Optimal. Leaf size=46 \[ \frac {2 \sinh (a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2636, 2641} \[ \frac {2 \sinh (a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2636
Rule 2641
Rubi steps
\begin {align*} \int \frac {1}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx &=\frac {2 \sinh (a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}+\frac {1}{3} \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx\\ &=-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b}+\frac {2 \sinh (a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.07, size = 84, normalized size = 1.83 \[ \frac {2 \left (\cosh (a+b x) \sqrt {\sinh (2 (a+b x))+\cosh (2 (a+b x))+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\cosh (2 (a+b x))-\sinh (2 (a+b x))\right )+\sinh (a+b x)\right )}{3 b \cosh ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\cosh \left (b x + a\right )^{\frac {5}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cosh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.33, size = 217, normalized size = 4.72 \[ \frac {2 \left (2 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right ) \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{3 \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )^{\frac {3}{2}} \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cosh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cosh ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________