Integrand size = 11, antiderivative size = 33 \[ \int \frac {\text {csch}(x)}{a+a \text {sech}(x)} \, dx=-\frac {\text {arctanh}(\cosh (x))}{2 a}-\frac {\coth (x) \text {csch}(x)}{2 a}+\frac {\text {csch}^2(x)}{2 a} \]
Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {\text {csch}(x)}{a+a \text {sech}(x)} \, dx=-\frac {\left (1+2 \cosh ^2\left (\frac {x}{2}\right ) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )\right ) \text {sech}(x)}{2 a (1+\text {sech}(x))} \]
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.636, Rules used = {3042, 26, 4359, 26, 25, 3042, 26, 3185, 26, 3042, 26, 3086, 15, 3091, 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.
\(\displaystyle \int \frac {\text {csch}(x)}{a \text {sech}(x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\cos \left (-\frac {\pi }{2}+i x\right ) \left (a-a \csc \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\cos \left (i x-\frac {\pi }{2}\right ) \left (a-a \csc \left (i x-\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 4359 |
\(\displaystyle i \int \frac {i \coth (x)}{-\cosh (x) a-a}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\int -\frac {\coth (x)}{\cosh (x) a+a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\coth (x)}{a \cosh (x)+a}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan \left (-\frac {\pi }{2}+i x\right )}{a-a \sin \left (-\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan \left (i x-\frac {\pi }{2}\right )}{a-a \sin \left (i x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3185 |
\(\displaystyle -i \left (\frac {\int i \coth ^2(x) \text {csch}(x)dx}{a}+\frac {\int -i \coth (x) \text {csch}^2(x)dx}{a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {i \int \coth ^2(x) \text {csch}(x)dx}{a}-\frac {i \int \coth (x) \text {csch}^2(x)dx}{a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {i \int -i \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}-\frac {i \int i \sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )dx}{a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )dx}{a}+\frac {\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}\right )\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}-\frac {i \int -i \text {csch}(x)d(-i \text {csch}(x))}{a}\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}+\frac {i \text {csch}^2(x)}{2 a}\right )\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle -i \left (\frac {-\frac {1}{2} \int -i \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)}{a}+\frac {i \text {csch}^2(x)}{2 a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {\frac {1}{2} i \int \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)}{a}+\frac {i \text {csch}^2(x)}{2 a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {\frac {1}{2} i \int i \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)}{a}+\frac {i \text {csch}^2(x)}{2 a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {-\frac {1}{2} \int \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)}{a}+\frac {i \text {csch}^2(x)}{2 a}\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -i \left (\frac {-\frac {1}{2} i \text {arctanh}(\cosh (x))-\frac {1}{2} i \coth (x) \text {csch}(x)}{a}+\frac {i \text {csch}^2(x)}{2 a}\right )\) |
3.1.56.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_.)*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_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[1/a Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x ] - Simp[1/(b*g) Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /; Fre eQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m _.), x_Symbol] :> Int[Cot[e + f*x]^p*(b + a*Sin[e + f*x])^m, x] /; FreeQ[{a , b, e, f, p}, x] && IntegerQ[m] && EqQ[m, p]
Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{2}}{2}+\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2 a}\) | \(20\) |
risch | \(-\frac {{\mathrm e}^{x}}{\left ({\mathrm e}^{x}+1\right )^{2} a}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2 a}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2 a}\) | \(35\) |
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (27) = 54\).
Time = 0.24 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.12 \[ \int \frac {\text {csch}(x)}{a+a \text {sech}(x)} \, dx=-\frac {{\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )}{2 \, {\left (a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + a\right )}} \]
-1/2*((cosh(x)^2 + 2*(cosh(x) + 1)*sinh(x) + sinh(x)^2 + 2*cosh(x) + 1)*lo g(cosh(x) + sinh(x) + 1) - (cosh(x)^2 + 2*(cosh(x) + 1)*sinh(x) + sinh(x)^ 2 + 2*cosh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 2*cosh(x) + 2*sinh(x))/(a* cosh(x)^2 + a*sinh(x)^2 + 2*a*cosh(x) + 2*(a*cosh(x) + a)*sinh(x) + a)
\[ \int \frac {\text {csch}(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\operatorname {csch}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {\text {csch}(x)}{a+a \text {sech}(x)} \, dx=-\frac {e^{\left (-x\right )}}{2 \, a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{2 \, a} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{2 \, a} \]
Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58 \[ \int \frac {\text {csch}(x)}{a+a \text {sech}(x)} \, dx=-\frac {\log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, a} + \frac {\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, a} + \frac {e^{\left (-x\right )} + e^{x} - 2}{4 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}} \]
-1/4*log(e^(-x) + e^x + 2)/a + 1/4*log(e^(-x) + e^x - 2)/a + 1/4*(e^(-x) + e^x - 2)/(a*(e^(-x) + e^x + 2))
Time = 2.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {\text {csch}(x)}{a+a \text {sech}(x)} \, dx=\frac {1}{a\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}-\frac {1}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{\sqrt {-a^2}} \]