Integrand size = 13, antiderivative size = 34 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\coth ^3(x)}{3 a}-\frac {\coth ^5(x)}{5 a}+\frac {\text {csch}^5(x)}{5 a} \]
Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {(-15-6 \cosh (x)-2 \cosh (2 x)+2 \cosh (3 x)+\cosh (4 x)) \text {csch}^3(x)}{60 a (1+\cosh (x))} \]
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.18, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 4360, 25, 25, 3042, 25, 3318, 25, 3042, 25, 3086, 15, 3087, 244, 2009}
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.
\(\displaystyle \int \frac {\text {csch}^4(x)}{a \text {sech}(x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos \left (-\frac {\pi }{2}+i x\right )^4 \left (a-a \csc \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\frac {\coth (x) \text {csch}^3(x)}{a (-\cosh (x))-a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\coth (x) \text {csch}^3(x)}{\cosh (x) a+a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\coth (x) \text {csch}^3(x)}{a \cosh (x)+a}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sin \left (-\frac {\pi }{2}+i x\right )}{\cos \left (-\frac {\pi }{2}+i x\right )^4 \left (a-a \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\sin \left (i x-\frac {\pi }{2}\right )}{\cos \left (i x-\frac {\pi }{2}\right )^4 \left (a-a \sin \left (i x-\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 3318 |
\(\displaystyle -\frac {\int -\coth ^2(x) \text {csch}^4(x)dx}{a}-\frac {\int \coth (x) \text {csch}^5(x)dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \coth ^2(x) \text {csch}^4(x)dx}{a}-\frac {\int \coth (x) \text {csch}^5(x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\sec \left (i x-\frac {\pi }{2}\right )^4 \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}-\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^5 \tan \left (i x-\frac {\pi }{2}\right )dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^5 \tan \left (i x-\frac {\pi }{2}\right )dx}{a}-\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^4 \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \frac {i \int \text {csch}^4(x)d(-i \text {csch}(x))}{a}-\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^4 \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\text {csch}^5(x)}{5 a}-\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^4 \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle \frac {\text {csch}^5(x)}{5 a}+\frac {i \int -\coth ^2(x) \left (1-\coth ^2(x)\right )d(i \coth (x))}{a}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\text {csch}^5(x)}{5 a}+\frac {i \int \left (\coth ^4(x)-\coth ^2(x)\right )d(i \coth (x))}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\text {csch}^5(x)}{5 a}+\frac {i \left (\frac {1}{5} i \coth ^5(x)-\frac {1}{3} i \coth ^3(x)\right )}{a}\) |
3.1.59.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 0.89 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{5}}{5}+\frac {2 \tanh \left (\frac {x}{2}\right )^{3}}{3}+\frac {2}{\tanh \left (\frac {x}{2}\right )}-\frac {1}{3 \tanh \left (\frac {x}{2}\right )^{3}}}{16 a}\) | \(39\) |
risch | \(-\frac {4 \left (15 \,{\mathrm e}^{4 x}+6 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x}-1\right )}{15 \left ({\mathrm e}^{x}+1\right )^{5} a \left ({\mathrm e}^{x}-1\right )^{3}}\) | \(42\) |
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (28) = 56\).
Time = 0.23 (sec) , antiderivative size = 219, normalized size of antiderivative = 6.44 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=-\frac {8 \, {\left (7 \, \cosh \left (x\right )^{2} + 4 \, {\left (4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 7 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )}}{15 \, {\left (a \cosh \left (x\right )^{6} + a \sinh \left (x\right )^{6} + 2 \, a \cosh \left (x\right )^{5} + 2 \, {\left (3 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{5} - 2 \, a \cosh \left (x\right )^{4} + {\left (15 \, a \cosh \left (x\right )^{2} + 10 \, a \cosh \left (x\right ) - 2 \, a\right )} \sinh \left (x\right )^{4} - 6 \, a \cosh \left (x\right )^{3} + 2 \, {\left (10 \, a \cosh \left (x\right )^{3} + 10 \, a \cosh \left (x\right )^{2} - 4 \, a \cosh \left (x\right ) - 3 \, a\right )} \sinh \left (x\right )^{3} - a \cosh \left (x\right )^{2} + {\left (15 \, a \cosh \left (x\right )^{4} + 20 \, a \cosh \left (x\right )^{3} - 12 \, a \cosh \left (x\right )^{2} - 18 \, a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + 2 \, {\left (3 \, a \cosh \left (x\right )^{5} + 5 \, a \cosh \left (x\right )^{4} - 4 \, a \cosh \left (x\right )^{3} - 9 \, a \cosh \left (x\right )^{2} + a \cosh \left (x\right ) + 4 \, a\right )} \sinh \left (x\right ) + 2 \, a\right )}} \]
-8/15*(7*cosh(x)^2 + 4*(4*cosh(x) + 1)*sinh(x) + 7*sinh(x)^2 + 2*cosh(x) + 1)/(a*cosh(x)^6 + a*sinh(x)^6 + 2*a*cosh(x)^5 + 2*(3*a*cosh(x) + a)*sinh( x)^5 - 2*a*cosh(x)^4 + (15*a*cosh(x)^2 + 10*a*cosh(x) - 2*a)*sinh(x)^4 - 6 *a*cosh(x)^3 + 2*(10*a*cosh(x)^3 + 10*a*cosh(x)^2 - 4*a*cosh(x) - 3*a)*sin h(x)^3 - a*cosh(x)^2 + (15*a*cosh(x)^4 + 20*a*cosh(x)^3 - 12*a*cosh(x)^2 - 18*a*cosh(x) - a)*sinh(x)^2 + 4*a*cosh(x) + 2*(3*a*cosh(x)^5 + 5*a*cosh(x )^4 - 4*a*cosh(x)^3 - 9*a*cosh(x)^2 + a*cosh(x) + 4*a)*sinh(x) + 2*a)
\[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\operatorname {csch}^{4}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (28) = 56\).
Time = 0.19 (sec) , antiderivative size = 292, normalized size of antiderivative = 8.59 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {8 \, e^{\left (-x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {8 \, e^{\left (-2 \, x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {8 \, e^{\left (-3 \, x\right )}}{5 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {4 \, e^{\left (-4 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a} + \frac {4}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} \]
8/15*e^(-x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a *e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a) - 8/15*e^(-2*x)/(2*a*e^(-x) - 2 *a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a) - 8/5*e^(-3*x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a) - 4*e^(-4*x) /(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a) + 4/15/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^ (-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {3 \, e^{\left (2 \, x\right )} - 12 \, e^{x} + 5}{24 \, a {\left (e^{x} - 1\right )}^{3}} - \frac {15 \, e^{\left (4 \, x\right )} + 60 \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} + 20 \, e^{x} + 7}{120 \, a {\left (e^{x} + 1\right )}^{5}} \]
1/24*(3*e^(2*x) - 12*e^x + 5)/(a*(e^x - 1)^3) - 1/120*(15*e^(4*x) + 60*e^( 3*x) + 10*e^(2*x) + 20*e^x + 7)/(a*(e^x + 1)^5)
Time = 1.99 (sec) , antiderivative size = 236, normalized size of antiderivative = 6.94 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {1}{6\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{40\,a}+\frac {{\mathrm {e}}^{3\,x}}{40\,a}+\frac {1}{40\,a}-\frac {{\mathrm {e}}^x}{8\,a}}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}-\frac {\frac {{\mathrm {e}}^{2\,x}}{40\,a}-\frac {1}{24\,a}+\frac {{\mathrm {e}}^x}{20\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {{\mathrm {e}}^{3\,x}}{10\,a}-\frac {{\mathrm {e}}^{2\,x}}{4\,a}+\frac {{\mathrm {e}}^{4\,x}}{40\,a}+\frac {1}{40\,a}+\frac {{\mathrm {e}}^x}{10\,a}}{10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{8\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{20\,a\,\left ({\mathrm {e}}^x+1\right )} \]
1/(6*a*(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1)) - ((3*exp(2*x))/(40*a) + ex p(3*x)/(40*a) + 1/(40*a) - exp(x)/(8*a))/(6*exp(2*x) + 4*exp(3*x) + exp(4* x) + 4*exp(x) + 1) - (exp(2*x)/(40*a) - 1/(24*a) + exp(x)/(20*a))/(3*exp(2 *x) + exp(3*x) + 3*exp(x) + 1) - (exp(3*x)/(10*a) - exp(2*x)/(4*a) + exp(4 *x)/(40*a) + 1/(40*a) + exp(x)/(10*a))/(10*exp(2*x) + 10*exp(3*x) + 5*exp( 4*x) + exp(5*x) + 5*exp(x) + 1) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) + 1/(8 *a*(exp(x) - 1)) - 1/(20*a*(exp(x) + 1))