Integrand size = 13, antiderivative size = 55 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a}-\frac {\coth (x) (15-8 \text {sech}(x))}{15 a}-\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a} \] Output:
x/a-1/15*coth(x)*(15-8*sech(x))/a-1/15*coth(x)^3*(5-4*sech(x))/a-1/5*coth( x)^5*(1-sech(x))/a
Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.25 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\text {csch}^3(x) \text {sech}(x) (-25+8 \cosh (x)+16 \cosh (2 x)-16 \cosh (3 x)-23 \cosh (4 x)-90 x \sinh (x)-30 x \sinh (2 x)+30 x \sinh (3 x)+15 x \sinh (4 x))}{120 a (1+\text {sech}(x))} \] Input:
Integrate[Coth[x]^4/(a + a*Sech[x]),x]
Output:
(Csch[x]^3*Sech[x]*(-25 + 8*Cosh[x] + 16*Cosh[2*x] - 16*Cosh[3*x] - 23*Cos h[4*x] - 90*x*Sinh[x] - 30*x*Sinh[2*x] + 30*x*Sinh[3*x] + 15*x*Sinh[4*x])) /(120*a*(1 + Sech[x]))
Time = 0.50 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {3042, 4376, 3042, 25, 4370, 25, 3042, 4370, 3042, 25, 4370, 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.
\(\displaystyle \int \frac {\coth ^4(x)}{a \text {sech}(x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cot \left (\frac {\pi }{2}+i x\right )^4 \left (a+a \csc \left (\frac {\pi }{2}+i x\right )\right )}dx\) |
\(\Big \downarrow \) 4376 |
\(\displaystyle \frac {\int \coth ^6(x) (a-a \text {sech}(x))dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\frac {a-a \csc \left (i x+\frac {\pi }{2}\right )}{\cot \left (i x+\frac {\pi }{2}\right )^6}dx}{a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {a-a \csc \left (i x+\frac {\pi }{2}\right )}{\cot \left (i x+\frac {\pi }{2}\right )^6}dx}{a^2}\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle -\frac {\frac {1}{5} \int -\coth ^4(x) (5 a-4 a \text {sech}(x))dx+\frac {1}{5} \coth ^5(x) (a-a \text {sech}(x))}{a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {1}{5} \coth ^5(x) (a-a \text {sech}(x))-\frac {1}{5} \int \coth ^4(x) (5 a-4 a \text {sech}(x))dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{5} \coth ^5(x) (a-a \text {sech}(x))-\frac {1}{5} \int \frac {5 a-4 a \csc \left (i x+\frac {\pi }{2}\right )}{\cot \left (i x+\frac {\pi }{2}\right )^4}dx}{a^2}\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \coth ^3(x) (5 a-4 a \text {sech}(x))-\frac {1}{3} \int \coth ^2(x) (15 a-8 a \text {sech}(x))dx\right )+\frac {1}{5} \coth ^5(x) (a-a \text {sech}(x))}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{5} \coth ^5(x) (a-a \text {sech}(x))+\frac {1}{5} \left (\frac {1}{3} \coth ^3(x) (5 a-4 a \text {sech}(x))-\frac {1}{3} \int -\frac {15 a-8 a \csc \left (i x+\frac {\pi }{2}\right )}{\cot \left (i x+\frac {\pi }{2}\right )^2}dx\right )}{a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {1}{5} \coth ^5(x) (a-a \text {sech}(x))+\frac {1}{5} \left (\frac {1}{3} \coth ^3(x) (5 a-4 a \text {sech}(x))+\frac {1}{3} \int \frac {15 a-8 a \csc \left (i x+\frac {\pi }{2}\right )}{\cot \left (i x+\frac {\pi }{2}\right )^2}dx\right )}{a^2}\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} (\int -15 adx+\coth (x) (15 a-8 a \text {sech}(x)))+\frac {1}{3} \coth ^3(x) (5 a-4 a \text {sech}(x))\right )+\frac {1}{5} \coth ^5(x) (a-a \text {sech}(x))}{a^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\frac {1}{5} \coth ^5(x) (a-a \text {sech}(x))+\frac {1}{5} \left (\frac {1}{3} \coth ^3(x) (5 a-4 a \text {sech}(x))+\frac {1}{3} (\coth (x) (15 a-8 a \text {sech}(x))-15 a x)\right )}{a^2}\) |
Input:
Int[Coth[x]^4/(a + a*Sech[x]),x]
Output:
-(((Coth[x]^5*(a - a*Sech[x]))/5 + ((Coth[x]^3*(5*a - 4*a*Sech[x]))/3 + (- 15*a*x + Coth[x]*(15*a - 8*a*Sech[x]))/3)/5)/a^2)
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-(e*Cot[c + d*x])^(m + 1))*((a + b*Csc[c + d*x])/( d*e*(m + 1))), x] - Simp[1/(e^2*(m + 1)) Int[(e*Cot[c + d*x])^(m + 2)*(a* (m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && L tQ[m, -1]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{5}}{5}-2 \tanh \left (\frac {x}{2}\right )^{3}-16 \tanh \left (\frac {x}{2}\right )-\frac {1}{3 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {6}{\tanh \left (\frac {x}{2}\right )}+16 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-16 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{16 a}\) | \(63\) |
risch | \(\frac {x}{a}+\frac {2 \,{\mathrm e}^{7 x}-2 \,{\mathrm e}^{6 x}-\frac {26 \,{\mathrm e}^{5 x}}{3}-\frac {10 \,{\mathrm e}^{4 x}}{3}+\frac {146 \,{\mathrm e}^{3 x}}{15}+\frac {62 \,{\mathrm e}^{2 x}}{15}-\frac {62 \,{\mathrm e}^{x}}{15}-\frac {46}{15}}{a \left (1+{\mathrm e}^{x}\right )^{5} \left ({\mathrm e}^{x}-1\right )^{3}}\) | \(66\) |
Input:
int(coth(x)^4/(a+a*sech(x)),x,method=_RETURNVERBOSE)
Output:
1/16/a*(-1/5*tanh(1/2*x)^5-2*tanh(1/2*x)^3-16*tanh(1/2*x)-1/3/tanh(1/2*x)^ 3-6/tanh(1/2*x)+16*ln(tanh(1/2*x)+1)-16*ln(tanh(1/2*x)-1))
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (47) = 94\).
Time = 0.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.75 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=-\frac {23 \, \cosh \left (x\right )^{4} - 2 \, {\left (2 \, {\left (15 \, x + 23\right )} \cosh \left (x\right ) + 15 \, x + 23\right )} \sinh \left (x\right )^{3} + 23 \, \sinh \left (x\right )^{4} + 16 \, \cosh \left (x\right )^{3} + 2 \, {\left (69 \, \cosh \left (x\right )^{2} + 24 \, \cosh \left (x\right ) - 8\right )} \sinh \left (x\right )^{2} - 16 \, \cosh \left (x\right )^{2} - 2 \, {\left (2 \, {\left (15 \, x + 23\right )} \cosh \left (x\right )^{3} + 3 \, {\left (15 \, x + 23\right )} \cosh \left (x\right )^{2} - 2 \, {\left (15 \, x + 23\right )} \cosh \left (x\right ) - 45 \, x - 69\right )} \sinh \left (x\right ) - 8 \, \cosh \left (x\right ) + 25}{30 \, {\left ({\left (2 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{3} + {\left (2 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} - 2 \, a \cosh \left (x\right ) - 3 \, a\right )} \sinh \left (x\right )\right )}} \] Input:
integrate(coth(x)^4/(a+a*sech(x)),x, algorithm="fricas")
Output:
-1/30*(23*cosh(x)^4 - 2*(2*(15*x + 23)*cosh(x) + 15*x + 23)*sinh(x)^3 + 23 *sinh(x)^4 + 16*cosh(x)^3 + 2*(69*cosh(x)^2 + 24*cosh(x) - 8)*sinh(x)^2 - 16*cosh(x)^2 - 2*(2*(15*x + 23)*cosh(x)^3 + 3*(15*x + 23)*cosh(x)^2 - 2*(1 5*x + 23)*cosh(x) - 45*x - 69)*sinh(x) - 8*cosh(x) + 25)/((2*a*cosh(x) + a )*sinh(x)^3 + (2*a*cosh(x)^3 + 3*a*cosh(x)^2 - 2*a*cosh(x) - 3*a)*sinh(x))
\[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\coth ^{4}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(coth(x)**4/(a+a*sech(x)),x)
Output:
Integral(coth(x)**4/(sech(x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (47) = 94\).
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.91 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} - \frac {2 \, {\left (31 \, e^{\left (-x\right )} - 31 \, e^{\left (-2 \, x\right )} - 73 \, e^{\left (-3 \, x\right )} + 25 \, e^{\left (-4 \, x\right )} + 65 \, e^{\left (-5 \, x\right )} + 15 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-7 \, x\right )} + 23\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} \] Input:
integrate(coth(x)^4/(a+a*sech(x)),x, algorithm="maxima")
Output:
x/a - 2/15*(31*e^(-x) - 31*e^(-2*x) - 73*e^(-3*x) + 25*e^(-4*x) + 65*e^(-5 *x) + 15*e^(-6*x) - 15*e^(-7*x) + 23)/(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)
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.16 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} - \frac {21 \, e^{\left (2 \, x\right )} - 36 \, e^{x} + 19}{24 \, a {\left (e^{x} - 1\right )}^{3}} + \frac {115 \, e^{\left (4 \, x\right )} + 380 \, e^{\left (3 \, x\right )} + 530 \, e^{\left (2 \, x\right )} + 340 \, e^{x} + 91}{40 \, a {\left (e^{x} + 1\right )}^{5}} \] Input:
integrate(coth(x)^4/(a+a*sech(x)),x, algorithm="giac")
Output:
x/a - 1/24*(21*e^(2*x) - 36*e^x + 19)/(a*(e^x - 1)^3) + 1/40*(115*e^(4*x) + 380*e^(3*x) + 530*e^(2*x) + 340*e^x + 91)/(a*(e^x + 1)^5)
Time = 2.46 (sec) , antiderivative size = 264, normalized size of antiderivative = 4.80 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\frac {9\,{\mathrm {e}}^{2\,x}}{4\,a}+\frac {3\,{\mathrm {e}}^{3\,x}}{2\,a}+\frac {23\,{\mathrm {e}}^{4\,x}}{40\,a}+\frac {23}{40\,a}+\frac {3\,{\mathrm {e}}^x}{2\,a}}{10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1}+\frac {\frac {9\,{\mathrm {e}}^{2\,x}}{8\,a}+\frac {23\,{\mathrm {e}}^{3\,x}}{40\,a}+\frac {3}{8\,a}+\frac {9\,{\mathrm {e}}^x}{8\,a}}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}+\frac {\frac {23\,{\mathrm {e}}^{2\,x}}{40\,a}+\frac {3}{8\,a}+\frac {3\,{\mathrm {e}}^x}{4\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}+\frac {\frac {3}{8\,a}+\frac {23\,{\mathrm {e}}^x}{40\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}+\frac {1}{6\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {x}{a}-\frac {7}{8\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {23}{40\,a\,\left ({\mathrm {e}}^x+1\right )} \] Input:
int(coth(x)^4/(a + a/cosh(x)),x)
Output:
((9*exp(2*x))/(4*a) + (3*exp(3*x))/(2*a) + (23*exp(4*x))/(40*a) + 23/(40*a ) + (3*exp(x))/(2*a))/(10*exp(2*x) + 10*exp(3*x) + 5*exp(4*x) + exp(5*x) + 5*exp(x) + 1) + ((9*exp(2*x))/(8*a) + (23*exp(3*x))/(40*a) + 3/(8*a) + (9 *exp(x))/(8*a))/(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1) + ((23 *exp(2*x))/(40*a) + 3/(8*a) + (3*exp(x))/(4*a))/(3*exp(2*x) + exp(3*x) + 3 *exp(x) + 1) + (3/(8*a) + (23*exp(x))/(40*a))/(exp(2*x) + 2*exp(x) + 1) + 1/(6*a*(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1)) - 1/(4*a*(exp(2*x) - 2*exp( x) + 1)) + x/a - 7/(8*a*(exp(x) - 1)) + 23/(40*a*(exp(x) + 1))
Time = 7.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.78 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {15 e^{8 x} x -15 e^{8 x}+30 e^{7 x} x -30 e^{6 x} x -90 e^{5 x} x -40 e^{5 x}-50 e^{4 x}+90 e^{3 x} x +56 e^{3 x}+30 e^{2 x} x +32 e^{2 x}-30 e^{x} x -32 e^{x}-15 x -31}{15 a \left (e^{8 x}+2 e^{7 x}-2 e^{6 x}-6 e^{5 x}+6 e^{3 x}+2 e^{2 x}-2 e^{x}-1\right )} \] Input:
int(coth(x)^4/(a+a*sech(x)),x)
Output:
(15*e**(8*x)*x - 15*e**(8*x) + 30*e**(7*x)*x - 30*e**(6*x)*x - 90*e**(5*x) *x - 40*e**(5*x) - 50*e**(4*x) + 90*e**(3*x)*x + 56*e**(3*x) + 30*e**(2*x) *x + 32*e**(2*x) - 30*e**x*x - 32*e**x - 15*x - 31)/(15*a*(e**(8*x) + 2*e* *(7*x) - 2*e**(6*x) - 6*e**(5*x) + 6*e**(3*x) + 2*e**(2*x) - 2*e**x - 1))