Optimal. Leaf size=62 \[ \frac {2 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{b}+\frac {2 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 62, 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 = {3853, 3856,
2719} \begin {gather*} \frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}+\frac {2 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 )}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2719
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int \text {sech}^{\frac {3}{2}}(a+b x) \, dx &=\frac {2 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{b}-\int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx\\ &=\frac {2 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{b}-\left (\sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \sqrt {\cosh (a+b x)} \, dx\\ &=\frac {2 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)}}{b}+\frac {2 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 49, normalized size = 0.79 \begin {gather*} \frac {2 \sqrt {\text {sech}(a+b x)} \left (i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )+\sinh (a+b x)\right )}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.68, size = 103, normalized size = 1.66
method | result | size |
default | \(\frac {4 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+2 \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b}\) | \(103\) |
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 = 96, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (\sqrt {2} \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}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )\right )}}{b} \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 \operatorname {sech}^{\frac {3}{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 {\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________