Optimal. Leaf size=72 \[ \frac{2}{3 b f \sqrt{b \tan (e+f x)} (d \sec (e+f x))^{3/2}}-\frac{8 \sqrt{d \sec (e+f x)}}{3 b d^2 f \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.108494, antiderivative size = 67, normalized size of antiderivative = 0.93, 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 = {2609, 2605} \[ -\frac{8 (b \tan (e+f x))^{3/2}}{3 b^3 f (d \sec (e+f x))^{3/2}}-\frac{2}{b f \sqrt{b \tan (e+f x)} (d \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2609
Rule 2605
Rubi steps
\begin{align*} \int \frac{1}{(d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}} \, dx &=-\frac{2}{b f (d \sec (e+f x))^{3/2} \sqrt{b \tan (e+f x)}}-\frac{4 \int \frac{\sqrt{b \tan (e+f x)}}{(d \sec (e+f x))^{3/2}} \, dx}{b^2}\\ &=-\frac{2}{b f (d \sec (e+f x))^{3/2} \sqrt{b \tan (e+f x)}}-\frac{8 (b \tan (e+f x))^{3/2}}{3 b^3 f (d \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.176048, size = 52, normalized size = 0.72 \[ \frac{(\cos (2 (e+f x))-7) \sec ^2(e+f x)}{3 b f \sqrt{b \tan (e+f x)} (d \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.147, size = 60, normalized size = 0.8 \begin{align*}{\frac{2\,\sin \left ( fx+e \right ) \left ( -4+ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) }{3\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \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{3}{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{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}} \left (b \tan \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67619, size = 161, normalized size = 2.24 \begin{align*} \frac{2 \,{\left (\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 )}}}{3 \, b^{2} d^{2} f \sin \left (f x + e\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{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}} \left (b \tan \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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