Integrand size = 15, antiderivative size = 38 \[ \int \text {csch}(a+b x) \text {sech}^4(a+b x) \, dx=-\frac {\text {arctanh}(\cosh (a+b x))}{b}+\frac {\text {sech}(a+b x)}{b}+\frac {\text {sech}^3(a+b x)}{3 b} \]
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.50 \[ \int \text {csch}(a+b x) \text {sech}^4(a+b x) \, dx=-\frac {\log \left (\cosh \left (\frac {1}{2} (a+b x)\right )\right )}{b}+\frac {\log \left (\sinh \left (\frac {1}{2} (a+b x)\right )\right )}{b}+\frac {\text {sech}(a+b x)}{b}+\frac {\text {sech}^3(a+b x)}{3 b} \]
-(Log[Cosh[(a + b*x)/2]]/b) + Log[Sinh[(a + b*x)/2]]/b + Sech[a + b*x]/b + Sech[a + b*x]^3/(3*b)
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 26, 3102, 25, 254, 2009}
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.
\(\displaystyle \int \text {csch}(a+b x) \text {sech}^4(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \csc (i a+i b x) \sec (i a+i b x)^4dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \csc (i a+i b x) \sec (i a+i b x)^4dx\) |
\(\Big \downarrow \) 3102 |
\(\displaystyle \frac {\int -\frac {\text {sech}^4(a+b x)}{1-\text {sech}^2(a+b x)}d\text {sech}(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\text {sech}^4(a+b x)}{1-\text {sech}^2(a+b x)}d\text {sech}(a+b x)}{b}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle -\frac {\int \left (-\text {sech}^2(a+b x)+\frac {1}{1-\text {sech}^2(a+b x)}-1\right )d\text {sech}(a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\text {arctanh}(\text {sech}(a+b x))+\frac {1}{3} \text {sech}^3(a+b x)+\text {sech}(a+b x)}{b}\) |
3.1.27.3.1 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[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_S ymbol] :> Simp[1/(f*a^n) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/ 2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1 )/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Time = 7.68 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {1}{3 \cosh \left (b x +a \right )^{3}}+\frac {1}{\cosh \left (b x +a \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(33\) |
default | \(\frac {\frac {1}{3 \cosh \left (b x +a \right )^{3}}+\frac {1}{\cosh \left (b x +a \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(33\) |
risch | \(\frac {2 \,{\mathrm e}^{b x +a} \left (3 \,{\mathrm e}^{4 b x +4 a}+10 \,{\mathrm e}^{2 b x +2 a}+3\right )}{3 b \left (1+{\mathrm e}^{2 b x +2 a}\right )^{3}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b}\) | \(77\) |
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (36) = 72\).
Time = 0.26 (sec) , antiderivative size = 697, normalized size of antiderivative = 18.34 \[ \int \text {csch}(a+b x) \text {sech}^4(a+b x) \, dx=\frac {6 \, \cosh \left (b x + a\right )^{5} + 30 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 6 \, \sinh \left (b x + a\right )^{5} + 20 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{3} + 20 \, \cosh \left (b x + a\right )^{3} + 60 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 3 \, {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{4} + 3 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 3 \, \cosh \left (b x + a\right )^{2} + 6 \, {\left (\cosh \left (b x + a\right )^{5} + 2 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 3 \, {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{4} + 3 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 3 \, \cosh \left (b x + a\right )^{2} + 6 \, {\left (\cosh \left (b x + a\right )^{5} + 2 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 6 \, {\left (5 \, \cosh \left (b x + a\right )^{4} + 10 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )}{3 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} + 3 \, b \cosh \left (b x + a\right )^{4} + 3 \, {\left (5 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} + 3 \, {\left (5 \, b \cosh \left (b x + a\right )^{4} + 6 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 6 \, {\left (b \cosh \left (b x + a\right )^{5} + 2 \, b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \]
1/3*(6*cosh(b*x + a)^5 + 30*cosh(b*x + a)*sinh(b*x + a)^4 + 6*sinh(b*x + a )^5 + 20*(3*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^3 + 20*cosh(b*x + a)^3 + 60 *(cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a)^2 - 3*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + 3*(5*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^4 + 3*cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 + 3*cosh(b* x + a))*sinh(b*x + a)^3 + 3*(5*cosh(b*x + a)^4 + 6*cosh(b*x + a)^2 + 1)*si nh(b*x + a)^2 + 3*cosh(b*x + a)^2 + 6*(cosh(b*x + a)^5 + 2*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) + 1)*log(cosh(b*x + a) + sinh(b*x + a) + 1 ) + 3*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + 3*(5*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^4 + 3*cosh(b*x + a)^4 + 4*(5*co sh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^3 + 3*(5*cosh(b*x + a)^4 + 6*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 3*cosh(b*x + a)^2 + 6*(cosh(b*x + a)^5 + 2*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) + 1)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 6*(5*cosh(b*x + a)^4 + 10*cosh(b*x + a)^2 + 1 )*sinh(b*x + a) + 6*cosh(b*x + a))/(b*cosh(b*x + a)^6 + 6*b*cosh(b*x + a)* sinh(b*x + a)^5 + b*sinh(b*x + a)^6 + 3*b*cosh(b*x + a)^4 + 3*(5*b*cosh(b* x + a)^2 + b)*sinh(b*x + a)^4 + 4*(5*b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a) )*sinh(b*x + a)^3 + 3*b*cosh(b*x + a)^2 + 3*(5*b*cosh(b*x + a)^4 + 6*b*cos h(b*x + a)^2 + b)*sinh(b*x + a)^2 + 6*(b*cosh(b*x + a)^5 + 2*b*cosh(b*x + a)^3 + b*cosh(b*x + a))*sinh(b*x + a) + b)
\[ \int \text {csch}(a+b x) \text {sech}^4(a+b x) \, dx=\int \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}^{4}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (36) = 72\).
Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.84 \[ \int \text {csch}(a+b x) \text {sech}^4(a+b x) \, dx=-\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, {\left (3 \, e^{\left (-b x - a\right )} + 10 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )}\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} \]
-log(e^(-b*x - a) + 1)/b + log(e^(-b*x - a) - 1)/b + 2/3*(3*e^(-b*x - a) + 10*e^(-3*b*x - 3*a) + 3*e^(-5*b*x - 5*a))/(b*(3*e^(-2*b*x - 2*a) + 3*e^(- 4*b*x - 4*a) + e^(-6*b*x - 6*a) + 1))
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (36) = 72\).
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.32 \[ \int \text {csch}(a+b x) \text {sech}^4(a+b x) \, dx=\frac {\frac {4 \, {\left (3 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}}{{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3}} - 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) + 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{6 \, b} \]
1/6*(4*(3*(e^(b*x + a) + e^(-b*x - a))^2 + 4)/(e^(b*x + a) + e^(-b*x - a)) ^3 - 3*log(e^(b*x + a) + e^(-b*x - a) + 2) + 3*log(e^(b*x + a) + e^(-b*x - a) - 2))/b
Time = 2.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.50 \[ \int \text {csch}(a+b x) \text {sech}^4(a+b x) \, dx=\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}+1\right )}+\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]