Optimal. Leaf size=18 \[ \frac {2 b \sqrt {b \sec (e+f x)}}{f} \]
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Rubi [A]
time = 0.02, antiderivative size = 18, 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 \sqrt {b \sec (e+f x)}}{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))^{3/2} \sin (e+f x) \, dx &=\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac {2 b \sqrt {b \sec (e+f x)}}{f}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 b \sqrt {b \sec (e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 17, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {2 b \sqrt {b \sec \left (f x +e \right )}}{f}\) | \(17\) |
default | \(\frac {2 b \sqrt {b \sec \left (f x +e \right )}}{f}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 25, normalized size = 1.39 \begin {gather*} \frac {2 \, \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {3}{2}} \cos \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 19, normalized size = 1.06 \begin {gather*} \frac {2 \, b \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}} \sin {\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.59, size = 27, normalized size = 1.50 \begin {gather*} \frac {2 \, b^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{\sqrt {b \cos \left (f x + e\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 18, normalized size = 1.00 \begin {gather*} \frac {2\,b\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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