Integrand size = 13, antiderivative size = 57 \[ \int \frac {\text {csch}^3(x)}{a+b \coth (x)} \, dx=\frac {a \text {arctanh}(\cosh (x))}{b^2}-\frac {\sqrt {a^2-b^2} \text {arctanh}\left (\frac {(b+a \coth (x)) \sinh (x)}{\sqrt {a^2-b^2}}\right )}{b^2}-\frac {\text {csch}(x)}{b} \] Output:
a*arctanh(cosh(x))/b^2-(a^2-b^2)^(1/2)*arctanh((b+a*coth(x))*sinh(x)/(a^2- b^2)^(1/2))/b^2-csch(x)/b
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.32 \[ \int \frac {\text {csch}^3(x)}{a+b \coth (x)} \, dx=-\frac {2 \sqrt {-a+b} \sqrt {a+b} \arctan \left (\frac {a+b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a+b} \sqrt {a+b}}\right )+b \text {csch}(x)+a \left (-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )}{b^2} \] Input:
Integrate[Csch[x]^3/(a + b*Coth[x]),x]
Output:
-((2*Sqrt[-a + b]*Sqrt[a + b]*ArcTan[(a + b*Tanh[x/2])/(Sqrt[-a + b]*Sqrt[ a + b])] + b*Csch[x] + a*(-Log[Cosh[x/2]] + Log[Sinh[x/2]]))/b^2)
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 26, 3989, 26, 3042, 26, 3967, 26, 3042, 26, 3988, 219, 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.
| \(\displaystyle \int \frac {\text {csch}^3(x)}{a+b \coth (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sec \left (-\frac {\pi }{2}+i x\right )^3}{a-i b \tan \left (-\frac {\pi }{2}+i x\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sec \left (i x-\frac {\pi }{2}\right )^3}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3989 |
| \(\displaystyle -i \left (\frac {\int -i (a-b \coth (x)) \text {csch}(x)dx}{b^2}-\frac {\left (a^2-b^2\right ) \int -\frac {i \text {csch}(x)}{a+b \coth (x)}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \left (a^2-b^2\right ) \int \frac {\text {csch}(x)}{a+b \coth (x)}dx}{b^2}-\frac {i \int (a-b \coth (x)) \text {csch}(x)dx}{b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i \left (a^2-b^2\right ) \int \frac {i \sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}-\frac {i \int i \sec \left (i x-\frac {\pi }{2}\right ) \left (a+i b \tan \left (i x-\frac {\pi }{2}\right )\right )dx}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right ) \left (a+i b \tan \left (i x-\frac {\pi }{2}\right )\right )dx}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle -i \left (\frac {a \int -i \text {csch}(x)dx-i b \text {csch}(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {-i a \int \text {csch}(x)dx-i b \text {csch}(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {-i a \int i \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {a \int \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 3988 |
| \(\displaystyle -i \left (\frac {a \int \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {i \left (a^2-b^2\right ) \int \frac {1}{a^2-b^2-(b+a \coth (x))^2 \sinh ^2(x)}d((b+a \coth (x)) \sinh (x))}{b^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -i \left (\frac {a \int \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {i \sqrt {a^2-b^2} \text {arctanh}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -i \left (\frac {i a \text {arctanh}(\cosh (x))-i b \text {csch}(x)}{b^2}-\frac {i \sqrt {a^2-b^2} \text {arctanh}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}\right )\) |
Input:
Int[Csch[x]^3/(a + b*Coth[x]),x]
Output:
(-I)*(((-I)*Sqrt[a^2 - b^2]*ArcTanh[((b + a*Coth[x])*Sinh[x])/Sqrt[a^2 - b ^2]])/b^2 + (I*a*ArcTanh[Cosh[x]] - I*b*Csch[x])/b^2)
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
Int[sec[(e_.) + (f_.)*(x_)]/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbo l] :> Simp[-f^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, (b - a*Tan[e + f *x])/Sec[e + f*x]], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_ Symbol] :> Simp[-(b^2)^(-1) Int[Sec[e + f*x]^(m - 2)*(a - b*Tan[e + f*x]) , x], x] + Simp[(a^2 + b^2)/b^2 Int[Sec[e + f*x]^(m - 2)/(a + b*Tan[e + f *x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0] && IGtQ[(m - 1) /2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.56 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(\frac {\tanh \left (\frac {x}{2}\right )}{2 b}+\frac {\left (4 a^{2}-4 b^{2}\right ) \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{2 b^{2} \sqrt {-a^{2}+b^{2}}}-\frac {1}{2 b \tanh \left (\frac {x}{2}\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{2}}\) | \(85\) |
| risch | \(-\frac {2 \,{\mathrm e}^{x}}{b \left ({\mathrm e}^{2 x}-1\right )}+\frac {a \ln \left (1+{\mathrm e}^{x}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{x}-1\right )}{b^{2}}+\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {\sqrt {a^{2}-b^{2}}}{a +b}\right )}{b^{2}}-\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {\sqrt {a^{2}-b^{2}}}{a +b}\right )}{b^{2}}\) | \(112\) |
Input:
int(csch(x)^3/(a+b*coth(x)),x,method=_RETURNVERBOSE)
Output:
1/2*tanh(1/2*x)/b+1/2/b^2*(4*a^2-4*b^2)/(-a^2+b^2)^(1/2)*arctan(1/2*(2*tan h(1/2*x)*b+2*a)/(-a^2+b^2)^(1/2))-1/2/b/tanh(1/2*x)-1/b^2*a*ln(tanh(1/2*x) )
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (53) = 106\).
Time = 0.11 (sec) , antiderivative size = 384, normalized size of antiderivative = 6.74 \[ \int \frac {\text {csch}^3(x)}{a+b \coth (x)} \, dx=\left [\frac {\sqrt {a^{2} - b^{2}} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + a - b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - a + b}\right ) - 2 \, b \cosh \left (x\right ) + {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, b \sinh \left (x\right )}{b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2} - b^{2}}, \frac {2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} + b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right ) - 2 \, b \cosh \left (x\right ) + {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, b \sinh \left (x\right )}{b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2} - b^{2}}\right ] \] Input:
integrate(csch(x)^3/(a+b*coth(x)),x, algorithm="fricas")
Output:
[(sqrt(a^2 - b^2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*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*b*cosh(x) + (a*cosh(x)^2 + 2* a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)*log(cosh(x) + sinh(x) + 1) - (a*cosh( x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)*log(cosh(x) + sinh(x) - 1) - 2*b*sinh(x))/(b^2*cosh(x)^2 + 2*b^2*cosh(x)*sinh(x) + b^2*sinh(x)^2 - b^2 ), (2*sqrt(-a^2 + b^2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*arc tan(sqrt(-a^2 + b^2)/((a + b)*cosh(x) + (a + b)*sinh(x))) - 2*b*cosh(x) + (a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)*log(cosh(x) + sinh(x ) + 1) - (a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)*log(cosh(x) + sinh(x) - 1) - 2*b*sinh(x))/(b^2*cosh(x)^2 + 2*b^2*cosh(x)*sinh(x) + b^ 2*sinh(x)^2 - b^2)]
\[ \int \frac {\text {csch}^3(x)}{a+b \coth (x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \] Input:
integrate(csch(x)**3/(a+b*coth(x)),x)
Output:
Integral(csch(x)**3/(a + b*coth(x)), x)
Exception generated. \[ \int \frac {\text {csch}^3(x)}{a+b \coth (x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(csch(x)^3/(a+b*coth(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*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.49 \[ \int \frac {\text {csch}^3(x)}{a+b \coth (x)} \, dx=\frac {a \log \left (e^{x} + 1\right )}{b^{2}} - \frac {a \log \left ({\left | e^{x} - 1 \right |}\right )}{b^{2}} + \frac {2 \, {\left (a^{2} - b^{2}\right )} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b^{2}} - \frac {2 \, e^{x}}{b {\left (e^{\left (2 \, x\right )} - 1\right )}} \] Input:
integrate(csch(x)^3/(a+b*coth(x)),x, algorithm="giac")
Output:
a*log(e^x + 1)/b^2 - a*log(abs(e^x - 1))/b^2 + 2*(a^2 - b^2)*arctan((a*e^x + b*e^x)/sqrt(-a^2 + b^2))/(sqrt(-a^2 + b^2)*b^2) - 2*e^x/(b*(e^(2*x) - 1 ))
Time = 2.75 (sec) , antiderivative size = 230, normalized size of antiderivative = 4.04 \[ \int \frac {\text {csch}^3(x)}{a+b \coth (x)} \, dx=\frac {2\,{\mathrm {e}}^x}{b-b\,{\mathrm {e}}^{2\,x}}-\frac {a\,\ln \left (32\,a\,b^2-64\,a^2\,b+32\,a^3-32\,a^3\,{\mathrm {e}}^x-32\,a\,b^2\,{\mathrm {e}}^x+64\,a^2\,b\,{\mathrm {e}}^x\right )}{b^2}+\frac {a\,\ln \left (32\,a\,b^2-64\,a^2\,b+32\,a^3+32\,a^3\,{\mathrm {e}}^x+32\,a\,b^2\,{\mathrm {e}}^x-64\,a^2\,b\,{\mathrm {e}}^x\right )}{b^2}+\frac {\ln \left (32\,a\,\sqrt {a^2-b^2}-32\,b\,\sqrt {a^2-b^2}-32\,a^2\,{\mathrm {e}}^x+32\,b^2\,{\mathrm {e}}^x\right )\,\sqrt {a^2-b^2}}{b^2}-\frac {\ln \left (32\,a\,\sqrt {a^2-b^2}-32\,b\,\sqrt {a^2-b^2}+32\,a^2\,{\mathrm {e}}^x-32\,b^2\,{\mathrm {e}}^x\right )\,\sqrt {a^2-b^2}}{b^2} \] Input:
int(1/(sinh(x)^3*(a + b*coth(x))),x)
Output:
(2*exp(x))/(b - b*exp(2*x)) - (a*log(32*a*b^2 - 64*a^2*b + 32*a^3 - 32*a^3 *exp(x) - 32*a*b^2*exp(x) + 64*a^2*b*exp(x)))/b^2 + (a*log(32*a*b^2 - 64*a ^2*b + 32*a^3 + 32*a^3*exp(x) + 32*a*b^2*exp(x) - 64*a^2*b*exp(x)))/b^2 + (log(32*a*(a^2 - b^2)^(1/2) - 32*b*(a^2 - b^2)^(1/2) - 32*a^2*exp(x) + 32* b^2*exp(x))*(a^2 - b^2)^(1/2))/b^2 - (log(32*a*(a^2 - b^2)^(1/2) - 32*b*(a ^2 - b^2)^(1/2) + 32*a^2*exp(x) - 32*b^2*exp(x))*(a^2 - b^2)^(1/2))/b^2
Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.51 \[ \int \frac {\text {csch}^3(x)}{a+b \coth (x)} \, dx=\frac {-2 e^{2 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {-a^{2}+b^{2}}}\right )+2 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {-a^{2}+b^{2}}}\right )-e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a +e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a -2 e^{x} b +\mathrm {log}\left (e^{x}-1\right ) a -\mathrm {log}\left (e^{x}+1\right ) a}{b^{2} \left (e^{2 x}-1\right )} \] Input:
int(csch(x)^3/(a+b*coth(x)),x)
Output:
( - 2*e**(2*x)*sqrt( - a**2 + b**2)*atan((e**x*a + e**x*b)/sqrt( - a**2 + b**2)) + 2*sqrt( - a**2 + b**2)*atan((e**x*a + e**x*b)/sqrt( - a**2 + b**2 )) - e**(2*x)*log(e**x - 1)*a + e**(2*x)*log(e**x + 1)*a - 2*e**x*b + log( e**x - 1)*a - log(e**x + 1)*a)/(b**2*(e**(2*x) - 1))