Optimal. Leaf size=31 \[ -\frac {4}{19} \text {sech}^{\frac {19}{4}}(x)+\frac {8}{11} \text {sech}^{\frac {11}{4}}(x)-\frac {4}{3} \text {sech}^{\frac {3}{4}}(x) \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2622, 270} \[ -\frac {4}{19} \text {sech}^{\frac {19}{4}}(x)+\frac {8}{11} \text {sech}^{\frac {11}{4}}(x)-\frac {4}{3} \text {sech}^{\frac {3}{4}}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 2622
Rubi steps
\begin {align*} \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx &=-\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{\sqrt [4]{x}} \, dx,x,\text {sech}(x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{x}}-2 x^{7/4}+x^{15/4}\right ) \, dx,x,\text {sech}(x)\right )\\ &=-\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)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 27, normalized size = 0.87 \[ \text {sech}^{\frac {3}{4}}(x) \left (-\frac {4}{19} \text {sech}^4(x)+\frac {8 \text {sech}^2(x)}{11}-\frac {4}{3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.67, size = 359, normalized size = 11.58 \[ -\frac {4 \cdot 2^{\frac {3}{4}} {\left (209 \, \cosh \relax (x)^{8} + 1672 \, \cosh \relax (x) \sinh \relax (x)^{7} + 209 \, \sinh \relax (x)^{8} + 76 \, {\left (77 \, \cosh \relax (x)^{2} + 5\right )} \sinh \relax (x)^{6} + 380 \, \cosh \relax (x)^{6} + 152 \, {\left (77 \, \cosh \relax (x)^{3} + 15 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 10 \, {\left (1463 \, \cosh \relax (x)^{4} + 570 \, \cosh \relax (x)^{2} + 87\right )} \sinh \relax (x)^{4} + 870 \, \cosh \relax (x)^{4} + 8 \, {\left (1463 \, \cosh \relax (x)^{5} + 950 \, \cosh \relax (x)^{3} + 435 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 4 \, {\left (1463 \, \cosh \relax (x)^{6} + 1425 \, \cosh \relax (x)^{4} + 1305 \, \cosh \relax (x)^{2} + 95\right )} \sinh \relax (x)^{2} + 380 \, \cosh \relax (x)^{2} + 8 \, {\left (209 \, \cosh \relax (x)^{7} + 285 \, \cosh \relax (x)^{5} + 435 \, \cosh \relax (x)^{3} + 95 \, \cosh \relax (x)\right )} \sinh \relax (x) + 209\right )} \left (\frac {\cosh \relax (x) + \sinh \relax (x)}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1}\right )^{\frac {3}{4}}}{627 \, {\left (\cosh \relax (x)^{8} + 8 \, \cosh \relax (x) \sinh \relax (x)^{7} + \sinh \relax (x)^{8} + 4 \, {\left (7 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{6} + 4 \, \cosh \relax (x)^{6} + 8 \, {\left (7 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 2 \, {\left (35 \, \cosh \relax (x)^{4} + 30 \, \cosh \relax (x)^{2} + 3\right )} \sinh \relax (x)^{4} + 6 \, \cosh \relax (x)^{4} + 8 \, {\left (7 \, \cosh \relax (x)^{5} + 10 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 4 \, {\left (7 \, \cosh \relax (x)^{6} + 15 \, \cosh \relax (x)^{4} + 9 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x)^{2} + 8 \, {\left (\cosh \relax (x)^{7} + 3 \, \cosh \relax (x)^{5} + 3 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {sech}\relax (x)^{\frac {3}{4}} \tanh \relax (x)^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[\int \mathrm {sech}\relax (x )^{\frac {3}{4}} \left (\tanh ^{5}\relax (x )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {sech}\relax (x)^{\frac {3}{4}} \tanh \relax (x)^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.17, size = 120, normalized size = 3.87 \[ \frac {32\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{11\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {1312\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{209\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^2}+\frac {128\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{19\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}-\frac {64\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{19\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^4}-\frac {4\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________