Integrand size = 13, antiderivative size = 207 \[ \int \frac {\coth ^4(x)}{a+b \text {sech}(x)} \, dx=-\frac {a b^2 x}{\left (a^2-b^2\right )^2}+\frac {b^4 x}{a \left (a^2-b^2\right )^2}+\frac {a x}{a^2-b^2}-\frac {2 b^5 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a (a-b)^{5/2} (a+b)^{5/2}}+\frac {a b^2 \coth (x)}{\left (a^2-b^2\right )^2}-\frac {a \coth (x)}{a^2-b^2}-\frac {a \coth ^3(x)}{3 \left (a^2-b^2\right )}-\frac {b^3 \text {csch}(x)}{\left (a^2-b^2\right )^2}+\frac {b \text {csch}(x)}{a^2-b^2}+\frac {b \text {csch}^3(x)}{3 \left (a^2-b^2\right )} \] Output:
-a*b^2*x/(a^2-b^2)^2+b^4*x/a/(a^2-b^2)^2+a*x/(a^2-b^2)-2*b^5*arctan((a-b)^ (1/2)*tanh(1/2*x)/(a+b)^(1/2))/a/(a-b)^(5/2)/(a+b)^(5/2)+a*b^2*coth(x)/(a^ 2-b^2)^2-a*coth(x)/(a^2-b^2)-a*coth(x)^3/(3*a^2-3*b^2)-b^3*csch(x)/(a^2-b^ 2)^2+b*csch(x)/(a^2-b^2)+b*csch(x)^3/(3*a^2-3*b^2)
Time = 0.59 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.80 \[ \int \frac {\coth ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {(b+a \cosh (x)) \text {sech}(x) \left (\frac {24 x}{a}+\frac {48 b^5 \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a \left (a^2-b^2\right )^{5/2}}-\frac {2 (8 a+11 b) \coth \left (\frac {x}{2}\right )}{(a+b)^2}+\frac {8 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )}{a-b}-\frac {\text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)}{2 (a+b)}-\frac {16 a \tanh \left (\frac {x}{2}\right )}{(a-b)^2}+\frac {22 b \tanh \left (\frac {x}{2}\right )}{(a-b)^2}\right )}{24 (a+b \text {sech}(x))} \] Input:
Integrate[Coth[x]^4/(a + b*Sech[x]),x]
Output:
((b + a*Cosh[x])*Sech[x]*((24*x)/a + (48*b^5*ArcTan[((-a + b)*Tanh[x/2])/S qrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2)) - (2*(8*a + 11*b)*Coth[x/2])/(a + b )^2 + (8*Csch[x]^3*Sinh[x/2]^4)/(a - b) - (Csch[x/2]^4*Sinh[x])/(2*(a + b) ) - (16*a*Tanh[x/2])/(a - b)^2 + (22*b*Tanh[x/2])/(a - b)^2))/(24*(a + b*S ech[x]))
Result contains complex when optimal does not.
Time = 1.39 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.87, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.077, Rules used = {3042, 4386, 3042, 25, 3381, 25, 3042, 25, 3086, 2009, 3381, 25, 3042, 25, 3086, 24, 3214, 3042, 3138, 218, 3954, 24, 25, 3042, 25, 3954, 24}
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.
| \(\displaystyle \int \frac {\coth ^4(x)}{a+b \text {sech}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot \left (\frac {\pi }{2}+i x\right )^4 \left (a+b \csc \left (\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 4386 |
| \(\displaystyle \int \frac {\cosh (x) \coth ^4(x)}{a \cosh (x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin \left (-\frac {\pi }{2}+i x\right )^5}{\cos \left (-\frac {\pi }{2}+i x\right )^4 \left (b-a \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin \left (i x-\frac {\pi }{2}\right )^5}{\cos \left (i x-\frac {\pi }{2}\right )^4 \left (b-a \sin \left (i x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 3381 |
| \(\displaystyle \frac {a \int \coth ^4(x)dx}{a^2-b^2}-\frac {b^2 \int \frac {\cosh (x) \coth ^2(x)}{b+a \cosh (x)}dx}{a^2-b^2}+\frac {b \int -\coth ^3(x) \text {csch}(x)dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \int \coth ^4(x)dx}{a^2-b^2}-\frac {b^2 \int \frac {\cosh (x) \coth ^2(x)}{b+a \cosh (x)}dx}{a^2-b^2}-\frac {b \int \coth ^3(x) \text {csch}(x)dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \int \frac {\sin \left (i x-\frac {\pi }{2}\right )^3}{\cos \left (i x-\frac {\pi }{2}\right )^2 \left (b-a \sin \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}-\frac {b \int -\sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^3dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \int \frac {\sin \left (i x-\frac {\pi }{2}\right )^3}{\cos \left (i x-\frac {\pi }{2}\right )^2 \left (b-a \sin \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}+\frac {b \int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^3dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {i b \int \left (-\text {csch}^2(x)-1\right )d(-i \text {csch}(x))}{a^2-b^2}-\frac {b^2 \int \frac {\sin \left (i x-\frac {\pi }{2}\right )^3}{\cos \left (i x-\frac {\pi }{2}\right )^2 \left (b-a \sin \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \int \frac {\sin \left (i x-\frac {\pi }{2}\right )^3}{\cos \left (i x-\frac {\pi }{2}\right )^2 \left (b-a \sin \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3381 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \left (\frac {b^2 \int -\frac {\cosh (x)}{b+a \cosh (x)}dx}{a^2-b^2}-\frac {a \int -\coth ^2(x)dx}{a^2-b^2}-\frac {b \int \coth (x) \text {csch}(x)dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \int \frac {\cosh (x)}{b+a \cosh (x)}dx}{a^2-b^2}+\frac {a \int \coth ^2(x)dx}{a^2-b^2}-\frac {b \int \coth (x) \text {csch}(x)dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \int \frac {\sin \left (i x+\frac {\pi }{2}\right )}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}+\frac {a \int -\tan \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}-\frac {b \int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \int \frac {\sin \left (i x+\frac {\pi }{2}\right )}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}-\frac {b \int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \int \frac {\sin \left (i x+\frac {\pi }{2}\right )}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}+\frac {i b \int 1d(-i \text {csch}(x))}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \int \frac {\sin \left (i x+\frac {\pi }{2}\right )}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \left (-\frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}-\frac {b^2 \left (\frac {x}{a}-\frac {b \int \frac {1}{b+a \cosh (x)}dx}{a}\right )}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {x}{a}-\frac {b \int \frac {1}{b+a \sin \left (i x+\frac {\pi }{2}\right )}dx}{a}\right )}{a^2-b^2}-\frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {b^2 \left (-\frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}-\frac {b^2 \left (\frac {x}{a}-\frac {2 b \int \frac {1}{(a-b) \tanh ^2\left (\frac {x}{2}\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {b^2 \left (-\frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}-\frac {b^2 \left (\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \int \tan \left (i x+\frac {\pi }{2}\right )^4dx}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle -\frac {b^2 \left (-\frac {a (\coth (x)-\int 1dx)}{a^2-b^2}-\frac {b^2 \left (\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (-\int -\coth ^2(x)dx-\frac {1}{3} \coth ^3(x)\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {a \left (-\int -\coth ^2(x)dx-\frac {1}{3} \coth ^3(x)\right )}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a^2-b^2}-\frac {a (\coth (x)-x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (\int \coth ^2(x)dx-\frac {\coth ^3(x)}{3}\right )}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a^2-b^2}-\frac {a (\coth (x)-x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {\coth ^3(x)}{3}+\int -\tan \left (i x+\frac {\pi }{2}\right )^2dx\right )}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a^2-b^2}-\frac {a (\coth (x)-x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (-\frac {1}{3} \coth ^3(x)-\int \tan \left (i x+\frac {\pi }{2}\right )^2dx\right )}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a^2-b^2}-\frac {a (\coth (x)-x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {a \left (\int 1dx-\frac {1}{3} \coth ^3(x)-\coth (x)\right )}{a^2-b^2}-\frac {b^2 \left (-\frac {b^2 \left (\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a^2-b^2}-\frac {a (\coth (x)-x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {b^2 \left (-\frac {b^2 \left (\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a^2-b^2}-\frac {a (\coth (x)-x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {a \left (x-\frac {1}{3} \coth ^3(x)-\coth (x)\right )}{a^2-b^2}-\frac {i b \left (\frac {1}{3} i \text {csch}^3(x)+i \text {csch}(x)\right )}{a^2-b^2}\) |
Input:
Int[Coth[x]^4/(a + b*Sech[x]),x]
Output:
(a*(x - Coth[x] - Coth[x]^3/3))/(a^2 - b^2) - (b^2*(-((b^2*(x/a - (2*b*Arc Tan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b])))/(a ^2 - b^2)) - (a*(-x + Coth[x]))/(a^2 - b^2) + (b*Csch[x])/(a^2 - b^2)))/(a ^2 - b^2) - (I*b*(I*Csch[x] + (I/3)*Csch[x]^3))/(a^2 - b^2)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a*(d^2/(a^2 - b^2)) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 2), x], x] + (-Simp[ b*(d/(a^2 - b^2)) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Simp[a^2*(d^2/(g^2*(a^2 - b^2))) Int[(g*Cos[e + f*x])^(p + 2)*((d*Sin[ e + f*x])^(n - 2)/(a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, d, e, f, g }, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 1 ]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Int[Cos[c + d*x]^m*((b + a*Sin[c + d*x])^n/Sin[c + d*x]^(m + n)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && IntegerQ[m] && (IntegerQ[m/2] || LeQ[m, 1])
Time = 1.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.74
| method | result | size |
| default | \(-\frac {\frac {a \tanh \left (\frac {x}{2}\right )^{3}}{3}-\frac {b \tanh \left (\frac {x}{2}\right )^{3}}{3}+5 a \tanh \left (\frac {x}{2}\right )-7 \tanh \left (\frac {x}{2}\right ) b}{8 \left (a -b \right )^{2}}-\frac {1}{24 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{3}}-\frac {5 a +7 b}{8 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {2 b^{5} \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a \left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}\) | \(153\) |
| risch | \(\frac {x}{a}-\frac {2 \left (-3 a^{2} b \,{\mathrm e}^{5 x}+6 \,{\mathrm e}^{5 x} b^{3}+6 a^{3} {\mathrm e}^{4 x}-9 a \,b^{2} {\mathrm e}^{4 x}+2 a^{2} b \,{\mathrm e}^{3 x}-8 b^{3} {\mathrm e}^{3 x}-6 a^{3} {\mathrm e}^{2 x}+12 a \,b^{2} {\mathrm e}^{2 x}-3 b \,{\mathrm e}^{x} a^{2}+6 b^{3} {\mathrm e}^{x}+4 a^{3}-7 a \,b^{2}\right )}{3 \left (a^{2}-b^{2}\right )^{2} \left ({\mathrm e}^{2 x}-1\right )^{3}}-\frac {b^{5} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a}+\frac {b^{5} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a}\) | \(274\) |
Input:
int(coth(x)^4/(a+b*sech(x)),x,method=_RETURNVERBOSE)
Output:
-1/8/(a-b)^2*(1/3*a*tanh(1/2*x)^3-1/3*b*tanh(1/2*x)^3+5*a*tanh(1/2*x)-7*ta nh(1/2*x)*b)-1/24/(a+b)/tanh(1/2*x)^3-1/8*(5*a+7*b)/(a+b)^2/tanh(1/2*x)+1/ a*ln(tanh(1/2*x)+1)-1/a*ln(tanh(1/2*x)-1)-2/a/(a-b)^2/(a+b)^2*b^5/((a-b)*( a+b))^(1/2)*arctan((a-b)*tanh(1/2*x)/((a-b)*(a+b))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1725 vs. \(2 (193) = 386\).
Time = 0.13 (sec) , antiderivative size = 3530, normalized size of antiderivative = 17.05 \[ \int \frac {\coth ^4(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \] Input:
integrate(coth(x)^4/(a+b*sech(x)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\coth ^4(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\coth ^{4}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \] Input:
integrate(coth(x)**4/(a+b*sech(x)),x)
Output:
Integral(coth(x)**4/(a + b*sech(x)), x)
Exception generated. \[ \int \frac {\coth ^4(x)}{a+b \text {sech}(x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(coth(x)^4/(a+b*sech(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 = 0.13 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.92 \[ \int \frac {\coth ^4(x)}{a+b \text {sech}(x)} \, dx=-\frac {2 \, b^{5} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {x}{a} + \frac {2 \, {\left (3 \, a^{2} b e^{\left (5 \, x\right )} - 6 \, b^{3} e^{\left (5 \, x\right )} - 6 \, a^{3} e^{\left (4 \, x\right )} + 9 \, a b^{2} e^{\left (4 \, x\right )} - 2 \, a^{2} b e^{\left (3 \, x\right )} + 8 \, b^{3} e^{\left (3 \, x\right )} + 6 \, a^{3} e^{\left (2 \, x\right )} - 12 \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} - 6 \, b^{3} e^{x} - 4 \, a^{3} + 7 \, a b^{2}\right )}}{3 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \] Input:
integrate(coth(x)^4/(a+b*sech(x)),x, algorithm="giac")
Output:
-2*b^5*arctan((a*e^x + b)/sqrt(a^2 - b^2))/((a^5 - 2*a^3*b^2 + a*b^4)*sqrt (a^2 - b^2)) + x/a + 2/3*(3*a^2*b*e^(5*x) - 6*b^3*e^(5*x) - 6*a^3*e^(4*x) + 9*a*b^2*e^(4*x) - 2*a^2*b*e^(3*x) + 8*b^3*e^(3*x) + 6*a^3*e^(2*x) - 12*a *b^2*e^(2*x) + 3*a^2*b*e^x - 6*b^3*e^x - 4*a^3 + 7*a*b^2)/((a^4 - 2*a^2*b^ 2 + b^4)*(e^(2*x) - 1)^3)
Time = 2.73 (sec) , antiderivative size = 713, normalized size of antiderivative = 3.44 \[ \int \frac {\coth ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {x}{a}-\frac {\frac {8\,a}{3\,\left (a^2-b^2\right )}-\frac {8\,b\,{\mathrm {e}}^x}{3\,\left (a^2-b^2\right )}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {\frac {2\,\left (2\,a^4-3\,a^2\,b^2\right )}{a\,{\left (a^2-b^2\right )}^2}-\frac {2\,{\mathrm {e}}^x\,\left (a^2\,b-2\,b^3\right )}{{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {4\,\left (a^4-a^2\,b^2\right )}{a\,{\left (a^2-b^2\right )}^2}-\frac {8\,{\mathrm {e}}^x\,\left (a^2\,b-b^3\right )}{3\,{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,b^5}{a^3\,{\left (a^2-b^2\right )}^2\,\sqrt {b^{10}}\,\left (a^5-2\,a^3\,b^2+a\,b^4\right )}+\frac {2\,\left (a\,b^5\,\sqrt {b^{10}}-2\,a^3\,b^3\,\sqrt {b^{10}}+a^5\,b\,\sqrt {b^{10}}\right )}{a^2\,b^4\,\sqrt {a^2\,{\left (a^2-b^2\right )}^5}\,\left (a^5-2\,a^3\,b^2+a\,b^4\right )\,\sqrt {a^{12}-5\,a^{10}\,b^2+10\,a^8\,b^4-10\,a^6\,b^6+5\,a^4\,b^8-a^2\,b^{10}}}\right )+\frac {2\,\left (a^6\,\sqrt {b^{10}}+a^2\,b^4\,\sqrt {b^{10}}-2\,a^4\,b^2\,\sqrt {b^{10}}\right )}{a^2\,b^4\,\sqrt {a^2\,{\left (a^2-b^2\right )}^5}\,\left (a^5-2\,a^3\,b^2+a\,b^4\right )\,\sqrt {a^{12}-5\,a^{10}\,b^2+10\,a^8\,b^4-10\,a^6\,b^6+5\,a^4\,b^8-a^2\,b^{10}}}\right )\,\left (\frac {a^6\,\sqrt {a^{12}-5\,a^{10}\,b^2+10\,a^8\,b^4-10\,a^6\,b^6+5\,a^4\,b^8-a^2\,b^{10}}}{2}+\frac {a^2\,b^4\,\sqrt {a^{12}-5\,a^{10}\,b^2+10\,a^8\,b^4-10\,a^6\,b^6+5\,a^4\,b^8-a^2\,b^{10}}}{2}-a^4\,b^2\,\sqrt {a^{12}-5\,a^{10}\,b^2+10\,a^8\,b^4-10\,a^6\,b^6+5\,a^4\,b^8-a^2\,b^{10}}\right )\right )\,\sqrt {b^{10}}}{\sqrt {a^{12}-5\,a^{10}\,b^2+10\,a^8\,b^4-10\,a^6\,b^6+5\,a^4\,b^8-a^2\,b^{10}}} \] Input:
int(coth(x)^4/(a + b/cosh(x)),x)
Output:
x/a - ((8*a)/(3*(a^2 - b^2)) - (8*b*exp(x))/(3*(a^2 - b^2)))/(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1) - ((2*(2*a^4 - 3*a^2*b^2))/(a*(a^2 - b^2)^2) - (2*exp(x)*(a^2*b - 2*b^3))/(a^2 - b^2)^2)/(exp(2*x) - 1) - ((4*(a^4 - a^2 *b^2))/(a*(a^2 - b^2)^2) - (8*exp(x)*(a^2*b - b^3))/(3*(a^2 - b^2)^2))/(ex p(4*x) - 2*exp(2*x) + 1) - (2*atan((exp(x)*((2*b^5)/(a^3*(a^2 - b^2)^2*(b^ 10)^(1/2)*(a*b^4 + a^5 - 2*a^3*b^2)) + (2*(a*b^5*(b^10)^(1/2) - 2*a^3*b^3* (b^10)^(1/2) + a^5*b*(b^10)^(1/2)))/(a^2*b^4*(a^2*(a^2 - b^2)^5)^(1/2)*(a* b^4 + a^5 - 2*a^3*b^2)*(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8* b^4 - 5*a^10*b^2)^(1/2))) + (2*(a^6*(b^10)^(1/2) + a^2*b^4*(b^10)^(1/2) - 2*a^4*b^2*(b^10)^(1/2)))/(a^2*b^4*(a^2*(a^2 - b^2)^5)^(1/2)*(a*b^4 + a^5 - 2*a^3*b^2)*(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^1 0*b^2)^(1/2)))*((a^6*(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^ 4 - 5*a^10*b^2)^(1/2))/2 + (a^2*b^4*(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6* b^6 + 10*a^8*b^4 - 5*a^10*b^2)^(1/2))/2 - a^4*b^2*(a^12 - a^2*b^10 + 5*a^4 *b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2)^(1/2)))*(b^10)^(1/2))/(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2)^(1/2)
Time = 19.31 (sec) , antiderivative size = 691, normalized size of antiderivative = 3.34 \[ \int \frac {\coth ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {-4 e^{6 x} a^{6}-9 a^{2} b^{4} x +6 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +b}{\sqrt {a^{2}-b^{2}}}\right ) b^{5}+3 e^{6 x} a^{6} x +9 e^{2 x} a^{6} x +9 a^{4} b^{2} x -3 a^{6} x +3 b^{6} x +6 e^{5 x} a^{5} b -18 e^{5 x} a^{3} b^{3}+12 e^{5 x} a \,b^{5}-9 e^{4 x} a^{6} x +9 e^{4 x} b^{6} x -4 e^{3 x} a^{5} b +20 e^{3 x} a^{3} b^{3}-16 e^{3 x} a \,b^{5}+6 e^{x} a^{5} b -18 e^{x} a^{3} b^{3}+18 e^{4 x} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +b}{\sqrt {a^{2}-b^{2}}}\right ) b^{5}+27 e^{4 x} a^{4} b^{2} x -27 e^{4 x} a^{2} b^{4} x -6 e^{6 x} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +b}{\sqrt {a^{2}-b^{2}}}\right ) b^{5}+10 e^{6 x} a^{4} b^{2}-6 e^{6 x} a^{2} b^{4}-3 e^{6 x} b^{6} x -9 e^{2 x} b^{6} x +12 e^{x} a \,b^{5}-4 a^{6}-18 e^{2 x} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +b}{\sqrt {a^{2}-b^{2}}}\right ) b^{5}-9 e^{6 x} a^{4} b^{2} x +9 e^{6 x} a^{2} b^{4} x -27 e^{2 x} a^{4} b^{2} x +27 e^{2 x} a^{2} b^{4} x -6 e^{2 x} a^{4} b^{2}+6 e^{2 x} a^{2} b^{4}+12 a^{4} b^{2}-8 a^{2} b^{4}}{3 a \left (e^{6 x} a^{6}-3 e^{6 x} a^{4} b^{2}+3 e^{6 x} a^{2} b^{4}-e^{6 x} b^{6}-3 e^{4 x} a^{6}+9 e^{4 x} a^{4} b^{2}-9 e^{4 x} a^{2} b^{4}+3 e^{4 x} b^{6}+3 e^{2 x} a^{6}-9 e^{2 x} a^{4} b^{2}+9 e^{2 x} a^{2} b^{4}-3 e^{2 x} b^{6}-a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}+b^{6}\right )} \] Input:
int(coth(x)^4/(a+b*sech(x)),x)
Output:
( - 6*e**(6*x)*sqrt(a**2 - b**2)*atan((e**x*a + b)/sqrt(a**2 - b**2))*b**5 + 18*e**(4*x)*sqrt(a**2 - b**2)*atan((e**x*a + b)/sqrt(a**2 - b**2))*b**5 - 18*e**(2*x)*sqrt(a**2 - b**2)*atan((e**x*a + b)/sqrt(a**2 - b**2))*b**5 + 6*sqrt(a**2 - b**2)*atan((e**x*a + b)/sqrt(a**2 - b**2))*b**5 + 3*e**(6 *x)*a**6*x - 4*e**(6*x)*a**6 - 9*e**(6*x)*a**4*b**2*x + 10*e**(6*x)*a**4*b **2 + 9*e**(6*x)*a**2*b**4*x - 6*e**(6*x)*a**2*b**4 - 3*e**(6*x)*b**6*x + 6*e**(5*x)*a**5*b - 18*e**(5*x)*a**3*b**3 + 12*e**(5*x)*a*b**5 - 9*e**(4*x )*a**6*x + 27*e**(4*x)*a**4*b**2*x - 27*e**(4*x)*a**2*b**4*x + 9*e**(4*x)* b**6*x - 4*e**(3*x)*a**5*b + 20*e**(3*x)*a**3*b**3 - 16*e**(3*x)*a*b**5 + 9*e**(2*x)*a**6*x - 27*e**(2*x)*a**4*b**2*x - 6*e**(2*x)*a**4*b**2 + 27*e* *(2*x)*a**2*b**4*x + 6*e**(2*x)*a**2*b**4 - 9*e**(2*x)*b**6*x + 6*e**x*a** 5*b - 18*e**x*a**3*b**3 + 12*e**x*a*b**5 - 3*a**6*x - 4*a**6 + 9*a**4*b**2 *x + 12*a**4*b**2 - 9*a**2*b**4*x - 8*a**2*b**4 + 3*b**6*x)/(3*a*(e**(6*x) *a**6 - 3*e**(6*x)*a**4*b**2 + 3*e**(6*x)*a**2*b**4 - e**(6*x)*b**6 - 3*e* *(4*x)*a**6 + 9*e**(4*x)*a**4*b**2 - 9*e**(4*x)*a**2*b**4 + 3*e**(4*x)*b** 6 + 3*e**(2*x)*a**6 - 9*e**(2*x)*a**4*b**2 + 9*e**(2*x)*a**2*b**4 - 3*e**( 2*x)*b**6 - a**6 + 3*a**4*b**2 - 3*a**2*b**4 + b**6))