Integrand size = 13, antiderivative size = 82 \[ \int \frac {\text {csch}^3(x)}{a+b \tanh (x)} \, dx=\frac {b \sqrt {a^2-b^2} \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{a^3}+\frac {\text {arctanh}(\cosh (x))}{2 a}-\frac {b^2 \text {arctanh}(\cosh (x))}{a^3}+\frac {b \text {csch}(x)}{a^2}-\frac {\coth (x) \text {csch}(x)}{2 a} \] Output:
b*(a^2-b^2)^(1/2)*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/a^3+1/2*ar ctanh(cosh(x))/a-b^2*arctanh(cosh(x))/a^3+b*csch(x)/a^2-1/2*coth(x)*csch(x )/a
Time = 0.49 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.79 \[ \int \frac {\text {csch}^3(x)}{a+b \tanh (x)} \, dx=-\frac {-16 \sqrt {a-b} b \sqrt {a+b} \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )-4 a b \coth \left (\frac {x}{2}\right )+a^2 \text {csch}^2\left (\frac {x}{2}\right )-4 a^2 \log \left (\cosh \left (\frac {x}{2}\right )\right )+8 b^2 \log \left (\cosh \left (\frac {x}{2}\right )\right )+4 a^2 \log \left (\sinh \left (\frac {x}{2}\right )\right )-8 b^2 \log \left (\sinh \left (\frac {x}{2}\right )\right )+a^2 \text {sech}^2\left (\frac {x}{2}\right )+4 a b \tanh \left (\frac {x}{2}\right )}{8 a^3} \] Input:
Integrate[Csch[x]^3/(a + b*Tanh[x]),x]
Output:
-1/8*(-16*Sqrt[a - b]*b*Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]* Sqrt[a + b])] - 4*a*b*Coth[x/2] + a^2*Csch[x/2]^2 - 4*a^2*Log[Cosh[x/2]] + 8*b^2*Log[Cosh[x/2]] + 4*a^2*Log[Sinh[x/2]] - 8*b^2*Log[Sinh[x/2]] + a^2* Sech[x/2]^2 + 4*a*b*Tanh[x/2])/a^3
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3042, 26, 4001, 26, 3042, 26, 3589, 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.
| \(\displaystyle \int \frac {\text {csch}^3(x)}{a+b \tanh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i x)^3 (a-i b \tan (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\sin (i x)^3 (a-i b \tan (i x))}dx\) |
| \(\Big \downarrow \) 4001 |
| \(\displaystyle -i \int \frac {i \coth (x) \text {csch}^2(x)}{a \cosh (x)+b \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\coth (x) \text {csch}^2(x)}{a \cosh (x)+b \sinh (x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \cos (i x)}{\sin (i x)^3 (a \cos (i x)-i b \sin (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\cos (i x)}{\sin (i x)^3 (a \cos (i x)-i b \sin (i x))}dx\) |
| \(\Big \downarrow \) 3589 |
| \(\displaystyle -i \int \left (-\frac {i \text {sech}^2(x) b^3}{a^3 (a \cosh (x)+b \sinh (x))}+\frac {i \text {csch}(x) \text {sech}^2(x) b^2}{a^3}-\frac {i \text {csch}^2(x) \text {sech}(x) b}{a^2}+\frac {i \text {csch}^3(x)}{a}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -i \left (-\frac {i b^2 \text {arctanh}(\cosh (x))}{a^3}+\frac {i b \text {csch}(x)}{a^2}+\frac {i b \sqrt {a^2-b^2} \arctan \left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{a^3}+\frac {i \text {arctanh}(\cosh (x))}{2 a}-\frac {i \coth (x) \text {csch}(x)}{2 a}\right )\) |
Input:
Int[Csch[x]^3/(a + b*Tanh[x]),x]
Output:
(-I)*((I*b*Sqrt[a^2 - b^2]*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]] )/a^3 + ((I/2)*ArcTanh[Cosh[x]])/a - (I*b^2*ArcTanh[Cosh[x]])/a^3 + (I*b*C sch[x])/a^2 - ((I/2)*Coth[x]*Csch[x])/a)
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_)]), x_Symbol] :> Int[Ex pandTrig[cos[c + d*x]^m*(sin[c + d*x]^n/(a*cos[c + d*x] + b*sin[c + d*x])), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegersQ[m, n]
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 = 1.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.34
| method | result | size |
| default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{2} a}{2}-2 b \tanh \left (\frac {x}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (-2 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {x}{2}\right )}+\frac {2 b \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{3}}\) | \(110\) |
| risch | \(-\frac {{\mathrm e}^{x} \left ({\mathrm e}^{2 x} a -2 \,{\mathrm e}^{2 x} b +a +2 b \right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} a^{2}}-\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2 a}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) b^{2}}{a^{3}}+\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2 a}-\frac {\ln \left ({\mathrm e}^{x}+1\right ) b^{2}}{a^{3}}+\frac {\sqrt {-a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{x}+\frac {\sqrt {-a^{2}+b^{2}}}{a +b}\right )}{a^{3}}-\frac {\sqrt {-a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{x}-\frac {\sqrt {-a^{2}+b^{2}}}{a +b}\right )}{a^{3}}\) | \(156\) |
Input:
int(csch(x)^3/(a+b*tanh(x)),x,method=_RETURNVERBOSE)
Output:
1/4/a^2*(1/2*tanh(1/2*x)^2*a-2*b*tanh(1/2*x))-1/8/a/tanh(1/2*x)^2+1/4/a^3* (-2*a^2+4*b^2)*ln(tanh(1/2*x))+1/2/a^2*b/tanh(1/2*x)+2*b*(a^2-b^2)^(1/2)/a ^3*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (74) = 148\).
Time = 0.13 (sec) , antiderivative size = 1165, normalized size of antiderivative = 14.21 \[ \int \frac {\text {csch}^3(x)}{a+b \tanh (x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^3/(a+b*tanh(x)),x, algorithm="fricas")
Output:
[-1/2*(2*(a^2 - 2*a*b)*cosh(x)^3 + 6*(a^2 - 2*a*b)*cosh(x)*sinh(x)^2 + 2*( a^2 - 2*a*b)*sinh(x)^3 - 2*(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x )^4 - 2*b*cosh(x)^2 + 2*(3*b*cosh(x)^2 - b)*sinh(x)^2 + 4*(b*cosh(x)^3 - b *cosh(x))*sinh(x) + b)*sqrt(-a^2 + b^2)*log(((a + b)*cosh(x)^2 + 2*(a + b) *cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh( x)) - a + b)/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh (x)^2 + a - b)) + 2*(a^2 + 2*a*b)*cosh(x) - ((a^2 - 2*b^2)*cosh(x)^4 + 4*( a^2 - 2*b^2)*cosh(x)*sinh(x)^3 + (a^2 - 2*b^2)*sinh(x)^4 - 2*(a^2 - 2*b^2) *cosh(x)^2 + 2*(3*(a^2 - 2*b^2)*cosh(x)^2 - a^2 + 2*b^2)*sinh(x)^2 + a^2 - 2*b^2 + 4*((a^2 - 2*b^2)*cosh(x)^3 - (a^2 - 2*b^2)*cosh(x))*sinh(x))*log( cosh(x) + sinh(x) + 1) + ((a^2 - 2*b^2)*cosh(x)^4 + 4*(a^2 - 2*b^2)*cosh(x )*sinh(x)^3 + (a^2 - 2*b^2)*sinh(x)^4 - 2*(a^2 - 2*b^2)*cosh(x)^2 + 2*(3*( a^2 - 2*b^2)*cosh(x)^2 - a^2 + 2*b^2)*sinh(x)^2 + a^2 - 2*b^2 + 4*((a^2 - 2*b^2)*cosh(x)^3 - (a^2 - 2*b^2)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) + 2*(3*(a^2 - 2*a*b)*cosh(x)^2 + a^2 + 2*a*b)*sinh(x))/(a^3*cosh(x)^4 + 4*a^3*cosh(x)*sinh(x)^3 + a^3*sinh(x)^4 - 2*a^3*cosh(x)^2 + a^3 + 2*(3*a ^3*cosh(x)^2 - a^3)*sinh(x)^2 + 4*(a^3*cosh(x)^3 - a^3*cosh(x))*sinh(x)), -1/2*(2*(a^2 - 2*a*b)*cosh(x)^3 + 6*(a^2 - 2*a*b)*cosh(x)*sinh(x)^2 + 2*(a ^2 - 2*a*b)*sinh(x)^3 + 4*(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x) ^4 - 2*b*cosh(x)^2 + 2*(3*b*cosh(x)^2 - b)*sinh(x)^2 + 4*(b*cosh(x)^3 -...
\[ \int \frac {\text {csch}^3(x)}{a+b \tanh (x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \] Input:
integrate(csch(x)**3/(a+b*tanh(x)),x)
Output:
Integral(csch(x)**3/(a + b*tanh(x)), x)
Exception generated. \[ \int \frac {\text {csch}^3(x)}{a+b \tanh (x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(csch(x)^3/(a+b*tanh(x)),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.52 \[ \int \frac {\text {csch}^3(x)}{a+b \tanh (x)} \, dx=\frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, a^{3}} - \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{2 \, a^{3}} + \frac {2 \, {\left (a^{2} b - b^{3}\right )} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{3}} - \frac {a e^{\left (3 \, x\right )} - 2 \, b e^{\left (3 \, x\right )} + a e^{x} + 2 \, b e^{x}}{a^{2} {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \] Input:
integrate(csch(x)^3/(a+b*tanh(x)),x, algorithm="giac")
Output:
1/2*(a^2 - 2*b^2)*log(e^x + 1)/a^3 - 1/2*(a^2 - 2*b^2)*log(abs(e^x - 1))/a ^3 + 2*(a^2*b - b^3)*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/(sqrt(a^2 - b ^2)*a^3) - (a*e^(3*x) - 2*b*e^(3*x) + a*e^x + 2*b*e^x)/(a^2*(e^(2*x) - 1)^ 2)
Time = 2.48 (sec) , antiderivative size = 506, normalized size of antiderivative = 6.17 \[ \int \frac {\text {csch}^3(x)}{a+b \tanh (x)} \, dx=\frac {\ln \left (8\,a^3\,b-16\,a\,b^3-4\,a^4+8\,b^4+4\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^x+8\,b^4\,{\mathrm {e}}^x-16\,a\,b^3\,{\mathrm {e}}^x+8\,a^3\,b\,{\mathrm {e}}^x+4\,a^2\,b^2\,{\mathrm {e}}^x\right )}{2\,a}-\frac {2\,{\mathrm {e}}^x}{a-2\,a\,{\mathrm {e}}^{2\,x}+a\,{\mathrm {e}}^{4\,x}}-\frac {\ln \left (16\,a\,b^3-8\,a^3\,b+4\,a^4-8\,b^4-4\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^x+8\,b^4\,{\mathrm {e}}^x-16\,a\,b^3\,{\mathrm {e}}^x+8\,a^3\,b\,{\mathrm {e}}^x+4\,a^2\,b^2\,{\mathrm {e}}^x\right )}{2\,a}-\frac {b^2\,\ln \left (8\,a^3\,b-16\,a\,b^3-4\,a^4+8\,b^4+4\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^x+8\,b^4\,{\mathrm {e}}^x-16\,a\,b^3\,{\mathrm {e}}^x+8\,a^3\,b\,{\mathrm {e}}^x+4\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^3}+\frac {b^2\,\ln \left (16\,a\,b^3-8\,a^3\,b+4\,a^4-8\,b^4-4\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^x+8\,b^4\,{\mathrm {e}}^x-16\,a\,b^3\,{\mathrm {e}}^x+8\,a^3\,b\,{\mathrm {e}}^x+4\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^3}-\frac {a\,{\mathrm {e}}^x}{a^2\,{\mathrm {e}}^{2\,x}-a^2}+\frac {2\,b\,{\mathrm {e}}^x}{a^2\,{\mathrm {e}}^{2\,x}-a^2}-\frac {b\,\ln \left (8\,b^2\,\sqrt {b^2-a^2}-8\,b^3\,{\mathrm {e}}^x+8\,a^2\,b\,{\mathrm {e}}^x-8\,a\,b\,\sqrt {b^2-a^2}\right )\,\sqrt {b^2-a^2}}{a^3}+\frac {b\,\ln \left (8\,b^2\,\sqrt {b^2-a^2}+8\,b^3\,{\mathrm {e}}^x-8\,a^2\,b\,{\mathrm {e}}^x-8\,a\,b\,\sqrt {b^2-a^2}\right )\,\sqrt {b^2-a^2}}{a^3} \] Input:
int(1/(sinh(x)^3*(a + b*tanh(x))),x)
Output:
log(8*a^3*b - 16*a*b^3 - 4*a^4 + 8*b^4 + 4*a^2*b^2 - 4*a^4*exp(x) + 8*b^4* exp(x) - 16*a*b^3*exp(x) + 8*a^3*b*exp(x) + 4*a^2*b^2*exp(x))/(2*a) - (2*e xp(x))/(a - 2*a*exp(2*x) + a*exp(4*x)) - log(16*a*b^3 - 8*a^3*b + 4*a^4 - 8*b^4 - 4*a^2*b^2 - 4*a^4*exp(x) + 8*b^4*exp(x) - 16*a*b^3*exp(x) + 8*a^3* b*exp(x) + 4*a^2*b^2*exp(x))/(2*a) - (b^2*log(8*a^3*b - 16*a*b^3 - 4*a^4 + 8*b^4 + 4*a^2*b^2 - 4*a^4*exp(x) + 8*b^4*exp(x) - 16*a*b^3*exp(x) + 8*a^3 *b*exp(x) + 4*a^2*b^2*exp(x)))/a^3 + (b^2*log(16*a*b^3 - 8*a^3*b + 4*a^4 - 8*b^4 - 4*a^2*b^2 - 4*a^4*exp(x) + 8*b^4*exp(x) - 16*a*b^3*exp(x) + 8*a^3 *b*exp(x) + 4*a^2*b^2*exp(x)))/a^3 - (a*exp(x))/(a^2*exp(2*x) - a^2) + (2* b*exp(x))/(a^2*exp(2*x) - a^2) - (b*log(8*b^2*(b^2 - a^2)^(1/2) - 8*b^3*ex p(x) + 8*a^2*b*exp(x) - 8*a*b*(b^2 - a^2)^(1/2))*(b^2 - a^2)^(1/2))/a^3 + (b*log(8*b^2*(b^2 - a^2)^(1/2) + 8*b^3*exp(x) - 8*a^2*b*exp(x) - 8*a*b*(b^ 2 - a^2)^(1/2))*(b^2 - a^2)^(1/2))/a^3
Time = 0.24 (sec) , antiderivative size = 350, normalized size of antiderivative = 4.27 \[ \int \frac {\text {csch}^3(x)}{a+b \tanh (x)} \, dx=\frac {4 e^{4 x} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {a^{2}-b^{2}}}\right ) b -8 e^{2 x} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {a^{2}-b^{2}}}\right ) b +4 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {a^{2}-b^{2}}}\right ) b -e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a^{2}+2 e^{4 x} \mathrm {log}\left (e^{x}-1\right ) b^{2}+e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a^{2}-2 e^{4 x} \mathrm {log}\left (e^{x}+1\right ) b^{2}-2 e^{3 x} a^{2}+4 e^{3 x} a b +2 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{2}-4 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) b^{2}-2 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{2}+4 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) b^{2}-2 e^{x} a^{2}-4 e^{x} a b -\mathrm {log}\left (e^{x}-1\right ) a^{2}+2 \,\mathrm {log}\left (e^{x}-1\right ) b^{2}+\mathrm {log}\left (e^{x}+1\right ) a^{2}-2 \,\mathrm {log}\left (e^{x}+1\right ) b^{2}}{2 a^{3} \left (e^{4 x}-2 e^{2 x}+1\right )} \] Input:
int(csch(x)^3/(a+b*tanh(x)),x)
Output:
(4*e**(4*x)*sqrt(a**2 - b**2)*atan((e**x*a + e**x*b)/sqrt(a**2 - b**2))*b - 8*e**(2*x)*sqrt(a**2 - b**2)*atan((e**x*a + e**x*b)/sqrt(a**2 - b**2))*b + 4*sqrt(a**2 - b**2)*atan((e**x*a + e**x*b)/sqrt(a**2 - b**2))*b - e**(4 *x)*log(e**x - 1)*a**2 + 2*e**(4*x)*log(e**x - 1)*b**2 + e**(4*x)*log(e**x + 1)*a**2 - 2*e**(4*x)*log(e**x + 1)*b**2 - 2*e**(3*x)*a**2 + 4*e**(3*x)* a*b + 2*e**(2*x)*log(e**x - 1)*a**2 - 4*e**(2*x)*log(e**x - 1)*b**2 - 2*e* *(2*x)*log(e**x + 1)*a**2 + 4*e**(2*x)*log(e**x + 1)*b**2 - 2*e**x*a**2 - 4*e**x*a*b - log(e**x - 1)*a**2 + 2*log(e**x - 1)*b**2 + log(e**x + 1)*a** 2 - 2*log(e**x + 1)*b**2)/(2*a**3*(e**(4*x) - 2*e**(2*x) + 1))