Integrand size = 19, antiderivative size = 37 \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=-\frac {\text {csch}^n(a+b x)}{b n}-\frac {\text {csch}^{2+n}(a+b x)}{b (2+n)} \]
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=-\frac {\text {csch}^n(a+b x) \left (2+n+n \text {csch}^2(a+b x)\right )}{b n (2+n)} \]
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3042, 3101, 25, 244, 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.
\(\displaystyle \int \cosh ^3(a+b x) \text {csch}^{n+3}(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(i \csc (i a+i b x))^{n+3}}{\sec (i a+i b x)^3}dx\) |
\(\Big \downarrow \) 3101 |
\(\displaystyle \frac {\int -\text {csch}^{n-1}(a+b x) \left (\text {csch}^2(a+b x)+1\right )d\text {csch}(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \text {csch}^{n-1}(a+b x) \left (\text {csch}^2(a+b x)+1\right )d\text {csch}(a+b x)}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {\int \left (\text {csch}^{n-1}(a+b x)+\text {csch}^{n+1}(a+b x)\right )d\text {csch}(a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\text {csch}^{n+2}(a+b x)}{n+2}-\frac {\text {csch}^n(a+b x)}{n}}{b}\) |
3.2.20.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_S ymbol] :> Simp[-(f*a^n)^(-1) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.86 (sec) , antiderivative size = 479, normalized size of antiderivative = 12.95
method | result | size |
risch | \(-\frac {\left (n \,{\mathrm e}^{4 b x +4 a}+2 \,{\mathrm e}^{4 b x +4 a}+2 n \,{\mathrm e}^{2 b x +2 a}-4 \,{\mathrm e}^{2 b x +2 a}+n +2\right ) 2^{n} \left ({\mathrm e}^{b x +a}-1\right )^{-n} \left ({\mathrm e}^{b x +a}\right )^{n} \left ({\mathrm e}^{b x +a}+1\right )^{-n} {\mathrm e}^{-\frac {i \pi n \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{b x +a}-1}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{b x +a}+1}\right ) \operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )-\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{b x +a}-1}\right ) {\operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )}^{2}+\operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left ({\mathrm e}^{b x +a}+1\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{3}-\operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left ({\mathrm e}^{b x +a}+1\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{b x +a}\right )-\operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left ({\mathrm e}^{b x +a}+1\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )+\operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left ({\mathrm e}^{b x +a}+1\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right ) \operatorname {csgn}\left (i {\mathrm e}^{b x +a}\right ) \operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )-\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{b x +a}+1}\right ) {\operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )}^{2}+{\operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{b x +a}-1\right ) \left ({\mathrm e}^{b x +a}+1\right )}\right )}^{3}\right )}{2}}}{b n \left (n +2\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}\) | \(479\) |
-(n*exp(4*b*x+4*a)+2*exp(4*b*x+4*a)+2*n*exp(2*b*x+2*a)-4*exp(2*b*x+2*a)+n+ 2)/b/n/(n+2)/(exp(2*b*x+2*a)-1)^2*2^n*(exp(b*x+a)-1)^(-n)*exp(b*x+a)^n*(ex p(b*x+a)+1)^(-n)*exp(-1/2*I*Pi*n*(csgn(I/(exp(b*x+a)-1))*csgn(I/(exp(b*x+a )+1))*csgn(I/(exp(b*x+a)-1)/(exp(b*x+a)+1))-csgn(I/(exp(b*x+a)-1))*csgn(I/ (exp(b*x+a)-1)/(exp(b*x+a)+1))^2+csgn(I*exp(b*x+a)/(exp(b*x+a)+1)/(exp(b*x +a)-1))^3-csgn(I*exp(b*x+a)/(exp(b*x+a)+1)/(exp(b*x+a)-1))^2*csgn(I*exp(b* x+a))-csgn(I*exp(b*x+a)/(exp(b*x+a)+1)/(exp(b*x+a)-1))^2*csgn(I/(exp(b*x+a )-1)/(exp(b*x+a)+1))+csgn(I*exp(b*x+a)/(exp(b*x+a)+1)/(exp(b*x+a)-1))*csgn (I*exp(b*x+a))*csgn(I/(exp(b*x+a)-1)/(exp(b*x+a)+1))-csgn(I/(exp(b*x+a)+1) )*csgn(I/(exp(b*x+a)-1)/(exp(b*x+a)+1))^2+csgn(I/(exp(b*x+a)-1)/(exp(b*x+a )+1))^3))
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (37) = 74\).
Time = 0.26 (sec) , antiderivative size = 216, normalized size of antiderivative = 5.84 \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=\frac {{\left ({\left (n + 2\right )} \cosh \left (b x + a\right )^{2} + {\left (n + 2\right )} \sinh \left (b x + a\right )^{2} + n - 2\right )} \cosh \left (n \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right ) + {\left ({\left (n + 2\right )} \cosh \left (b x + a\right )^{2} + {\left (n + 2\right )} \sinh \left (b x + a\right )^{2} + n - 2\right )} \sinh \left (n \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right )}{b n^{2} - {\left (b n^{2} + 2 \, b n\right )} \cosh \left (b x + a\right )^{2} - {\left (b n^{2} + 2 \, b n\right )} \sinh \left (b x + a\right )^{2} + 2 \, b n} \]
(((n + 2)*cosh(b*x + a)^2 + (n + 2)*sinh(b*x + a)^2 + n - 2)*cosh(n*log(2* (cosh(b*x + a) + sinh(b*x + a))/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b* x + a) + sinh(b*x + a)^2 - 1))) + ((n + 2)*cosh(b*x + a)^2 + (n + 2)*sinh( b*x + a)^2 + n - 2)*sinh(n*log(2*(cosh(b*x + a) + sinh(b*x + a))/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1))))/(b*n^2 - (b*n^2 + 2*b*n)*cosh(b*x + a)^2 - (b*n^2 + 2*b*n)*sinh(b*x + a)^2 + 2*b*n )
\[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=\int \coth ^{3}{\left (a + b x \right )} \operatorname {csch}^{n}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (37) = 74\).
Time = 0.34 (sec) , antiderivative size = 414, normalized size of antiderivative = 11.19 \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=-\frac {2^{n} n e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right )\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {{\left (2^{n + 1} n - 2^{n + 2}\right )} e^{\left (-{\left (b x + a\right )} n - 2 \, b x - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 2 \, a\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {{\left (2^{n} n + 2^{n + 1}\right )} e^{\left (-{\left (b x + a\right )} n - 4 \, b x - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 4 \, a\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {2^{n + 1} e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right )\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} \]
-2^n*n*e^(-(b*x + a)*n - n*log(e^(-b*x - a) + 1) - n*log(-e^(-b*x - a) + 1 ))/((n^2 - 2*(n^2 + 2*n)*e^(-2*b*x - 2*a) + (n^2 + 2*n)*e^(-4*b*x - 4*a) + 2*n)*b) - (2^(n + 1)*n - 2^(n + 2))*e^(-(b*x + a)*n - 2*b*x - n*log(e^(-b *x - a) + 1) - n*log(-e^(-b*x - a) + 1) - 2*a)/((n^2 - 2*(n^2 + 2*n)*e^(-2 *b*x - 2*a) + (n^2 + 2*n)*e^(-4*b*x - 4*a) + 2*n)*b) - (2^n*n + 2^(n + 1)) *e^(-(b*x + a)*n - 4*b*x - n*log(e^(-b*x - a) + 1) - n*log(-e^(-b*x - a) + 1) - 4*a)/((n^2 - 2*(n^2 + 2*n)*e^(-2*b*x - 2*a) + (n^2 + 2*n)*e^(-4*b*x - 4*a) + 2*n)*b) - 2^(n + 1)*e^(-(b*x + a)*n - n*log(e^(-b*x - a) + 1) - n *log(-e^(-b*x - a) + 1))/((n^2 - 2*(n^2 + 2*n)*e^(-2*b*x - 2*a) + (n^2 + 2 *n)*e^(-4*b*x - 4*a) + 2*n)*b)
\[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=\int { \operatorname {csch}\left (b x + a\right )^{n} \coth \left (b x + a\right )^{3} \,d x } \]
Time = 2.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.70 \[ \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx=-\frac {{\left (\frac {1}{\frac {{\mathrm {e}}^{a+b\,x}}{2}-\frac {{\mathrm {e}}^{-a-b\,x}}{2}}\right )}^n\,\left (\frac {1}{b\,n}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{b\,n}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\left (2\,n-4\right )}{b\,n\,\left (n+2\right )}\right )}{{\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1} \]