Integrand size = 15, antiderivative size = 36 \[ \int \cosh (a+b x) \text {csch}^2(c+b x) \, dx=-\frac {\cosh (a-c) \text {csch}(c+b x)}{b}-\frac {\text {arctanh}(\cosh (c+b x)) \sinh (a-c)}{b} \]
Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.50 \[ \int \cosh (a+b x) \text {csch}^2(c+b x) \, dx=-\frac {\cosh (a-c) \text {csch}(c+b x)}{b}-\frac {2 i \arctan \left (\frac {(\cosh (c)-\sinh (c)) \left (\cosh (c) \cosh \left (\frac {b x}{2}\right )+\sinh (c) \sinh \left (\frac {b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac {b x}{2}\right )-i \cosh \left (\frac {b x}{2}\right ) \sinh (c)}\right ) \sinh (a-c)}{b} \]
-((Cosh[a - c]*Csch[c + b*x])/b) - ((2*I)*ArcTan[((Cosh[c] - Sinh[c])*(Cos h[c]*Cosh[(b*x)/2] + Sinh[c]*Sinh[(b*x)/2]))/(I*Cosh[c]*Cosh[(b*x)/2] - I* Cosh[(b*x)/2]*Sinh[c])]*Sinh[a - c])/b
Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6159, 3042, 26, 3086, 24, 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.
\(\displaystyle \int \cosh (a+b x) \text {csch}^2(b x+c) \, dx\) |
\(\Big \downarrow \) 6159 |
\(\displaystyle \sinh (a-c) \int \text {csch}(c+b x)dx+\cosh (a-c) \int \coth (c+b x) \text {csch}(c+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sinh (a-c) \int i \csc (i c+i b x)dx+\cosh (a-c) \int \sec \left (i c+i b x-\frac {\pi }{2}\right ) \tan \left (i c+i b x-\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \sinh (a-c) \int \csc (i c+i b x)dx+\cosh (a-c) \int \sec \left (i c+i b x-\frac {\pi }{2}\right ) \tan \left (i c+i b x-\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle i \sinh (a-c) \int \csc (i c+i b x)dx-\frac {i \cosh (a-c) \int 1d(-i \text {csch}(c+b x))}{b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\cosh (a-c) \text {csch}(b x+c)}{b}+i \sinh (a-c) \int \csc (i c+i b x)dx\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {\sinh (a-c) \text {arctanh}(\cosh (b x+c))}{b}-\frac {\cosh (a-c) \text {csch}(b x+c)}{b}\) |
3.2.65.3.1 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_.)*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[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Cosh[v_]*Csch[w_]^(n_.), x_Symbol] :> Simp[Cosh[v - w] Int[Coth[w]*Cs ch[w]^(n - 1), x], x] + Simp[Sinh[v - w] Int[Csch[w]^(n - 1), x], x] /; G tQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
Leaf count of result is larger than twice the leaf count of optimal. \(169\) vs. \(2(36)=72\).
Time = 0.44 (sec) , antiderivative size = 170, normalized size of antiderivative = 4.72
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 a}+{\mathrm e}^{2 c}\right )}{b \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}\) | \(170\) |
1/b*exp(b*x+a)*(exp(2*a)+exp(2*c))/(-exp(2*b*x+2*a+2*c)+exp(2*a))+1/2*ln(e xp(b*x+a)-exp(a-c))/b*exp(-a-c)*exp(2*a)-1/2*ln(exp(b*x+a)-exp(a-c))/b*exp (-a-c)*exp(2*c)-1/2*ln(exp(b*x+a)+exp(a-c))/b*exp(-a-c)*exp(2*a)+1/2*ln(ex p(b*x+a)+exp(a-c))/b*exp(-a-c)*exp(2*c)
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (36) = 72\).
Time = 0.25 (sec) , antiderivative size = 617, normalized size of antiderivative = 17.14 \[ \int \cosh (a+b x) \text {csch}^2(c+b x) \, dx=\frac {4 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - 2 \, \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - 2 \, {\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right ) - {\left ({\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right )^{2} + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \sinh \left (b x + c\right )^{2} + {\left (\cosh \left (b x + c\right )^{2} - 1\right )} \sinh \left (-a + c\right )^{2} - \cosh \left (-a + c\right )^{2} - 2 \, {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right )\right )} \sinh \left (b x + c\right ) - 2 \, {\left (\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) - \cosh \left (-a + c\right )\right )} \sinh \left (-a + c\right ) + 1\right )} \log \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right ) + 1\right ) + {\left ({\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right )^{2} + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \sinh \left (b x + c\right )^{2} + {\left (\cosh \left (b x + c\right )^{2} - 1\right )} \sinh \left (-a + c\right )^{2} - \cosh \left (-a + c\right )^{2} - 2 \, {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right )\right )} \sinh \left (b x + c\right ) - 2 \, {\left (\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) - \cosh \left (-a + c\right )\right )} \sinh \left (-a + c\right ) + 1\right )} \log \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right ) - 1\right ) - 2 \, {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} + 1\right )} \sinh \left (b x + c\right )}{2 \, {\left (b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) + {\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right )^{2} - b \cosh \left (-a + c\right ) + 2 \, {\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) - {\left (b \cosh \left (b x + c\right )^{2} - b\right )} \sinh \left (-a + c\right )\right )}} \]
1/2*(4*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) - 2*cosh(b*x + c)*sinh(-a + c)^2 - 2*(cosh(-a + c)^2 + 1)*cosh(b*x + c) - ((cosh(-a + c)^2 - 1)*cosh( b*x + c)^2 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^ 2 - 1)*sinh(b*x + c)^2 + (cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh(-a + c)^2 - 2*(2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(- a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c))*sinh(b*x + c) - 2*(cosh(b*x + c)^2*cosh(-a + c) - cosh(-a + c))*sinh(-a + c) + 1)*log(cosh(b*x + c) + sinh(b*x + c) + 1) + ((cosh(-a + c)^2 - 1)*cosh(b*x + c)^2 + (cosh(-a + c )^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c)^2 + (cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh(-a + c)^2 - 2*(2*cosh(b*x + c) *cosh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^ 2 - 1)*cosh(b*x + c))*sinh(b*x + c) - 2*(cosh(b*x + c)^2*cosh(-a + c) - co sh(-a + c))*sinh(-a + c) + 1)*log(cosh(b*x + c) + sinh(b*x + c) - 1) - 2*( cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 + 1)*sinh(b* x + c))/(b*cosh(b*x + c)^2*cosh(-a + c) + (b*cosh(-a + c) - b*sinh(-a + c) )*sinh(b*x + c)^2 - b*cosh(-a + c) + 2*(b*cosh(b*x + c)*cosh(-a + c) - b*c osh(b*x + c)*sinh(-a + c))*sinh(b*x + c) - (b*cosh(b*x + c)^2 - b)*sinh(-a + c))
\[ \int \cosh (a+b x) \text {csch}^2(c+b x) \, dx=\int \cosh {\left (a + b x \right )} \operatorname {csch}^{2}{\left (b x + c \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (36) = 72\).
Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.92 \[ \int \cosh (a+b x) \text {csch}^2(c+b x) \, dx=-\frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} + \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-b x - a\right )}}{b {\left (e^{\left (-2 \, b x\right )} - e^{\left (2 \, c\right )}\right )}} \]
-1/2*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) + e^c)/b + 1/2*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) - e^c)/b + (e^(2*a) + e^(2*c))*e^(-b*x - a)/(b*(e^(-2*b*x) - e^(2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (36) = 72\).
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.94 \[ \int \cosh (a+b x) \text {csch}^2(c+b x) \, dx=-\frac {{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\right ) - {\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + c\right )} - 1 \right |}\right ) + \frac {2 \, {\left (e^{\left (b x + 2 \, a\right )} + e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-a\right )}}{e^{\left (2 \, b x + 2 \, c\right )} - 1}}{2 \, b} \]
-1/2*((e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(e^(b*x + c) + 1) - (e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(abs(e^(b*x + c) - 1)) + 2*(e^(b*x + 2*a) + e^(b*x + 2*c))*e^(-a)/(e^(2*b*x + 2*c) - 1))/b
Time = 2.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 4.33 \[ \int \cosh (a+b x) \text {csch}^2(c+b x) \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {-b^2}-{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {-b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{2\,c}\,\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}-2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left ({\mathrm {e}}^{4\,a-4\,c}-2\,{\mathrm {e}}^{2\,a-2\,c}+1\right )}}{\sqrt {-b^2}}+\frac {{\mathrm {e}}^{a+b\,x}\,\left ({\mathrm {e}}^{2\,a-2\,c}+1\right )}{b\,\left ({\mathrm {e}}^{2\,a-2\,c}-{\mathrm {e}}^{2\,a+2\,b\,x}\right )} \]
(atan((exp(-a)*exp(2*c)*exp(b*x)*((-b^2)^(1/2) - exp(2*a)*exp(-2*c)*(-b^2) ^(1/2)))/(b*(exp(-2*a)*exp(2*c)*(exp(4*a)*exp(-4*c) - 2*exp(2*a)*exp(-2*c) + 1))^(1/2)))*(exp(2*c - 2*a)*(exp(4*a - 4*c) - 2*exp(2*a - 2*c) + 1))^(1 /2))/(-b^2)^(1/2) + (exp(a + b*x)*(exp(2*a - 2*c) + 1))/(b*(exp(2*a - 2*c) - exp(2*a + 2*b*x)))