Optimal. Leaf size=72 \[ \frac{8 (b \tan (e+f x))^{3/2}}{21 b d^2 f (d \sec (e+f x))^{3/2}}+\frac{2 (b \tan (e+f x))^{3/2}}{7 b f (d \sec (e+f x))^{7/2}} \]
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Rubi [A] time = 0.103374, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2612, 2605} \[ \frac{8 (b \tan (e+f x))^{3/2}}{21 b d^2 f (d \sec (e+f x))^{3/2}}+\frac{2 (b \tan (e+f x))^{3/2}}{7 b f (d \sec (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2612
Rule 2605
Rubi steps
\begin{align*} \int \frac{\sqrt{b \tan (e+f x)}}{(d \sec (e+f x))^{7/2}} \, dx &=\frac{2 (b \tan (e+f x))^{3/2}}{7 b f (d \sec (e+f x))^{7/2}}+\frac{4 \int \frac{\sqrt{b \tan (e+f x)}}{(d \sec (e+f x))^{3/2}} \, dx}{7 d^2}\\ &=\frac{2 (b \tan (e+f x))^{3/2}}{7 b f (d \sec (e+f x))^{7/2}}+\frac{8 (b \tan (e+f x))^{3/2}}{21 b d^2 f (d \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.169996, size = 53, normalized size = 0.74 \[ \frac{(19 \sin (e+f x)+3 \sin (3 (e+f x))) \sqrt{b \tan (e+f x)}}{42 d^3 f \sqrt{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.191, size = 62, normalized size = 0.9 \begin{align*}{\frac{ \left ( 6\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+8 \right ) \sin \left ( fx+e \right ) }{21\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}}\sqrt{{\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}} \left ({\frac{d}{\cos \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \tan \left (f x + e\right )}}{\left (d \sec \left (f x + e\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7139, size = 159, normalized size = 2.21 \begin{align*} \frac{2 \,{\left (3 \, \cos \left (f x + e\right )^{3} + 4 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt{\frac{d}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{21 \, d^{4} f} \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{\sqrt{b \tan \left (f x + e\right )}}{\left (d \sec \left (f x + e\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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