)/(a*cosh(x) + a*sinh(x) + a)

Sympy [F]

\[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\cosh ^{3}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(cosh(x)**3/(a+a*sech(x)),x)

[Out]

Integral(cosh(x)**3/(sech(x) + 1), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 \, x}{2 \, a} - \frac {21 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}}{24 \, a} - \frac {2 \, e^{\left (-x\right )} - 18 \, e^{\left (-2 \, x\right )} - 69 \, e^{\left (-3 \, x\right )} - 1}{24 \, {\left (a e^{\left (-3 \, x\right )} + a e^{\left (-4 \, x\right )}\right )}} \]

[In]

integrate(cosh(x)^3/(a+a*sech(x)),x, algorithm="maxima")

[Out]

-3/2*x/a - 1/24*(21*e^(-x) - 3*e^(-2*x) + e^(-3*x))/a - 1/24*(2*e^(-x) - 18*e^(-2*x) - 69*e^(-3*x) - 1)/(a*e^(
-3*x) + a*e^(-4*x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 \, x}{2 \, a} - \frac {{\left (69 \, e^{\left (3 \, x\right )} + 18 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1\right )} e^{\left (-3 \, x\right )}}{24 \, a {\left (e^{x} + 1\right )}} + \frac {a^{2} e^{\left (3 \, x\right )} - 3 \, a^{2} e^{\left (2 \, x\right )} + 21 \, a^{2} e^{x}}{24 \, a^{3}} \]

[In]

integrate(cosh(x)^3/(a+a*sech(x)),x, algorithm="giac")

[Out]

-3/2*x/a - 1/24*(69*e^(3*x) + 18*e^(2*x) - 2*e^x + 1)*e^(-3*x)/(a*(e^x + 1)) + 1/24*(a^2*e^(3*x) - 3*a^2*e^(2*
x) + 21*a^2*e^x)/a^3

Mupad [B] (verification not implemented)

Time = 2.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {7\,{\mathrm {e}}^{-x}}{8\,a}-\frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {3\,x}{2\,a}-\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}+\frac {7\,{\mathrm {e}}^x}{8\,a} \]

[In]

int(cosh(x)^3/(a + a/cosh(x)),x)

[Out]

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