Integrand size = 13, antiderivative size = 30 \[ \int \frac {\tanh ^5(x)}{a+a \cosh (x)} \, dx=-\frac {\text {sech}(x)}{a}+\frac {\text {sech}^3(x)}{3 a}-\frac {\tanh ^4(x)}{4 a} \] Output:
-sech(x)/a+1/3*sech(x)^3/a-1/4*tanh(x)^4/a
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {\tanh ^5(x)}{a+a \cosh (x)} \, dx=\frac {2 (3+5 \cosh (x)) \text {sech}^4(x) \sinh ^6\left (\frac {x}{2}\right )}{3 a} \] Input:
Integrate[Tanh[x]^5/(a + a*Cosh[x]),x]
Output:
(2*(3 + 5*Cosh[x])*Sech[x]^4*Sinh[x/2]^6)/(3*a)
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 26, 3185, 26, 3042, 26, 3086, 2009, 3087, 15}
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.
\(\displaystyle \int \frac {\tanh ^5(x)}{a \cosh (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\tan \left (-\frac {\pi }{2}+i x\right )^5 \left (a-a \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\left (a-a \sin \left (i x-\frac {\pi }{2}\right )\right ) \tan \left (i x-\frac {\pi }{2}\right )^5}dx\) |
\(\Big \downarrow \) 3185 |
\(\displaystyle i \left (\frac {\int i \text {sech}^2(x) \tanh ^3(x)dx}{a}+\frac {\int -i \text {sech}(x) \tanh ^3(x)dx}{a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {i \int \text {sech}^2(x) \tanh ^3(x)dx}{a}-\frac {i \int \text {sech}(x) \tanh ^3(x)dx}{a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (\frac {i \int i \sec (i x)^2 \tan (i x)^3dx}{a}-\frac {i \int i \sec (i x) \tan (i x)^3dx}{a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {\int \sec (i x) \tan (i x)^3dx}{a}-\frac {\int \sec (i x)^2 \tan (i x)^3dx}{a}\right )\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle i \left (-\frac {i \int \left (\text {sech}^2(x)-1\right )d\text {sech}(x)}{a}-\frac {\int \sec (i x)^2 \tan (i x)^3dx}{a}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (-\frac {\int \sec (i x)^2 \tan (i x)^3dx}{a}-\frac {i \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a}\right )\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle i \left (\frac {i \int -i \tanh ^3(x)d(i \tanh (x))}{a}-\frac {i \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a}\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle i \left (\frac {i \tanh ^4(x)}{4 a}-\frac {i \left (\frac {\text {sech}^3(x)}{3}-\text {sech}(x)\right )}{a}\right )\) |
Input:
Int[Tanh[x]^5/(a + a*Cosh[x]),x]
Output:
I*(((-I)*(-Sech[x] + Sech[x]^3/3))/a + ((I/4)*Tanh[x]^4)/a)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[1/a Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x ] - Simp[1/(b*g) Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /; Fre eQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]
Time = 0.84 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43
method | result | size |
default | \(\frac {-\frac {4}{\left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )^{4}}-\frac {8}{\left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )^{2}}+\frac {32}{3 \left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )^{3}}}{a}\) | \(43\) |
risch | \(-\frac {2 \,{\mathrm e}^{x} \left (3 \,{\mathrm e}^{6 x}-3 \,{\mathrm e}^{5 x}+5 \,{\mathrm e}^{4 x}+5 \,{\mathrm e}^{2 x}-3 \,{\mathrm e}^{x}+3\right )}{3 \left ({\mathrm e}^{2 x}+1\right )^{4} a}\) | \(46\) |
Input:
int(tanh(x)^5/(a+cosh(x)*a),x,method=_RETURNVERBOSE)
Output:
32/a*(-1/8/(tanh(1/2*x)^2+1)^4-1/4/(tanh(1/2*x)^2+1)^2+1/3/(tanh(1/2*x)^2+ 1)^3)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (26) = 52\).
Time = 0.07 (sec) , antiderivative size = 174, normalized size of antiderivative = 5.80 \[ \int \frac {\tanh ^5(x)}{a+a \cosh (x)} \, dx=-\frac {2 \, {\left (3 \, \cosh \left (x\right )^{4} + 3 \, {\left (4 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + 3 \, \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{3} + {\left (18 \, \cosh \left (x\right )^{2} - 9 \, \cosh \left (x\right ) + 8\right )} \sinh \left (x\right )^{2} + 8 \, \cosh \left (x\right )^{2} + {\left (12 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 3\right )} \sinh \left (x\right ) - 3 \, \cosh \left (x\right ) + 5\right )}}{3 \, {\left (a \cosh \left (x\right )^{5} + 5 \, a \cosh \left (x\right ) \sinh \left (x\right )^{4} + a \sinh \left (x\right )^{5} + 5 \, a \cosh \left (x\right )^{3} + {\left (10 \, a \cosh \left (x\right )^{2} + 3 \, a\right )} \sinh \left (x\right )^{3} + 5 \, {\left (2 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 10 \, a \cosh \left (x\right ) + {\left (5 \, a \cosh \left (x\right )^{4} + 9 \, a \cosh \left (x\right )^{2} + 2 \, a\right )} \sinh \left (x\right )\right )}} \] Input:
integrate(tanh(x)^5/(a+a*cosh(x)),x, algorithm="fricas")
Output:
-2/3*(3*cosh(x)^4 + 3*(4*cosh(x) - 1)*sinh(x)^3 + 3*sinh(x)^4 - 3*cosh(x)^ 3 + (18*cosh(x)^2 - 9*cosh(x) + 8)*sinh(x)^2 + 8*cosh(x)^2 + (12*cosh(x)^3 - 9*cosh(x)^2 + 4*cosh(x) + 3)*sinh(x) - 3*cosh(x) + 5)/(a*cosh(x)^5 + 5* a*cosh(x)*sinh(x)^4 + a*sinh(x)^5 + 5*a*cosh(x)^3 + (10*a*cosh(x)^2 + 3*a) *sinh(x)^3 + 5*(2*a*cosh(x)^3 + 3*a*cosh(x))*sinh(x)^2 + 10*a*cosh(x) + (5 *a*cosh(x)^4 + 9*a*cosh(x)^2 + 2*a)*sinh(x))
\[ \int \frac {\tanh ^5(x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\tanh ^{5}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(tanh(x)**5/(a+a*cosh(x)),x)
Output:
Integral(tanh(x)**5/(cosh(x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (26) = 52\).
Time = 0.03 (sec) , antiderivative size = 223, normalized size of antiderivative = 7.43 \[ \int \frac {\tanh ^5(x)}{a+a \cosh (x)} \, dx=-\frac {2 \, e^{\left (-x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} + \frac {2 \, e^{\left (-2 \, x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} - \frac {10 \, e^{\left (-3 \, x\right )}}{3 \, {\left (4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a\right )}} - \frac {10 \, e^{\left (-5 \, x\right )}}{3 \, {\left (4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a\right )}} + \frac {2 \, e^{\left (-6 \, x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} - \frac {2 \, e^{\left (-7 \, x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} \] Input:
integrate(tanh(x)^5/(a+a*cosh(x)),x, algorithm="maxima")
Output:
-2*e^(-x)/(4*a*e^(-2*x) + 6*a*e^(-4*x) + 4*a*e^(-6*x) + a*e^(-8*x) + a) + 2*e^(-2*x)/(4*a*e^(-2*x) + 6*a*e^(-4*x) + 4*a*e^(-6*x) + a*e^(-8*x) + a) - 10/3*e^(-3*x)/(4*a*e^(-2*x) + 6*a*e^(-4*x) + 4*a*e^(-6*x) + a*e^(-8*x) + a) - 10/3*e^(-5*x)/(4*a*e^(-2*x) + 6*a*e^(-4*x) + 4*a*e^(-6*x) + a*e^(-8*x ) + a) + 2*e^(-6*x)/(4*a*e^(-2*x) + 6*a*e^(-4*x) + 4*a*e^(-6*x) + a*e^(-8* x) + a) - 2*e^(-7*x)/(4*a*e^(-2*x) + 6*a*e^(-4*x) + 4*a*e^(-6*x) + a*e^(-8 *x) + a)
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {\tanh ^5(x)}{a+a \cosh (x)} \, dx=-\frac {2 \, {\left (3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4 \, e^{\left (-x\right )} - 4 \, e^{x} + 6\right )}}{3 \, a {\left (e^{\left (-x\right )} + e^{x}\right )}^{4}} \] Input:
integrate(tanh(x)^5/(a+a*cosh(x)),x, algorithm="giac")
Output:
-2/3*(3*(e^(-x) + e^x)^3 - 3*(e^(-x) + e^x)^2 - 4*e^(-x) - 4*e^x + 6)/(a*( e^(-x) + e^x)^4)
Time = 1.93 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.90 \[ \int \frac {\tanh ^5(x)}{a+a \cosh (x)} \, dx=\frac {\frac {8}{a}-\frac {8\,{\mathrm {e}}^x}{3\,a}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {6}{a}-\frac {8\,{\mathrm {e}}^x}{3\,a}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {\frac {2}{a}-\frac {2\,{\mathrm {e}}^x}{a}}{{\mathrm {e}}^{2\,x}+1}-\frac {4}{a\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )} \] Input:
int(tanh(x)^5/(a + a*cosh(x)),x)
Output:
(8/a - (8*exp(x))/(3*a))/(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1) - (6/a - (8*exp(x))/(3*a))/(2*exp(2*x) + exp(4*x) + 1) + (2/a - (2*exp(x))/a)/(exp (2*x) + 1) - 4/(a*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57 \[ \int \frac {\tanh ^5(x)}{a+a \cosh (x)} \, dx=\frac {-3 e^{8 x}-12 e^{7 x}-20 e^{5 x}-18 e^{4 x}-20 e^{3 x}-12 e^{x}-3}{6 a \left (e^{8 x}+4 e^{6 x}+6 e^{4 x}+4 e^{2 x}+1\right )} \] Input:
int(tanh(x)^5/(a+a*cosh(x)),x)
Output:
( - 3*e**(8*x) - 12*e**(7*x) - 20*e**(5*x) - 18*e**(4*x) - 20*e**(3*x) - 1 2*e**x - 3)/(6*a*(e**(8*x) + 4*e**(6*x) + 6*e**(4*x) + 4*e**(2*x) + 1))