Integrand size = 13, antiderivative size = 46 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=-\frac {3 \text {arctanh}(\cosh (x))}{8 a}+\frac {\coth ^4(x)}{4 a}-\frac {3 \coth (x) \text {csch}(x)}{8 a}-\frac {\coth ^3(x) \text {csch}(x)}{4 a} \] Output:
-3/8*arctanh(cosh(x))/a+1/4*coth(x)^4/a-3/8*coth(x)*csch(x)/a-1/4*coth(x)^ 3*csch(x)/a
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=\frac {-8-2 \coth ^2\left (\frac {x}{2}\right )-12 \cosh ^2\left (\frac {x}{2}\right ) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )+\text {sech}^2\left (\frac {x}{2}\right )}{16 a (1+\cosh (x))} \] Input:
Integrate[Coth[x]^3/(a + a*Cosh[x]),x]
Output:
(-8 - 2*Coth[x/2]^2 - 12*Cosh[x/2]^2*(Log[Cosh[x/2]] - Log[Sinh[x/2]]) + S ech[x/2]^2)/(16*a*(1 + Cosh[x]))
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.308, Rules used = {3042, 26, 3185, 26, 3042, 26, 3087, 15, 3091, 26, 3042, 26, 3091, 26, 3042, 26, 4257}
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.
\(\displaystyle \int \frac {\coth ^3(x)}{a \cosh (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \tan \left (-\frac {\pi }{2}+i x\right )^3}{a-a \sin \left (-\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\tan \left (i x-\frac {\pi }{2}\right )^3}{a-a \sin \left (i x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3185 |
\(\displaystyle i \left (\frac {\int -i \coth ^4(x) \text {csch}(x)dx}{a}+\frac {\int i \coth ^3(x) \text {csch}^2(x)dx}{a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {i \int \coth ^3(x) \text {csch}^2(x)dx}{a}-\frac {i \int \coth ^4(x) \text {csch}(x)dx}{a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (\frac {i \int -i \sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )^3dx}{a}-\frac {i \int i \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^4dx}{a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )^3dx}{a}+\frac {\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^4dx}{a}\right )\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^4dx}{a}-\frac {i \int -i \coth ^3(x)d(i \coth (x))}{a}\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^4dx}{a}-\frac {i \coth ^4(x)}{4 a}\right )\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle i \left (\frac {\frac {1}{4} i \coth ^3(x) \text {csch}(x)-\frac {3}{4} \int i \coth ^2(x) \text {csch}(x)dx}{a}-\frac {i \coth ^4(x)}{4 a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {\frac {1}{4} i \coth ^3(x) \text {csch}(x)-\frac {3}{4} i \int \coth ^2(x) \text {csch}(x)dx}{a}-\frac {i \coth ^4(x)}{4 a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (\frac {\frac {1}{4} i \coth ^3(x) \text {csch}(x)-\frac {3}{4} i \int -i \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}-\frac {i \coth ^4(x)}{4 a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {\frac {1}{4} i \coth ^3(x) \text {csch}(x)-\frac {3}{4} \int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}-\frac {i \coth ^4(x)}{4 a}\right )\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle i \left (\frac {\frac {1}{4} i \coth ^3(x) \text {csch}(x)-\frac {3}{4} \left (-\frac {1}{2} \int -i \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )}{a}-\frac {i \coth ^4(x)}{4 a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {\frac {1}{4} i \coth ^3(x) \text {csch}(x)-\frac {3}{4} \left (\frac {1}{2} i \int \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )}{a}-\frac {i \coth ^4(x)}{4 a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (\frac {\frac {1}{4} i \coth ^3(x) \text {csch}(x)-\frac {3}{4} \left (\frac {1}{2} i \int i \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )}{a}-\frac {i \coth ^4(x)}{4 a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {\frac {1}{4} i \coth ^3(x) \text {csch}(x)-\frac {3}{4} \left (-\frac {1}{2} \int \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )}{a}-\frac {i \coth ^4(x)}{4 a}\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle i \left (\frac {\frac {1}{4} i \coth ^3(x) \text {csch}(x)-\frac {3}{4} \left (-\frac {1}{2} i \text {arctanh}(\cosh (x))-\frac {1}{2} i \coth (x) \text {csch}(x)\right )}{a}-\frac {i \coth ^4(x)}{4 a}\right )\) |
Input:
Int[Coth[x]^3/(a + a*Cosh[x]),x]
Output:
I*(((-1/4*I)*Coth[x]^4)/a + ((I/4)*Coth[x]^3*Csch[x] - (3*((-1/2*I)*ArcTan h[Cosh[x]] - (I/2)*Coth[x]*Csch[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[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[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{4}}{4}+\frac {3 \tanh \left (\frac {x}{2}\right )^{2}}{2}+3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {1}{2 \tanh \left (\frac {x}{2}\right )^{2}}}{8 a}\) | \(38\) |
risch | \(-\frac {{\mathrm e}^{x} \left (5 \,{\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}+5\right )}{4 \left ({\mathrm e}^{x}+1\right )^{4} a \left ({\mathrm e}^{x}-1\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8 a}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8 a}\) | \(65\) |
Input:
int(coth(x)^3/(a+cosh(x)*a),x,method=_RETURNVERBOSE)
Output:
1/8/a*(1/4*tanh(1/2*x)^4+3/2*tanh(1/2*x)^2+3*ln(tanh(1/2*x))-1/2/tanh(1/2* x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (38) = 76\).
Time = 0.08 (sec) , antiderivative size = 631, normalized size of antiderivative = 13.72 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=\text {Too large to display} \] Input:
integrate(coth(x)^3/(a+a*cosh(x)),x, algorithm="fricas")
Output:
-1/8*(10*cosh(x)^5 + 2*(25*cosh(x) + 2)*sinh(x)^4 + 10*sinh(x)^5 + 4*cosh( x)^4 + 4*(25*cosh(x)^2 + 4*cosh(x) + 1)*sinh(x)^3 + 4*cosh(x)^3 + 4*(25*co sh(x)^3 + 6*cosh(x)^2 + 3*cosh(x) + 1)*sinh(x)^2 + 4*cosh(x)^2 + 3*(cosh(x )^6 + 2*(3*cosh(x) + 1)*sinh(x)^5 + sinh(x)^6 + 2*cosh(x)^5 + (15*cosh(x)^ 2 + 10*cosh(x) - 1)*sinh(x)^4 - cosh(x)^4 + 4*(5*cosh(x)^3 + 5*cosh(x)^2 - cosh(x) - 1)*sinh(x)^3 - 4*cosh(x)^3 + (15*cosh(x)^4 + 20*cosh(x)^3 - 6*c osh(x)^2 - 12*cosh(x) - 1)*sinh(x)^2 - cosh(x)^2 + 2*(3*cosh(x)^5 + 5*cosh (x)^4 - 2*cosh(x)^3 - 6*cosh(x)^2 - cosh(x) + 1)*sinh(x) + 2*cosh(x) + 1)* log(cosh(x) + sinh(x) + 1) - 3*(cosh(x)^6 + 2*(3*cosh(x) + 1)*sinh(x)^5 + sinh(x)^6 + 2*cosh(x)^5 + (15*cosh(x)^2 + 10*cosh(x) - 1)*sinh(x)^4 - cosh (x)^4 + 4*(5*cosh(x)^3 + 5*cosh(x)^2 - cosh(x) - 1)*sinh(x)^3 - 4*cosh(x)^ 3 + (15*cosh(x)^4 + 20*cosh(x)^3 - 6*cosh(x)^2 - 12*cosh(x) - 1)*sinh(x)^2 - cosh(x)^2 + 2*(3*cosh(x)^5 + 5*cosh(x)^4 - 2*cosh(x)^3 - 6*cosh(x)^2 - cosh(x) + 1)*sinh(x) + 2*cosh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 2*(25*c osh(x)^4 + 8*cosh(x)^3 + 6*cosh(x)^2 + 4*cosh(x) + 5)*sinh(x) + 10*cosh(x) )/(a*cosh(x)^6 + a*sinh(x)^6 + 2*a*cosh(x)^5 + 2*(3*a*cosh(x) + a)*sinh(x) ^5 - a*cosh(x)^4 + (15*a*cosh(x)^2 + 10*a*cosh(x) - a)*sinh(x)^4 - 4*a*cos h(x)^3 + 4*(5*a*cosh(x)^3 + 5*a*cosh(x)^2 - a*cosh(x) - a)*sinh(x)^3 - a*c osh(x)^2 + (15*a*cosh(x)^4 + 20*a*cosh(x)^3 - 6*a*cosh(x)^2 - 12*a*cosh(x) - a)*sinh(x)^2 + 2*a*cosh(x) + 2*(3*a*cosh(x)^5 + 5*a*cosh(x)^4 - 2*a*...
\[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\coth ^{3}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(coth(x)**3/(a+a*cosh(x)),x)
Output:
Integral(coth(x)**3/(cosh(x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (38) = 76\).
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.24 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=-\frac {5 \, e^{\left (-x\right )} + 2 \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )} + 2 \, e^{\left (-4 \, x\right )} + 5 \, e^{\left (-5 \, x\right )}}{4 \, {\left (2 \, a e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - 4 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + 2 \, a e^{\left (-5 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} - \frac {3 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} + \frac {3 \, \log \left (e^{\left (-x\right )} - 1\right )}{8 \, a} \] Input:
integrate(coth(x)^3/(a+a*cosh(x)),x, algorithm="maxima")
Output:
-1/4*(5*e^(-x) + 2*e^(-2*x) + 2*e^(-3*x) + 2*e^(-4*x) + 5*e^(-5*x))/(2*a*e ^(-x) - a*e^(-2*x) - 4*a*e^(-3*x) - a*e^(-4*x) + 2*a*e^(-5*x) + a*e^(-6*x) + a) - 3/8*log(e^(-x) + 1)/a + 3/8*log(e^(-x) - 1)/a
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (38) = 76\).
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.04 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=-\frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} + \frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} - \frac {3 \, e^{\left (-x\right )} + 3 \, e^{x} - 2}{16 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} + \frac {9 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 4 \, e^{\left (-x\right )} + 4 \, e^{x} - 12}{32 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{2}} \] Input:
integrate(coth(x)^3/(a+a*cosh(x)),x, algorithm="giac")
Output:
-3/16*log(e^(-x) + e^x + 2)/a + 3/16*log(e^(-x) + e^x - 2)/a - 1/16*(3*e^( -x) + 3*e^x - 2)/(a*(e^(-x) + e^x - 2)) + 1/32*(9*(e^(-x) + e^x)^2 + 4*e^( -x) + 4*e^x - 12)/(a*(e^(-x) + e^x + 2)^2)
Time = 1.94 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.87 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=\frac {3}{2\,a\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{2\,a\,\left (6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{4\,\sqrt {-a^2}}-\frac {1}{a\,\left (3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1\right )} \] Input:
int(coth(x)^3/(a + a*cosh(x)),x)
Output:
3/(2*a*(exp(2*x) + 2*exp(x) + 1)) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) + 1/ (2*a*(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1)) - 1/(4*a*(exp(x) - 1)) - 1/(a*(exp(x) + 1)) - (3*atan((exp(x)*(-a^2)^(1/2))/a))/(4*(-a^2)^ (1/2)) - 1/(a*(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1))
Time = 0.25 (sec) , antiderivative size = 245, normalized size of antiderivative = 5.33 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=\frac {3 e^{6 x} \mathrm {log}\left (e^{x}-1\right )-3 e^{6 x} \mathrm {log}\left (e^{x}+1\right )+5 e^{6 x}+6 e^{5 x} \mathrm {log}\left (e^{x}-1\right )-6 e^{5 x} \mathrm {log}\left (e^{x}+1\right )-3 e^{4 x} \mathrm {log}\left (e^{x}-1\right )+3 e^{4 x} \mathrm {log}\left (e^{x}+1\right )-9 e^{4 x}-12 e^{3 x} \mathrm {log}\left (e^{x}-1\right )+12 e^{3 x} \mathrm {log}\left (e^{x}+1\right )-24 e^{3 x}-3 e^{2 x} \mathrm {log}\left (e^{x}-1\right )+3 e^{2 x} \mathrm {log}\left (e^{x}+1\right )-9 e^{2 x}+6 e^{x} \mathrm {log}\left (e^{x}-1\right )-6 e^{x} \mathrm {log}\left (e^{x}+1\right )+3 \,\mathrm {log}\left (e^{x}-1\right )-3 \,\mathrm {log}\left (e^{x}+1\right )+5}{8 a \left (e^{6 x}+2 e^{5 x}-e^{4 x}-4 e^{3 x}-e^{2 x}+2 e^{x}+1\right )} \] Input:
int(coth(x)^3/(a+a*cosh(x)),x)
Output:
(3*e**(6*x)*log(e**x - 1) - 3*e**(6*x)*log(e**x + 1) + 5*e**(6*x) + 6*e**( 5*x)*log(e**x - 1) - 6*e**(5*x)*log(e**x + 1) - 3*e**(4*x)*log(e**x - 1) + 3*e**(4*x)*log(e**x + 1) - 9*e**(4*x) - 12*e**(3*x)*log(e**x - 1) + 12*e* *(3*x)*log(e**x + 1) - 24*e**(3*x) - 3*e**(2*x)*log(e**x - 1) + 3*e**(2*x) *log(e**x + 1) - 9*e**(2*x) + 6*e**x*log(e**x - 1) - 6*e**x*log(e**x + 1) + 3*log(e**x - 1) - 3*log(e**x + 1) + 5)/(8*a*(e**(6*x) + 2*e**(5*x) - e** (4*x) - 4*e**(3*x) - e**(2*x) + 2*e**x + 1))