Optimal. Leaf size=22 \[ \frac{2 (d \tan (e+f x))^{7/2}}{7 d f} \]
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Rubi [A] time = 0.0427842, antiderivative size = 22, normalized size of antiderivative = 1., 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} \[ \frac{2 (d \tan (e+f x))^{7/2}}{7 d f} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int (d x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 (d \tan (e+f x))^{7/2}}{7 d f}\\ \end{align*}
Mathematica [A] time = 0.0548, size = 22, normalized size = 1. \[ \frac{2 (d \tan (e+f x))^{7/2}}{7 d f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 19, normalized size = 0.9 \begin{align*}{\frac{2}{7\,df} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940318, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \, \left (d \tan \left (f x + e\right )\right )^{\frac{7}{2}}}{7 \, d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74948, size = 134, normalized size = 6.09 \begin{align*} -\frac{2 \,{\left (d^{2} \cos \left (f x + e\right )^{2} - d^{2}\right )} \sqrt{\frac{d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{7 \, f \cos \left (f x + e\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33562, size = 38, normalized size = 1.73 \begin{align*} \frac{2 \, \sqrt{d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right )^{3}}{7 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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