Integrand size = 13, antiderivative size = 59 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {a \text {arctanh}(\cosh (x))}{b^2}-\frac {2 a^2 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}-\frac {\coth (x)}{b} \]
a*arctanh(cosh(x))/b^2-coth(x)/b-2*a^2*arctanh((a-b*tanh(1/2*x))/(a^2+b^2) ^(1/2))/b^2/(a^2+b^2)^(1/2)
Time = 0.48 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.56 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {\text {csch}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \left (-b \cosh (x)+a \left (\frac {2 a \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \sinh (x)\right )}{2 b^2} \]
(Csch[x/2]*Sech[x/2]*(-(b*Cosh[x]) + a*((2*a*ArcTan[(a - b*Tanh[x/2])/Sqrt [-a^2 - b^2]])/Sqrt[-a^2 - b^2] + Log[Cosh[x/2]] - Log[Sinh[x/2]])*Sinh[x] ))/(2*b^2)
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.34, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.231, Rules used = {3042, 26, 4277, 25, 3042, 25, 4276, 26, 3042, 26, 4257, 4318, 3042, 3139, 1083, 219}
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.
| \(\displaystyle \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \csc (i x)^3}{a+i b \csc (i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\csc (i x)^3}{a+i b \csc (i x)}dx\) |
| \(\Big \downarrow \) 4277 |
| \(\displaystyle -i \left (\frac {i a \int -\frac {\text {csch}^2(x)}{a+b \text {csch}(x)}dx}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (-\frac {i a \int \frac {\text {csch}^2(x)}{a+b \text {csch}(x)}dx}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-\frac {i a \int -\frac {\csc (i x)^2}{a+i b \csc (i x)}dx}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {i a \int \frac {\csc (i x)^2}{a+i b \csc (i x)}dx}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 4276 |
| \(\displaystyle -i \left (\frac {i a \left (\frac {i a \int -\frac {i \text {csch}(x)}{a+b \text {csch}(x)}dx}{b}-\frac {i \int -i \text {csch}(x)dx}{b}\right )}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i a \left (\frac {a \int \frac {\text {csch}(x)}{a+b \text {csch}(x)}dx}{b}-\frac {\int \text {csch}(x)dx}{b}\right )}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i a \left (\frac {a \int \frac {i \csc (i x)}{a+i b \csc (i x)}dx}{b}-\frac {\int i \csc (i x)dx}{b}\right )}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i a \left (\frac {i a \int \frac {\csc (i x)}{a+i b \csc (i x)}dx}{b}-\frac {i \int \csc (i x)dx}{b}\right )}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -i \left (\frac {i a \left (\frac {\text {arctanh}(\cosh (x))}{b}+\frac {i a \int \frac {\csc (i x)}{a+i b \csc (i x)}dx}{b}\right )}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle -i \left (\frac {i a \left (\frac {a \int \frac {1}{\frac {a \sinh (x)}{b}+1}dx}{b^2}+\frac {\text {arctanh}(\cosh (x))}{b}\right )}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i a \left (\frac {\text {arctanh}(\cosh (x))}{b}+\frac {a \int \frac {1}{1-\frac {i a \sin (i x)}{b}}dx}{b^2}\right )}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -i \left (\frac {i a \left (\frac {2 a \int \frac {1}{-\tanh ^2\left (\frac {x}{2}\right )+\frac {2 a \tanh \left (\frac {x}{2}\right )}{b}+1}d\tanh \left (\frac {x}{2}\right )}{b^2}+\frac {\text {arctanh}(\cosh (x))}{b}\right )}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -i \left (\frac {i a \left (\frac {\text {arctanh}(\cosh (x))}{b}-\frac {4 a \int \frac {1}{4 \left (\frac {a^2}{b^2}+1\right )-\left (\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )^2}d\left (\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )}{b^2}\right )}{b}-\frac {i \coth (x)}{b}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -i \left (\frac {i a \left (\frac {\text {arctanh}(\cosh (x))}{b}-\frac {2 a \text {arctanh}\left (\frac {b \left (\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )}{2 \sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}}\right )}{b}-\frac {i \coth (x)}{b}\right )\) |
(-I)*((I*a*(ArcTanh[Cosh[x]]/b - (2*a*ArcTanh[(b*((2*a)/b - 2*Tanh[x/2]))/ (2*Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2])))/b - (I*Coth[x])/b)
3.1.82.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Sym bol] :> Simp[1/b Int[Csc[e + f*x], x], x] - Simp[a/b Int[Csc[e + f*x]/( a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]
Int[csc[(e_.) + (f_.)*(x_)]^3/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Sym bol] :> Simp[-Cot[e + f*x]/(b*f), x] - Simp[a/b Int[Csc[e + f*x]^2/(a + b *Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24
| method | result | size |
| default | \(-\frac {\tanh \left (\frac {x}{2}\right )}{2 b}-\frac {2 a^{2} \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{2} \sqrt {a^{2}+b^{2}}}-\frac {1}{2 b \tanh \left (\frac {x}{2}\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{2}}\) | \(73\) |
| risch | \(-\frac {2}{b \left ({\mathrm e}^{2 x}-1\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, b^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, b^{2}}+\frac {a \ln \left ({\mathrm e}^{x}+1\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{x}-1\right )}{b^{2}}\) | \(143\) |
-1/2/b*tanh(1/2*x)-2*a^2/b^2/(a^2+b^2)^(1/2)*arctanh(1/2*(-2*b*tanh(1/2*x) +2*a)/(a^2+b^2)^(1/2))-1/2/b/tanh(1/2*x)-a/b^2*ln(tanh(1/2*x))
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (55) = 110\).
Time = 0.27 (sec) , antiderivative size = 345, normalized size of antiderivative = 5.85 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {2 \, a^{2} b + 2 \, b^{3} - {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right ) + {\left (a^{3} + a b^{2} - {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{3} + a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (a^{3} + a b^{2} - {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{3} + a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{2} b^{2} + b^{4} - {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}} \]
(2*a^2*b + 2*b^3 - (a^2*cosh(x)^2 + 2*a^2*cosh(x)*sinh(x) + a^2*sinh(x)^2 - a^2)*sqrt(a^2 + b^2)*log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh(x) + a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 + b^2)*(a*cosh( x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*cosh( x) + b)*sinh(x) - a)) + (a^3 + a*b^2 - (a^3 + a*b^2)*cosh(x)^2 - 2*(a^3 + a*b^2)*cosh(x)*sinh(x) - (a^3 + a*b^2)*sinh(x)^2)*log(cosh(x) + sinh(x) + 1) - (a^3 + a*b^2 - (a^3 + a*b^2)*cosh(x)^2 - 2*(a^3 + a*b^2)*cosh(x)*sinh (x) - (a^3 + a*b^2)*sinh(x)^2)*log(cosh(x) + sinh(x) - 1))/(a^2*b^2 + b^4 - (a^2*b^2 + b^4)*cosh(x)^2 - 2*(a^2*b^2 + b^4)*cosh(x)*sinh(x) - (a^2*b^2 + b^4)*sinh(x)^2)
\[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.69 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {a^{2} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{2}} + \frac {a \log \left (e^{\left (-x\right )} + 1\right )}{b^{2}} - \frac {a \log \left (e^{\left (-x\right )} - 1\right )}{b^{2}} + \frac {2}{b e^{\left (-2 \, x\right )} - b} \]
a^2*log((a*e^(-x) - b - sqrt(a^2 + b^2))/(a*e^(-x) - b + sqrt(a^2 + b^2))) /(sqrt(a^2 + b^2)*b^2) + a*log(e^(-x) + 1)/b^2 - a*log(e^(-x) - 1)/b^2 + 2 /(b*e^(-2*x) - b)
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.66 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {a^{2} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{2}} + \frac {a \log \left (e^{x} + 1\right )}{b^{2}} - \frac {a \log \left ({\left | e^{x} - 1 \right |}\right )}{b^{2}} - \frac {2}{b {\left (e^{\left (2 \, x\right )} - 1\right )}} \]
a^2*log(abs(2*a*e^x + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^x + 2*b + 2*sqrt( a^2 + b^2)))/(sqrt(a^2 + b^2)*b^2) + a*log(e^x + 1)/b^2 - a*log(abs(e^x - 1))/b^2 - 2/(b*(e^(2*x) - 1))
Time = 2.50 (sec) , antiderivative size = 292, normalized size of antiderivative = 4.95 \[ \int \frac {\text {csch}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {2}{b-b\,{\mathrm {e}}^{2\,x}}-\frac {a\,\ln \left (32\,{\mathrm {e}}^x-32\right )}{b^2}+\frac {a\,\ln \left (32\,{\mathrm {e}}^x+32\right )}{b^2}+\frac {a^2\,\ln \left (32\,a^4\,{\mathrm {e}}^x-64\,a\,b^3-64\,a^3\,b-32\,a^3\,\sqrt {a^2+b^2}+128\,b^4\,{\mathrm {e}}^x+128\,b^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^x-64\,a\,b^2\,\sqrt {a^2+b^2}+96\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2\,b^2+b^4}-\frac {a^2\,\ln \left (32\,a^3\,\sqrt {a^2+b^2}-64\,a\,b^3-64\,a^3\,b+32\,a^4\,{\mathrm {e}}^x+128\,b^4\,{\mathrm {e}}^x-128\,b^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^x+64\,a\,b^2\,\sqrt {a^2+b^2}-96\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2\,b^2+b^4} \]
2/(b - b*exp(2*x)) - (a*log(32*exp(x) - 32))/b^2 + (a*log(32*exp(x) + 32)) /b^2 + (a^2*log(32*a^4*exp(x) - 64*a*b^3 - 64*a^3*b - 32*a^3*(a^2 + b^2)^( 1/2) + 128*b^4*exp(x) + 128*b^3*exp(x)*(a^2 + b^2)^(1/2) + 160*a^2*b^2*exp (x) - 64*a*b^2*(a^2 + b^2)^(1/2) + 96*a^2*b*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(b^4 + a^2*b^2) - (a^2*log(32*a^3*(a^2 + b^2)^(1/2) - 64*a* b^3 - 64*a^3*b + 32*a^4*exp(x) + 128*b^4*exp(x) - 128*b^3*exp(x)*(a^2 + b^ 2)^(1/2) + 160*a^2*b^2*exp(x) + 64*a*b^2*(a^2 + b^2)^(1/2) - 96*a^2*b*exp( x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(b^4 + a^2*b^2)