Optimal. Leaf size=22 \[ \frac {2 (d \tan (e+f x))^{3/2}}{3 d f} \]
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Rubi [A] time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2607, 32} \[ \frac {2 (d \tan (e+f x))^{3/2}}{3 d f} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2607
Rubi steps
\begin {align*} \int \sec ^2(e+f x) \sqrt {d \tan (e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {d x} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 (d \tan (e+f x))^{3/2}}{3 d f}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 22, normalized size = 1.00 \[ \frac {2 (d \tan (e+f x))^{3/2}}{3 d f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 37, normalized size = 1.68 \[ \frac {2 \, \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{3 \, f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 23, normalized size = 1.05 \[ \frac {2 \, \sqrt {d \tan \left (f x + e\right )} \tan \left (f x + e\right )}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 19, normalized size = 0.86 \[ \frac {2 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 18, normalized size = 0.82 \[ \frac {2 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{3 \, d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.57, size = 53, normalized size = 2.41 \[ \frac {2\,\sin \left (2\,e+2\,f\,x\right )\,\sqrt {\frac {d\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}}{3\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
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 \[ \int \sqrt {d \tan {\left (e + f x \right )}} \sec ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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