Optimal. Leaf size=66 \[ -\frac {6 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{5 b}+\frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3854, 3856,
2719} \begin {gather*} \frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}-\frac {6 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2719
Rule 3854
Rule 3856
Rubi steps
\begin {align*} \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx &=\frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}+\frac {3}{5} \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx\\ &=\frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}+\frac {1}{5} \left (3 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \sqrt {\cosh (a+b x)} \, dx\\ &=-\frac {6 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{5 b}+\frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 59, normalized size = 0.89 \begin {gather*} \frac {\sqrt {\text {sech}(a+b x)} \left (-12 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )+\sinh (a+b x)+\sinh (3 (a+b x))\right )}{10 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(187\) vs.
\(2(82)=164\).
time = 1.66, size = 188, normalized size = 2.85
method | result | size |
default | \(\frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (8 \left (\cosh ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-16 \left (\cosh ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+10 \left (\cosh ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-3 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b}\) | \(188\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 370, normalized size = 5.61 \begin {gather*} \frac {\sqrt {2} {\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} + {\left (15 \, \cosh \left (b x + a\right )^{2} - 11\right )} \sinh \left (b x + a\right )^{4} - 11 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - 11 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} - 66 \, \cosh \left (b x + a\right )^{2} - 13\right )} \sinh \left (b x + a\right )^{2} - 13 \, \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} - 22 \, \cosh \left (b x + a\right )^{3} - 13 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\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}} - 24 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{3} + 3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sqrt {2} \sinh \left (b x + a\right )^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )}{20 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\operatorname {sech}^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________