Optimal. Leaf size=20 \[ \frac {2 b (b \sec (e+f x))^{3/2}}{3 f} \]
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Rubi [A]
time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2702, 30}
\begin {gather*} \frac {2 b (b \sec (e+f x))^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2702
Rubi steps
\begin {align*} \int (b \sec (e+f x))^{5/2} \sin (e+f x) \, dx &=\frac {b \text {Subst}\left (\int \sqrt {x} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac {2 b (b \sec (e+f x))^{3/2}}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 b (b \sec (e+f x))^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 17, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {2 b \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f}\) | \(17\) |
default | \(\frac {2 b \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 25, normalized size = 1.25 \begin {gather*} \frac {2 \, \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {5}{2}} \cos \left (f x + e\right )}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 30, normalized size = 1.50 \begin {gather*} \frac {2 \, b^{2} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{3 \, f \cos \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs.
\(2 (17) = 34\).
time = 4.58, size = 36, normalized size = 1.80 \begin {gather*} \frac {2 \, b^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{3 \, \sqrt {b \cos \left (f x + e\right )} f \cos \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 39, normalized size = 1.95 \begin {gather*} \frac {4\,b^2\,\cos \left (e+f\,x\right )\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{3\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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