Optimal. Leaf size=34 \[ -\frac{2 (d \sec (e+f x))^{3/2}}{3 b f (b \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.0568758, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2605} \[ -\frac{2 (d \sec (e+f x))^{3/2}}{3 b f (b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2605
Rubi steps
\begin{align*} \int \frac{(d \sec (e+f x))^{3/2}}{(b \tan (e+f x))^{5/2}} \, dx &=-\frac{2 (d \sec (e+f x))^{3/2}}{3 b f (b \tan (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.156271, size = 34, normalized size = 1. \[ -\frac{2 (d \sec (e+f x))^{3/2}}{3 b f (b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.133, size = 50, normalized size = 1.5 \begin{align*} -{\frac{2\,\sin \left ( fx+e \right ) }{3\,f\cos \left ( fx+e \right ) } \left ({\frac{d}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}} \left ({\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62757, size = 143, normalized size = 4.21 \begin{align*} \frac{2 \, d \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt{\frac{d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{3 \,{\left (b^{3} f \cos \left (f x + e\right )^{2} - b^{3} f\right )}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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