Integrand size = 11, antiderivative size = 31 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=-\frac {4}{3} \text {sech}^{\frac {3}{4}}(x)+\frac {8}{11} \text {sech}^{\frac {11}{4}}(x)-\frac {4}{19} \text {sech}^{\frac {19}{4}}(x) \]
Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=\text {sech}^{\frac {3}{4}}(x) \left (-\frac {4}{3}+\frac {8 \text {sech}^2(x)}{11}-\frac {4 \text {sech}^4(x)}{19}\right ) \]
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 26, 3102, 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.
\(\displaystyle \int \sinh ^5(x) \text {sech}^{\frac {23}{4}}(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sec (i x)^{23/4}}{\csc (i x)^5}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sec (i x)^{23/4}}{\csc (i x)^5}dx\) |
\(\Big \downarrow \) 3102 |
\(\displaystyle -\int \frac {\left (1-\text {sech}^2(x)\right )^2}{\sqrt [4]{\text {sech}(x)}}d\text {sech}(x)\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\int \left (\text {sech}^{\frac {15}{4}}(x)-2 \text {sech}^{\frac {7}{4}}(x)+\frac {1}{\sqrt [4]{\text {sech}(x)}}\right )d\text {sech}(x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4}{19} \text {sech}^{\frac {19}{4}}(x)+\frac {8}{11} \text {sech}^{\frac {11}{4}}(x)-\frac {4}{3} \text {sech}^{\frac {3}{4}}(x)\) |
3.6.82.3.1 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])
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(-\frac {4 \operatorname {sech}\left (x \right )^{\frac {3}{4}}}{3}+\frac {8 \operatorname {sech}\left (x \right )^{\frac {11}{4}}}{11}-\frac {4 \operatorname {sech}\left (x \right )^{\frac {19}{4}}}{19}\) | \(20\) |
default | \(-\frac {4 \operatorname {sech}\left (x \right )^{\frac {3}{4}}}{3}+\frac {8 \operatorname {sech}\left (x \right )^{\frac {11}{4}}}{11}-\frac {4 \operatorname {sech}\left (x \right )^{\frac {19}{4}}}{19}\) | \(20\) |
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (19) = 38\).
Time = 0.24 (sec) , antiderivative size = 359, normalized size of antiderivative = 11.58 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=-\frac {4 \cdot 2^{\frac {3}{4}} {\left (209 \, \cosh \left (x\right )^{8} + 1672 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + 209 \, \sinh \left (x\right )^{8} + 76 \, {\left (77 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{6} + 380 \, \cosh \left (x\right )^{6} + 152 \, {\left (77 \, \cosh \left (x\right )^{3} + 15 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 10 \, {\left (1463 \, \cosh \left (x\right )^{4} + 570 \, \cosh \left (x\right )^{2} + 87\right )} \sinh \left (x\right )^{4} + 870 \, \cosh \left (x\right )^{4} + 8 \, {\left (1463 \, \cosh \left (x\right )^{5} + 950 \, \cosh \left (x\right )^{3} + 435 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (1463 \, \cosh \left (x\right )^{6} + 1425 \, \cosh \left (x\right )^{4} + 1305 \, \cosh \left (x\right )^{2} + 95\right )} \sinh \left (x\right )^{2} + 380 \, \cosh \left (x\right )^{2} + 8 \, {\left (209 \, \cosh \left (x\right )^{7} + 285 \, \cosh \left (x\right )^{5} + 435 \, \cosh \left (x\right )^{3} + 95 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 209\right )} \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}\right )^{\frac {3}{4}}}{627 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{6} + 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} + 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} + 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
-4/627*2^(3/4)*(209*cosh(x)^8 + 1672*cosh(x)*sinh(x)^7 + 209*sinh(x)^8 + 7 6*(77*cosh(x)^2 + 5)*sinh(x)^6 + 380*cosh(x)^6 + 152*(77*cosh(x)^3 + 15*co sh(x))*sinh(x)^5 + 10*(1463*cosh(x)^4 + 570*cosh(x)^2 + 87)*sinh(x)^4 + 87 0*cosh(x)^4 + 8*(1463*cosh(x)^5 + 950*cosh(x)^3 + 435*cosh(x))*sinh(x)^3 + 4*(1463*cosh(x)^6 + 1425*cosh(x)^4 + 1305*cosh(x)^2 + 95)*sinh(x)^2 + 380 *cosh(x)^2 + 8*(209*cosh(x)^7 + 285*cosh(x)^5 + 435*cosh(x)^3 + 95*cosh(x) )*sinh(x) + 209)*((cosh(x) + sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sin h(x)^2 + 1))^(3/4)/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cos h(x)^2 + 1)*sinh(x)^6 + 4*cosh(x)^6 + 8*(7*cosh(x)^3 + 3*cosh(x))*sinh(x)^ 5 + 2*(35*cosh(x)^4 + 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cos h(x)^5 + 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 + 15*cosh(x) ^4 + 9*cosh(x)^2 + 1)*sinh(x)^2 + 4*cosh(x)^2 + 8*(cosh(x)^7 + 3*cosh(x)^5 + 3*cosh(x)^3 + cosh(x))*sinh(x) + 1)
Time = 39.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=- \frac {4 \tanh ^{4}{\left (x \right )} \operatorname {sech}^{\frac {3}{4}}{\left (x \right )}}{19} - \frac {64 \tanh ^{2}{\left (x \right )} \operatorname {sech}^{\frac {3}{4}}{\left (x \right )}}{209} - \frac {512 \operatorname {sech}^{\frac {3}{4}}{\left (x \right )}}{627} \]
\[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=\int { \operatorname {sech}\left (x\right )^{\frac {3}{4}} \tanh \left (x\right )^{5} \,d x } \]
\[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=\int { \operatorname {sech}\left (x\right )^{\frac {3}{4}} \tanh \left (x\right )^{5} \,d x } \]
Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.87 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=\frac {32\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{11\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {1312\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{209\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^2}+\frac {128\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{19\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}-\frac {64\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{19\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^4}-\frac {4\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{3} \]