Integrand size = 13, antiderivative size = 54 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 x}{2 a}+\frac {4 \sinh (x)}{a}-\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh ^2(x) \sinh (x)}{a+a \text {sech}(x)}+\frac {4 \sinh ^3(x)}{3 a} \]
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\text {sech}\left (\frac {x}{2}\right ) \left (-36 x \cosh \left (\frac {x}{2}\right )+45 \sinh \left (\frac {x}{2}\right )+18 \sinh \left (\frac {3 x}{2}\right )-2 \sinh \left (\frac {5 x}{2}\right )+\sinh \left (\frac {7 x}{2}\right )\right )}{24 a} \]
(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)
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 4306, 25, 3042, 4274, 3042, 3113, 2009, 3115, 24}
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.
\(\displaystyle \int \frac {\cosh ^3(x)}{a \text {sech}(x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc \left (\frac {\pi }{2}+i x\right )^3 \left (a+a \csc \left (\frac {\pi }{2}+i x\right )\right )}dx\) |
\(\Big \downarrow \) 4306 |
\(\displaystyle -\frac {\int -\cosh ^3(x) (4 a-3 a \text {sech}(x))dx}{a^2}-\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cosh ^3(x) (4 a-3 a \text {sech}(x))dx}{a^2}-\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a}+\frac {\int \frac {4 a-3 a \csc \left (i x+\frac {\pi }{2}\right )}{\csc \left (i x+\frac {\pi }{2}\right )^3}dx}{a^2}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle \frac {4 a \int \cosh ^3(x)dx-3 a \int \cosh ^2(x)dx}{a^2}-\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a}+\frac {4 a \int \sin \left (i x+\frac {\pi }{2}\right )^3dx-3 a \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle -\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a}+\frac {4 i a \int \left (\sinh ^2(x)+1\right )d(-i \sinh (x))-3 a \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a}+\frac {4 i a \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )-3 a \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a}+\frac {-3 a \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )+4 i a \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a}+\frac {-3 a \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )+4 i a \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2}\) |
-((Cosh[x]^2*Sinh[x])/(a + a*Sech[x])) + (-3*a*(x/2 + (Cosh[x]*Sinh[x])/2) + (4*I)*a*((-I)*Sinh[x] - (I/3)*Sinh[x]^3))/a^2
3.1.69.3.1 Defintions of rubi rules used
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
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] - Simp[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 ]
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 | \(\frac {\tanh \left (\frac {x}{2}\right )-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {5}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {5}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}}{a}\) | \(86\) |
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 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\cosh \left (x\right )^{4} + {\left (4 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{3} + {\left (6 \, \cosh \left (x\right )^{2} - 9 \, \cosh \left (x\right ) + 20\right )} \sinh \left (x\right )^{2} - 3 \, {\left (12 \, x - 1\right )} \cosh \left (x\right ) + 20 \, \cosh \left (x\right )^{2} + {\left (4 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} - 36 \, x + 32 \, \cosh \left (x\right ) + 39\right )} \sinh \left (x\right ) - 36 \, x - 69}{24 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + a\right )}} \]
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)
\[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\cosh ^{3}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 \, x}{2 \, a} - \frac {21 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}}{24 \, a} - \frac {2 \, e^{\left (-x\right )} - 18 \, e^{\left (-2 \, x\right )} - 69 \, e^{\left (-3 \, x\right )} - 1}{24 \, {\left (a e^{\left (-3 \, x\right )} + a e^{\left (-4 \, x\right )}\right )}} \]
-3/2*x/a - 1/24*(21*e^(-x) - 3*e^(-2*x) + e^(-3*x))/a - 1/24*(2*e^(-x) - 1 8*e^(-2*x) - 69*e^(-3*x) - 1)/(a*e^(-3*x) + a*e^(-4*x))
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 \, x}{2 \, a} - \frac {{\left (69 \, e^{\left (3 \, x\right )} + 18 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1\right )} e^{\left (-3 \, x\right )}}{24 \, a {\left (e^{x} + 1\right )}} + \frac {a^{2} e^{\left (3 \, x\right )} - 3 \, a^{2} e^{\left (2 \, x\right )} + 21 \, a^{2} e^{x}}{24 \, a^{3}} \]
-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
Time = 2.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {7\,{\mathrm {e}}^{-x}}{8\,a}-\frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {3\,x}{2\,a}-\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}+\frac {7\,{\mathrm {e}}^x}{8\,a} \]