Integrand size = 13, antiderivative size = 109 \[ \int \frac {\text {csch}^5(x)}{a+b \coth (x)} \, dx=-\frac {a \text {arctanh}(\cosh (x))}{2 b^2}+\frac {a \left (a^2-b^2\right ) \text {arctanh}(\cosh (x))}{b^4}-\frac {\left (a^2-b^2\right )^{3/2} \text {arctanh}\left (\frac {(b+a \coth (x)) \sinh (x)}{\sqrt {a^2-b^2}}\right )}{b^4}-\frac {\left (a^2-b^2\right ) \text {csch}(x)}{b^3}+\frac {a \coth (x) \text {csch}(x)}{2 b^2}-\frac {\text {csch}^3(x)}{3 b} \] Output:
-1/2*a*arctanh(cosh(x))/b^2+a*(a^2-b^2)*arctanh(cosh(x))/b^4-(a^2-b^2)^(3/ 2)*arctanh((b+a*coth(x))*sinh(x)/(a^2-b^2)^(1/2))/b^4-(a^2-b^2)*csch(x)/b^ 3+1/2*a*coth(x)*csch(x)/b^2-1/3*csch(x)^3/b
Leaf count is larger than twice the leaf count of optimal. \(259\) vs. \(2(109)=218\).
Time = 0.72 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.38 \[ \int \frac {\text {csch}^5(x)}{a+b \coth (x)} \, dx=\frac {-96 a^2 \sqrt {-a+b} \sqrt {a+b} \arctan \left (\frac {a+b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a+b} \sqrt {a+b}}\right )+96 b^2 \sqrt {-a+b} \sqrt {a+b} \arctan \left (\frac {a+b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a+b} \sqrt {a+b}}\right )+4 b \left (-6 a^2+7 b^2\right ) \coth \left (\frac {x}{2}\right )+6 a b^2 \text {csch}^2\left (\frac {x}{2}\right )+48 a^3 \log \left (\cosh \left (\frac {x}{2}\right )\right )-72 a b^2 \log \left (\cosh \left (\frac {x}{2}\right )\right )-48 a^3 \log \left (\sinh \left (\frac {x}{2}\right )\right )+72 a b^2 \log \left (\sinh \left (\frac {x}{2}\right )\right )+6 a b^2 \text {sech}^2\left (\frac {x}{2}\right )-16 b^3 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-b^3 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)+24 a^2 b \tanh \left (\frac {x}{2}\right )-28 b^3 \tanh \left (\frac {x}{2}\right )}{48 b^4} \] Input:
Integrate[Csch[x]^5/(a + b*Coth[x]),x]
Output:
(-96*a^2*Sqrt[-a + b]*Sqrt[a + b]*ArcTan[(a + b*Tanh[x/2])/(Sqrt[-a + b]*S qrt[a + b])] + 96*b^2*Sqrt[-a + b]*Sqrt[a + b]*ArcTan[(a + b*Tanh[x/2])/(S qrt[-a + b]*Sqrt[a + b])] + 4*b*(-6*a^2 + 7*b^2)*Coth[x/2] + 6*a*b^2*Csch[ x/2]^2 + 48*a^3*Log[Cosh[x/2]] - 72*a*b^2*Log[Cosh[x/2]] - 48*a^3*Log[Sinh [x/2]] + 72*a*b^2*Log[Sinh[x/2]] + 6*a*b^2*Sech[x/2]^2 - 16*b^3*Csch[x]^3* Sinh[x/2]^4 - b^3*Csch[x/2]^4*Sinh[x] + 24*a^2*b*Tanh[x/2] - 28*b^3*Tanh[x /2])/(48*b^4)
Result contains complex when optimal does not.
Time = 0.97 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.923, Rules used = {3042, 26, 3989, 26, 3042, 26, 3967, 26, 3042, 26, 3989, 26, 3042, 26, 3967, 26, 3042, 26, 3988, 219, 4255, 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.
| \(\displaystyle \int \frac {\text {csch}^5(x)}{a+b \coth (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sec \left (-\frac {\pi }{2}+i x\right )^5}{a-i b \tan \left (-\frac {\pi }{2}+i x\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sec \left (i x-\frac {\pi }{2}\right )^5}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3989 |
| \(\displaystyle i \left (\frac {\int i (a-b \coth (x)) \text {csch}^3(x)dx}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {i \text {csch}^3(x)}{a+b \coth (x)}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i \int (a-b \coth (x)) \text {csch}^3(x)dx}{b^2}-\frac {i \left (a^2-b^2\right ) \int \frac {\text {csch}^3(x)}{a+b \coth (x)}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i \int -i \sec \left (i x-\frac {\pi }{2}\right )^3 \left (a+i b \tan \left (i x-\frac {\pi }{2}\right )\right )dx}{b^2}-\frac {i \left (a^2-b^2\right ) \int -\frac {i \sec \left (i x-\frac {\pi }{2}\right )^3}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^3 \left (a+i b \tan \left (i x-\frac {\pi }{2}\right )\right )dx}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )^3}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle i \left (\frac {a \int i \text {csch}^3(x)dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )^3}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a \int \text {csch}^3(x)dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )^3}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a \int -i \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )^3}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )^3}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )\) |
| \(\Big \downarrow \) 3989 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {\int -i (a-b \coth (x)) \text {csch}(x)dx}{b^2}-\frac {\left (a^2-b^2\right ) \int -\frac {i \text {csch}(x)}{a+b \coth (x)}dx}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {i \left (a^2-b^2\right ) \int \frac {\text {csch}(x)}{a+b \coth (x)}dx}{b^2}-\frac {i \int (a-b \coth (x)) \text {csch}(x)dx}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {i \left (a^2-b^2\right ) \int \frac {i \sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}-\frac {i \int i \sec \left (i x-\frac {\pi }{2}\right ) \left (a+i b \tan \left (i x-\frac {\pi }{2}\right )\right )dx}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right ) \left (a+i b \tan \left (i x-\frac {\pi }{2}\right )\right )dx}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {a \int -i \text {csch}(x)dx-i b \text {csch}(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {-i a \int \text {csch}(x)dx-i b \text {csch}(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {-i a \int i \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {a \int \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 3988 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {a \int \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {i \left (a^2-b^2\right ) \int \frac {1}{a^2-b^2-(b+a \coth (x))^2 \sinh ^2(x)}d((b+a \coth (x)) \sinh (x))}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle i \left (\frac {a \int \csc (i x)^3dx+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {a \int \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {i \sqrt {a^2-b^2} \text {arctanh}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle i \left (\frac {a \left (\frac {1}{2} \int -i \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {a \int \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {i \sqrt {a^2-b^2} \text {arctanh}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {a \left (-\frac {1}{2} i \int \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {a \int \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {i \sqrt {a^2-b^2} \text {arctanh}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {a \left (-\frac {1}{2} i \int i \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {a \int \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {i \sqrt {a^2-b^2} \text {arctanh}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {a \left (\frac {1}{2} \int \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {a \int \csc (i x)dx-i b \text {csch}(x)}{b^2}-\frac {i \sqrt {a^2-b^2} \text {arctanh}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}\right )}{b^2}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle i \left (\frac {a \left (\frac {1}{2} i \text {arctanh}(\cosh (x))-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {1}{3} i b \text {csch}^3(x)}{b^2}-\frac {\left (a^2-b^2\right ) \left (\frac {i a \text {arctanh}(\cosh (x))-i b \text {csch}(x)}{b^2}-\frac {i \sqrt {a^2-b^2} \text {arctanh}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}\right )}{b^2}\right )\) |
Input:
Int[Csch[x]^5/(a + b*Coth[x]),x]
Output:
I*(-(((a^2 - b^2)*(((-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))/b^2) + ((I/3)*b*Csch[x]^3 + a*((I/2)*ArcTanh[Cosh[x]] - (I/2)*Coth[x]*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_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 3.14 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.60
| method | result | size |
| default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{3} b^{2}}{3}-\tanh \left (\frac {x}{2}\right )^{2} a b +4 \tanh \left (\frac {x}{2}\right ) a^{2}-5 \tanh \left (\frac {x}{2}\right ) b^{2}}{8 b^{3}}+\frac {\left (16 a^{4}-32 a^{2} b^{2}+16 b^{4}\right ) \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{8 b^{4} \sqrt {-a^{2}+b^{2}}}-\frac {1}{24 b \tanh \left (\frac {x}{2}\right )^{3}}-\frac {4 a^{2}-5 b^{2}}{8 b^{3} \tanh \left (\frac {x}{2}\right )}+\frac {a}{8 b^{2} \tanh \left (\frac {x}{2}\right )^{2}}-\frac {a \left (2 a^{2}-3 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2 b^{4}}\) | \(174\) |
| risch | \(-\frac {{\mathrm e}^{x} \left (6 \,{\mathrm e}^{4 x} a^{2}-3 \,{\mathrm e}^{4 x} a b -6 b^{2} {\mathrm e}^{4 x}-12 \,{\mathrm e}^{2 x} a^{2}+20 b^{2} {\mathrm e}^{2 x}+6 a^{2}+3 a b -6 b^{2}\right )}{3 b^{3} \left ({\mathrm e}^{2 x}-1\right )^{3}}+\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {\sqrt {a^{2}-b^{2}}}{a +b}\right ) a^{2}}{b^{4}}-\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {\sqrt {a^{2}-b^{2}}}{a +b}\right )}{b^{2}}-\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {\sqrt {a^{2}-b^{2}}}{a +b}\right ) a^{2}}{b^{4}}+\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {\sqrt {a^{2}-b^{2}}}{a +b}\right )}{b^{2}}+\frac {a^{3} \ln \left (1+{\mathrm e}^{x}\right )}{b^{4}}-\frac {3 a \ln \left (1+{\mathrm e}^{x}\right )}{2 b^{2}}-\frac {a^{3} \ln \left ({\mathrm e}^{x}-1\right )}{b^{4}}+\frac {3 a \ln \left ({\mathrm e}^{x}-1\right )}{2 b^{2}}\) | \(277\) |
Input:
int(csch(x)^5/(a+b*coth(x)),x,method=_RETURNVERBOSE)
Output:
1/8/b^3*(1/3*tanh(1/2*x)^3*b^2-tanh(1/2*x)^2*a*b+4*tanh(1/2*x)*a^2-5*tanh( 1/2*x)*b^2)+1/8/b^4*(16*a^4-32*a^2*b^2+16*b^4)/(-a^2+b^2)^(1/2)*arctan(1/2 *(2*tanh(1/2*x)*b+2*a)/(-a^2+b^2)^(1/2))-1/24/b/tanh(1/2*x)^3-1/8*(4*a^2-5 *b^2)/b^3/tanh(1/2*x)+1/8/b^2*a/tanh(1/2*x)^2-1/2/b^4*a*(2*a^2-3*b^2)*ln(t anh(1/2*x))
Leaf count of result is larger than twice the leaf count of optimal. 1296 vs. \(2 (99) = 198\).
Time = 0.17 (sec) , antiderivative size = 2647, normalized size of antiderivative = 24.28 \[ \int \frac {\text {csch}^5(x)}{a+b \coth (x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^5/(a+b*coth(x)),x, algorithm="fricas")
Output:
[-1/6*(6*(2*a^2*b - a*b^2 - 2*b^3)*cosh(x)^5 + 30*(2*a^2*b - a*b^2 - 2*b^3 )*cosh(x)*sinh(x)^4 + 6*(2*a^2*b - a*b^2 - 2*b^3)*sinh(x)^5 - 8*(3*a^2*b - 5*b^3)*cosh(x)^3 - 4*(6*a^2*b - 10*b^3 - 15*(2*a^2*b - a*b^2 - 2*b^3)*cos h(x)^2)*sinh(x)^3 + 12*(5*(2*a^2*b - a*b^2 - 2*b^3)*cosh(x)^3 - 2*(3*a^2*b - 5*b^3)*cosh(x))*sinh(x)^2 + 6*((a^2 - b^2)*cosh(x)^6 + 6*(a^2 - b^2)*co sh(x)*sinh(x)^5 + (a^2 - b^2)*sinh(x)^6 - 3*(a^2 - b^2)*cosh(x)^4 + 3*(5*( a^2 - b^2)*cosh(x)^2 - a^2 + b^2)*sinh(x)^4 + 4*(5*(a^2 - b^2)*cosh(x)^3 - 3*(a^2 - b^2)*cosh(x))*sinh(x)^3 + 3*(a^2 - b^2)*cosh(x)^2 + 3*(5*(a^2 - b^2)*cosh(x)^4 - 6*(a^2 - b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^2 - a^2 + b^ 2 + 6*((a^2 - b^2)*cosh(x)^5 - 2*(a^2 - b^2)*cosh(x)^3 + (a^2 - b^2)*cosh( x))*sinh(x))*sqrt(a^2 - b^2)*log(((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*si nh(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)) + 6*(2*a^2*b + a*b^2 - 2*b^3)*cosh(x) - 3*((2*a^3 - 3*a*b^2)*cosh(x)^6 + 6*(2*a^3 - 3*a*b^2)*cosh(x)*sinh(x)^5 + (2*a^3 - 3*a*b^2)*sinh(x)^6 - 3 *(2*a^3 - 3*a*b^2)*cosh(x)^4 - 3*(2*a^3 - 3*a*b^2 - 5*(2*a^3 - 3*a*b^2)*co sh(x)^2)*sinh(x)^4 + 4*(5*(2*a^3 - 3*a*b^2)*cosh(x)^3 - 3*(2*a^3 - 3*a*b^2 )*cosh(x))*sinh(x)^3 - 2*a^3 + 3*a*b^2 + 3*(2*a^3 - 3*a*b^2)*cosh(x)^2 + 3 *(5*(2*a^3 - 3*a*b^2)*cosh(x)^4 + 2*a^3 - 3*a*b^2 - 6*(2*a^3 - 3*a*b^2)*co sh(x)^2)*sinh(x)^2 + 6*((2*a^3 - 3*a*b^2)*cosh(x)^5 - 2*(2*a^3 - 3*a*b^...
\[ \int \frac {\text {csch}^5(x)}{a+b \coth (x)} \, dx=\int \frac {\operatorname {csch}^{5}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \] Input:
integrate(csch(x)**5/(a+b*coth(x)),x)
Output:
Integral(csch(x)**5/(a + b*coth(x)), x)
Exception generated. \[ \int \frac {\text {csch}^5(x)}{a+b \coth (x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(csch(x)^5/(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 = 0.14 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.62 \[ \int \frac {\text {csch}^5(x)}{a+b \coth (x)} \, dx=\frac {{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, b^{4}} - \frac {{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{2 \, b^{4}} + \frac {2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b^{4}} - \frac {6 \, a^{2} e^{\left (5 \, x\right )} - 3 \, a b e^{\left (5 \, x\right )} - 6 \, b^{2} e^{\left (5 \, x\right )} - 12 \, a^{2} e^{\left (3 \, x\right )} + 20 \, b^{2} e^{\left (3 \, x\right )} + 6 \, a^{2} e^{x} + 3 \, a b e^{x} - 6 \, b^{2} e^{x}}{3 \, b^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \] Input:
integrate(csch(x)^5/(a+b*coth(x)),x, algorithm="giac")
Output:
1/2*(2*a^3 - 3*a*b^2)*log(e^x + 1)/b^4 - 1/2*(2*a^3 - 3*a*b^2)*log(abs(e^x - 1))/b^4 + 2*(a^4 - 2*a^2*b^2 + b^4)*arctan((a*e^x + b*e^x)/sqrt(-a^2 + b^2))/(sqrt(-a^2 + b^2)*b^4) - 1/3*(6*a^2*e^(5*x) - 3*a*b*e^(5*x) - 6*b^2* e^(5*x) - 12*a^2*e^(3*x) + 20*b^2*e^(3*x) + 6*a^2*e^x + 3*a*b*e^x - 6*b^2* e^x)/(b^3*(e^(2*x) - 1)^3)
Time = 3.40 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.36 \[ \int \frac {\text {csch}^5(x)}{a+b \coth (x)} \, dx=\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (3\,a\,b^2-2\,a^3\right )}{2\,b^4}-\frac {8\,{\mathrm {e}}^x}{3\,b\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (3\,a\,b^2-2\,a^3\right )}{2\,b^4}-\frac {\ln \left (\sqrt {{\left (a+b\right )}^3\,{\left (a-b\right )}^3}+a^3\,{\mathrm {e}}^x-b^3\,{\mathrm {e}}^x-a\,b^2\,{\mathrm {e}}^x+a^2\,b\,{\mathrm {e}}^x\right )\,\sqrt {{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{b^4}+\frac {\ln \left (\sqrt {{\left (a+b\right )}^3\,{\left (a-b\right )}^3}-a^3\,{\mathrm {e}}^x+b^3\,{\mathrm {e}}^x+a\,b^2\,{\mathrm {e}}^x-a^2\,b\,{\mathrm {e}}^x\right )\,\sqrt {{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{b^4}+\frac {2\,{\mathrm {e}}^x\,\left (3\,a-4\,b\right )}{3\,b^2\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}+\frac {{\mathrm {e}}^x\,\left (-2\,a^2+a\,b+2\,b^2\right )}{b^3\,\left ({\mathrm {e}}^{2\,x}-1\right )} \] Input:
int(1/(sinh(x)^5*(a + b*coth(x))),x)
Output:
(log(exp(x) - 1)*(3*a*b^2 - 2*a^3))/(2*b^4) - (8*exp(x))/(3*b*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (log(exp(x) + 1)*(3*a*b^2 - 2*a^3))/(2*b^4 ) - (log(((a + b)^3*(a - b)^3)^(1/2) + a^3*exp(x) - b^3*exp(x) - a*b^2*exp (x) + a^2*b*exp(x))*((a + b)^3*(a - b)^3)^(1/2))/b^4 + (log(((a + b)^3*(a - b)^3)^(1/2) - a^3*exp(x) + b^3*exp(x) + a*b^2*exp(x) - a^2*b*exp(x))*((a + b)^3*(a - b)^3)^(1/2))/b^4 + (2*exp(x)*(3*a - 4*b))/(3*b^2*(exp(4*x) - 2*exp(2*x) + 1)) + (exp(x)*(a*b - 2*a^2 + 2*b^2))/(b^3*(exp(2*x) - 1))
Time = 0.20 (sec) , antiderivative size = 702, normalized size of antiderivative = 6.44 \[ \int \frac {\text {csch}^5(x)}{a+b \coth (x)} \, dx=\frac {12 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}-12 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {-a^{2}+b^{2}}}\right ) b^{2}-6 e^{6 x} \mathrm {log}\left (e^{x}-1\right ) a^{3}+6 e^{6 x} \mathrm {log}\left (e^{x}+1\right ) a^{3}-12 e^{5 x} a^{2} b +6 e^{5 x} a \,b^{2}+18 e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a^{3}-18 e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a^{3}+24 e^{3 x} a^{2} b -18 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{3}+27 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a \,b^{2}-27 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a \,b^{2}-12 e^{6 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}+12 e^{6 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {-a^{2}+b^{2}}}\right ) b^{2}+36 e^{4 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}-36 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^{2}-36 e^{2 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}+36 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^{2}+9 e^{6 x} \mathrm {log}\left (e^{x}-1\right ) a \,b^{2}-9 e^{6 x} \mathrm {log}\left (e^{x}+1\right ) a \,b^{2}-27 e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a \,b^{2}+27 e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a \,b^{2}+12 e^{5 x} b^{3}-40 e^{3 x} b^{3}+12 e^{x} b^{3}+6 \,\mathrm {log}\left (e^{x}-1\right ) a^{3}-6 \,\mathrm {log}\left (e^{x}+1\right ) a^{3}+18 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{3}-12 e^{x} a^{2} b -6 e^{x} a \,b^{2}-9 \,\mathrm {log}\left (e^{x}-1\right ) a \,b^{2}+9 \,\mathrm {log}\left (e^{x}+1\right ) a \,b^{2}}{6 b^{4} \left (e^{6 x}-3 e^{4 x}+3 e^{2 x}-1\right )} \] Input:
int(csch(x)^5/(a+b*coth(x)),x)
Output:
( - 12*e**(6*x)*sqrt( - a**2 + b**2)*atan((e**x*a + e**x*b)/sqrt( - a**2 + b**2))*a**2 + 12*e**(6*x)*sqrt( - a**2 + b**2)*atan((e**x*a + e**x*b)/sqr t( - a**2 + b**2))*b**2 + 36*e**(4*x)*sqrt( - a**2 + b**2)*atan((e**x*a + e**x*b)/sqrt( - a**2 + b**2))*a**2 - 36*e**(4*x)*sqrt( - a**2 + b**2)*atan ((e**x*a + e**x*b)/sqrt( - a**2 + b**2))*b**2 - 36*e**(2*x)*sqrt( - a**2 + b**2)*atan((e**x*a + e**x*b)/sqrt( - a**2 + b**2))*a**2 + 36*e**(2*x)*sqr t( - a**2 + b**2)*atan((e**x*a + e**x*b)/sqrt( - a**2 + b**2))*b**2 + 12*s qrt( - a**2 + b**2)*atan((e**x*a + e**x*b)/sqrt( - a**2 + b**2))*a**2 - 12 *sqrt( - a**2 + b**2)*atan((e**x*a + e**x*b)/sqrt( - a**2 + b**2))*b**2 - 6*e**(6*x)*log(e**x - 1)*a**3 + 9*e**(6*x)*log(e**x - 1)*a*b**2 + 6*e**(6* x)*log(e**x + 1)*a**3 - 9*e**(6*x)*log(e**x + 1)*a*b**2 - 12*e**(5*x)*a**2 *b + 6*e**(5*x)*a*b**2 + 12*e**(5*x)*b**3 + 18*e**(4*x)*log(e**x - 1)*a**3 - 27*e**(4*x)*log(e**x - 1)*a*b**2 - 18*e**(4*x)*log(e**x + 1)*a**3 + 27* e**(4*x)*log(e**x + 1)*a*b**2 + 24*e**(3*x)*a**2*b - 40*e**(3*x)*b**3 - 18 *e**(2*x)*log(e**x - 1)*a**3 + 27*e**(2*x)*log(e**x - 1)*a*b**2 + 18*e**(2 *x)*log(e**x + 1)*a**3 - 27*e**(2*x)*log(e**x + 1)*a*b**2 - 12*e**x*a**2*b - 6*e**x*a*b**2 + 12*e**x*b**3 + 6*log(e**x - 1)*a**3 - 9*log(e**x - 1)*a *b**2 - 6*log(e**x + 1)*a**3 + 9*log(e**x + 1)*a*b**2)/(6*b**4*(e**(6*x) - 3*e**(4*x) + 3*e**(2*x) - 1))