Integrand size = 19, antiderivative size = 36 \[ \int \text {sech}^{3+n}(a+b x) \sinh ^3(a+b x) \, dx=-\frac {\text {sech}^n(a+b x)}{b n}+\frac {\text {sech}^{2+n}(a+b x)}{b (2+n)} \]
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \text {sech}^{3+n}(a+b x) \sinh ^3(a+b x) \, dx=\frac {\text {sech}^n(a+b x) \left (-\frac {1}{n}+\frac {\text {sech}^2(a+b x)}{2+n}\right )}{b} \]
Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3042, 26, 3102, 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.
\(\displaystyle \int \sinh ^3(a+b x) \text {sech}^{n+3}(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sec (i a+i b x)^{n+3}}{\csc (i a+i b x)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\sec (i a+i b x)^{n+3}}{\csc (i a+i b x)^3}dx\) |
\(\Big \downarrow \) 3102 |
\(\displaystyle \frac {\int -\text {sech}^{n-1}(a+b x) \left (1-\text {sech}^2(a+b x)\right )d\text {sech}(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \text {sech}^{n-1}(a+b x) \left (1-\text {sech}^2(a+b x)\right )d\text {sech}(a+b x)}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {\int \left (\text {sech}^{n-1}(a+b x)-\text {sech}^{n+1}(a+b x)\right )d\text {sech}(a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\text {sech}^{n+2}(a+b x)}{n+2}-\frac {\text {sech}^n(a+b x)}{n}}{b}\) |
3.1.89.3.1 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[((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_)]^(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])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.18 (sec) , antiderivative size = 209, normalized size of antiderivative = 5.81
method | result | size |
risch | \(-\frac {\left (n \,{\mathrm e}^{4 b x +4 a}+2 \,{\mathrm e}^{4 b x +4 a}-2 n \,{\mathrm e}^{2 b x +2 a}+4 \,{\mathrm e}^{2 b x +2 a}+n +2\right ) 2^{n} \left ({\mathrm e}^{b x +a}\right )^{n} \left (1+{\mathrm e}^{2 b x +2 a}\right )^{-n} {\mathrm e}^{-\frac {i \operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}\right ) \pi n \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}\right )+\operatorname {csgn}\left (\frac {i}{1+{\mathrm e}^{2 b x +2 a}}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}\right )+\operatorname {csgn}\left (i {\mathrm e}^{b x +a}\right )\right )}{2}}}{b n \left (n +2\right ) \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}\) | \(209\) |
-(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)/(1+exp(2*b*x+2*a))^2*2^n*exp(b*x+a)^n*(1+exp(2*b*x+2*a))^(-n) *exp(-1/2*I*csgn(I*exp(b*x+a)/(1+exp(2*b*x+2*a)))*Pi*n*(-csgn(I*exp(b*x+a) /(1+exp(2*b*x+2*a)))+csgn(I/(1+exp(2*b*x+2*a))))*(-csgn(I*exp(b*x+a)/(1+ex p(2*b*x+2*a)))+csgn(I*exp(b*x+a))))
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (36) = 72\).
Time = 0.24 (sec) , antiderivative size = 219, normalized size of antiderivative = 6.08 \[ \int \text {sech}^{3+n}(a+b x) \sinh ^3(a+b x) \, dx=-\frac {{\left ({\left (n + 2\right )} \cosh \left (b x + a\right )^{2} + {\left (n + 2\right )} \sinh \left (b x + a\right )^{2} - n + 2\right )} \cosh \left (n \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}\right )\right ) + {\left ({\left (n + 2\right )} \cosh \left (b x + a\right )^{2} + {\left (n + 2\right )} \sinh \left (b x + a\right )^{2} - n + 2\right )} \sinh \left (n \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}\right )\right )}{b n^{2} + {\left (b n^{2} + 2 \, b n\right )} \cosh \left (b x + a\right )^{2} + {\left (b n^{2} + 2 \, b n\right )} \sinh \left (b x + a\right )^{2} + 2 \, b n} \]
-(((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)
\[ \int \text {sech}^{3+n}(a+b x) \sinh ^3(a+b x) \, dx=\begin {cases} x \tanh ^{3}{\left (a \right )} \operatorname {sech}^{n}{\left (a \right )} & \text {for}\: b = 0 \\\int \frac {\tanh ^{3}{\left (a + b x \right )}}{\operatorname {sech}^{2}{\left (a + b x \right )}}\, dx & \text {for}\: n = -2 \\x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} - \frac {\tanh ^{2}{\left (a + b x \right )}}{2 b} & \text {for}\: n = 0 \\- \frac {n \tanh ^{2}{\left (a + b x \right )} \operatorname {sech}^{n}{\left (a + b x \right )}}{b n^{2} + 2 b n} - \frac {2 \operatorname {sech}^{n}{\left (a + b x \right )}}{b n^{2} + 2 b n} & \text {otherwise} \end {cases} \]
Piecewise((x*tanh(a)**3*sech(a)**n, Eq(b, 0)), (Integral(tanh(a + b*x)**3/ sech(a + b*x)**2, x), Eq(n, -2)), (x - log(tanh(a + b*x) + 1)/b - tanh(a + b*x)**2/(2*b), Eq(n, 0)), (-n*tanh(a + b*x)**2*sech(a + b*x)**n/(b*n**2 + 2*b*n) - 2*sech(a + b*x)**n/(b*n**2 + 2*b*n), True))
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (36) = 72\).
Time = 0.34 (sec) , antiderivative size = 345, normalized size of antiderivative = 9.58 \[ \int \text {sech}^{3+n}(a+b x) \sinh ^3(a+b x) \, dx=-\frac {2^{n} n e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )\right )}}{{\left (n^{2} + 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} + \frac {{\left (2^{n + 1} n - 2^{n + 2}\right )} e^{\left (-{\left (b x + a\right )} n - 2 \, b x - n \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right ) - 2 \, a\right )}}{{\left (n^{2} + 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {{\left (2^{n} n + 2^{n + 1}\right )} e^{\left (-{\left (b x + a\right )} n - 4 \, b x - n \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right ) - 4 \, a\right )}}{{\left (n^{2} + 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {2^{n + 1} e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )\right )}}{{\left (n^{2} + 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} \]
-2^n*n*e^(-(b*x + a)*n - n*log(e^(-2*b*x - 2*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^(-2*b*x - 2*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^(-2*b*x - 2 *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^(-2*b*x - 2*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)
\[ \int \text {sech}^{3+n}(a+b x) \sinh ^3(a+b x) \, dx=\int { \operatorname {sech}\left (b x + a\right )^{n} \tanh \left (b x + a\right )^{3} \,d x } \]
Time = 2.38 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.81 \[ \int \text {sech}^{3+n}(a+b x) \sinh ^3(a+b x) \, dx=-\frac {{\left (\frac {1}{\frac {{\mathrm {e}}^{a+b\,x}}{2}+\frac {{\mathrm {e}}^{-a-b\,x}}{2}}\right )}^n\,\left (\frac {1}{b\,n}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{b\,n}-\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\left (2\,n-4\right )}{b\,n\,\left (n+2\right )}\right )}{2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1} \]