Integrand size = 11, antiderivative size = 34 \[ \int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx=\frac {1}{8} \text {arctanh}(\cosh (x))-\frac {1}{8} \coth (x) \text {csch}(x)+\frac {\text {csch}^3(x)}{3}-\frac {1}{4} \coth (x) \text {csch}^3(x) \] Output:
1/8*arctanh(cosh(x))-1/8*coth(x)*csch(x)+1/3*csch(x)^3-1/4*coth(x)*csch(x) ^3
Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(34)=68\).
Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.03 \[ \int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx=\frac {1}{192} \text {csch}^4(x) \left (-42 \cosh (x)-6 \cosh (3 x)+2 \sinh (x) \left (32-9 \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \sinh (x)+3 \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \sinh (3 x)\right )\right ) \] Input:
Integrate[Csch[x]^5/(1 + Tanh[x]),x]
Output:
(Csch[x]^4*(-42*Cosh[x] - 6*Cosh[3*x] + 2*Sinh[x]*(32 - 9*(Log[Cosh[x/2]] - Log[Sinh[x/2]])*Sinh[x] + 3*(Log[Cosh[x/2]] - Log[Sinh[x/2]])*Sinh[3*x]) ))/192
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.091, Rules used = {3042, 26, 4001, 26, 3042, 26, 3587, 25, 3042, 26, 3586, 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.
\(\displaystyle \int \frac {\text {csch}^5(x)}{\tanh (x)+1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\sin (i x)^5 (1-i \tan (i x))}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\sin (i x)^5 (1-i \tan (i x))}dx\) |
\(\Big \downarrow \) 4001 |
\(\displaystyle i \int -\frac {i \coth (x) \text {csch}^4(x)}{\cosh (x)+\sinh (x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\coth (x) \text {csch}^4(x)}{\sinh (x)+\cosh (x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \cos (i x)}{\sin (i x)^5 (\cos (i x)-i \sin (i x))}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cos (i x)}{(\cos (i x)-i \sin (i x)) \sin (i x)^5}dx\) |
\(\Big \downarrow \) 3587 |
\(\displaystyle -\int -\coth (x) \text {csch}^4(x) (\cosh (x)-\sinh (x))dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \coth (x) \text {csch}^4(x) (\cosh (x)-\sinh (x))dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \cos (i x) (i \sin (i x)+\cos (i x))}{\sin (i x)^5}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cos (i x) (\cos (i x)+i \sin (i x))}{\sin (i x)^5}dx\) |
\(\Big \downarrow \) 3586 |
\(\displaystyle i \int \left (i \coth (x) \text {csch}^3(x)-i \coth ^2(x) \text {csch}^3(x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (-\frac {1}{8} i \text {arctanh}(\cosh (x))-\frac {1}{3} i \text {csch}^3(x)+\frac {1}{4} i \coth (x) \text {csch}^3(x)+\frac {1}{8} i \coth (x) \text {csch}(x)\right )\) |
Input:
Int[Csch[x]^5/(1 + Tanh[x]),x]
Output:
I*((-1/8*I)*ArcTanh[Cosh[x]] + (I/8)*Coth[x]*Csch[x] - (I/3)*Csch[x]^3 + ( I/4)*Coth[x]*Csch[x]^3)
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[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(p_.), x_Symbol] :> In t[ExpandTrig[cos[c + d*x]^m*sin[c + d*x]^n*(a*cos[c + d*x] + b*sin[c + d*x] )^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0]
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Sim p[a^p*b^p Int[(Cos[c + d*x]^m*Sin[c + d*x]^n)/(b*Cos[c + d*x] + a*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a^2 + b^2, 0] && IL tQ[p, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Int[Sin[e + f*x]^m*((a*Cos[e + f*x] + b*Sin[e + f*x])^n/C os[e + f*x]^n), x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && ILtQ [n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))
Time = 2.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41
method | result | size |
risch | \(-\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{6 x}-11 \,{\mathrm e}^{4 x}+53 \,{\mathrm e}^{2 x}+3\right )}{12 \left ({\mathrm e}^{2 x}-1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{x}-1\right )}{8}+\frac {\ln \left ({\mathrm e}^{x}+1\right )}{8}\) | \(48\) |
default | \(\frac {\tanh \left (\frac {x}{2}\right )^{4}}{64}-\frac {\tanh \left (\frac {x}{2}\right )^{3}}{24}+\frac {\tanh \left (\frac {x}{2}\right )}{8}-\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8}\) | \(55\) |
Input:
int(csch(x)^5/(1+tanh(x)),x,method=_RETURNVERBOSE)
Output:
-1/12*exp(x)*(3*exp(6*x)-11*exp(4*x)+53*exp(2*x)+3)/(exp(2*x)-1)^4-1/8*ln( exp(x)-1)+1/8*ln(exp(x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (26) = 52\).
Time = 0.09 (sec) , antiderivative size = 640, normalized size of antiderivative = 18.82 \[ \int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^5/(1+tanh(x)),x, algorithm="fricas")
Output:
-1/24*(6*cosh(x)^7 + 42*cosh(x)*sinh(x)^6 + 6*sinh(x)^7 + 2*(63*cosh(x)^2 - 11)*sinh(x)^5 - 22*cosh(x)^5 + 10*(21*cosh(x)^3 - 11*cosh(x))*sinh(x)^4 + 2*(105*cosh(x)^4 - 110*cosh(x)^2 + 53)*sinh(x)^3 + 106*cosh(x)^3 + 2*(63 *cosh(x)^5 - 110*cosh(x)^3 + 159*cosh(x))*sinh(x)^2 - 3*(cosh(x)^8 + 8*cos h(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 1)*sinh(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3 - 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 - 10*cosh(x)^3 + 3*cosh(x))*si nh(x)^3 + 4*(7*cosh(x)^6 - 15*cosh(x)^4 + 9*cosh(x)^2 - 1)*sinh(x)^2 - 4*c osh(x)^2 + 8*(cosh(x)^7 - 3*cosh(x)^5 + 3*cosh(x)^3 - cosh(x))*sinh(x) + 1 )*log(cosh(x) + sinh(x) + 1) + 3*(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x )^8 + 4*(7*cosh(x)^2 - 1)*sinh(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3 - 3*cos h(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x )^4 + 8*(7*cosh(x)^5 - 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^ 6 - 15*cosh(x)^4 + 9*cosh(x)^2 - 1)*sinh(x)^2 - 4*cosh(x)^2 + 8*(cosh(x)^7 - 3*cosh(x)^5 + 3*cosh(x)^3 - cosh(x))*sinh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 2*(21*cosh(x)^6 - 55*cosh(x)^4 + 159*cosh(x)^2 + 3)*sinh(x) + 6*co sh(x))/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 1)* sinh(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3 - 3*cosh(x))*sinh(x)^5 + 2*(35*co sh(x)^4 - 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 - 10* cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 15*cosh(x)^4 + 9*co...
\[ \int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx=\int \frac {\operatorname {csch}^{5}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \] Input:
integrate(csch(x)**5/(1+tanh(x)),x)
Output:
Integral(csch(x)**5/(tanh(x) + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (26) = 52\).
Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.18 \[ \int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx=\frac {3 \, e^{\left (-x\right )} - 11 \, e^{\left (-3 \, x\right )} + 53 \, e^{\left (-5 \, x\right )} + 3 \, e^{\left (-7 \, x\right )}}{12 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac {1}{8} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {1}{8} \, \log \left (e^{\left (-x\right )} - 1\right ) \] Input:
integrate(csch(x)^5/(1+tanh(x)),x, algorithm="maxima")
Output:
1/12*(3*e^(-x) - 11*e^(-3*x) + 53*e^(-5*x) + 3*e^(-7*x))/(4*e^(-2*x) - 6*e ^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) + 1/8*log(e^(-x) + 1) - 1/8*log(e^(-x ) - 1)
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx=-\frac {3 \, e^{\left (7 \, x\right )} - 11 \, e^{\left (5 \, x\right )} + 53 \, e^{\left (3 \, x\right )} + 3 \, e^{x}}{12 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} + \frac {1}{8} \, \log \left (e^{x} + 1\right ) - \frac {1}{8} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(csch(x)^5/(1+tanh(x)),x, algorithm="giac")
Output:
-1/12*(3*e^(7*x) - 11*e^(5*x) + 53*e^(3*x) + 3*e^x)/(e^(2*x) - 1)^4 + 1/8* log(e^x + 1) - 1/8*log(abs(e^x - 1))
Time = 2.06 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.44 \[ \int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx=\frac {\ln \left (\frac {{\mathrm {e}}^x}{4}+\frac {1}{4}\right )}{8}-\frac {\ln \left (\frac {{\mathrm {e}}^x}{4}-\frac {1}{4}\right )}{8}-\frac {{\mathrm {e}}^x}{4\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{3\,x}+2\,{\mathrm {e}}^x}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {4\,{\mathrm {e}}^x}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}+\frac {{\mathrm {e}}^x}{6\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )} \] Input:
int(1/(sinh(x)^5*(tanh(x) + 1)),x)
Output:
log(exp(x)/4 + 1/4)/8 - log(exp(x)/4 - 1/4)/8 - exp(x)/(4*(exp(2*x) - 1)) - (2*exp(3*x) + 2*exp(x))/(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1) - (4*exp(x))/(3*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) + exp(x)/( 6*(exp(4*x) - 2*exp(2*x) + 1))
Time = 0.23 (sec) , antiderivative size = 180, normalized size of antiderivative = 5.29 \[ \int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx=\frac {-3 e^{8 x} \mathrm {log}\left (e^{x}-1\right )+3 e^{8 x} \mathrm {log}\left (e^{x}+1\right )-6 e^{7 x}+12 e^{6 x} \mathrm {log}\left (e^{x}-1\right )-12 e^{6 x} \mathrm {log}\left (e^{x}+1\right )+22 e^{5 x}-18 e^{4 x} \mathrm {log}\left (e^{x}-1\right )+18 e^{4 x} \mathrm {log}\left (e^{x}+1\right )-106 e^{3 x}+12 e^{2 x} \mathrm {log}\left (e^{x}-1\right )-12 e^{2 x} \mathrm {log}\left (e^{x}+1\right )-6 e^{x}-3 \,\mathrm {log}\left (e^{x}-1\right )+3 \,\mathrm {log}\left (e^{x}+1\right )}{24 e^{8 x}-96 e^{6 x}+144 e^{4 x}-96 e^{2 x}+24} \] Input:
int(csch(x)^5/(1+tanh(x)),x)
Output:
( - 3*e**(8*x)*log(e**x - 1) + 3*e**(8*x)*log(e**x + 1) - 6*e**(7*x) + 12* e**(6*x)*log(e**x - 1) - 12*e**(6*x)*log(e**x + 1) + 22*e**(5*x) - 18*e**( 4*x)*log(e**x - 1) + 18*e**(4*x)*log(e**x + 1) - 106*e**(3*x) + 12*e**(2*x )*log(e**x - 1) - 12*e**(2*x)*log(e**x + 1) - 6*e**x - 3*log(e**x - 1) + 3 *log(e**x + 1))/(24*(e**(8*x) - 4*e**(6*x) + 6*e**(4*x) - 4*e**(2*x) + 1))