Optimal. Leaf size=103 \[ \frac{8 b \sqrt{b \tan (e+f x)}}{45 d^4 f \sqrt{d \sec (e+f x)}}+\frac{2 b \sqrt{b \tan (e+f x)}}{45 d^2 f (d \sec (e+f x))^{5/2}}-\frac{2 b \sqrt{b \tan (e+f x)}}{9 f (d \sec (e+f x))^{9/2}} \]
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Rubi [A] time = 0.162904, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2610, 2612, 2605} \[ \frac{8 b \sqrt{b \tan (e+f x)}}{45 d^4 f \sqrt{d \sec (e+f x)}}+\frac{2 b \sqrt{b \tan (e+f x)}}{45 d^2 f (d \sec (e+f x))^{5/2}}-\frac{2 b \sqrt{b \tan (e+f x)}}{9 f (d \sec (e+f x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2610
Rule 2612
Rule 2605
Rubi steps
\begin{align*} \int \frac{(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{9/2}} \, dx &=-\frac{2 b \sqrt{b \tan (e+f x)}}{9 f (d \sec (e+f x))^{9/2}}+\frac{b^2 \int \frac{1}{(d \sec (e+f x))^{5/2} \sqrt{b \tan (e+f x)}} \, dx}{9 d^2}\\ &=-\frac{2 b \sqrt{b \tan (e+f x)}}{9 f (d \sec (e+f x))^{9/2}}+\frac{2 b \sqrt{b \tan (e+f x)}}{45 d^2 f (d \sec (e+f x))^{5/2}}+\frac{\left (4 b^2\right ) \int \frac{1}{\sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}} \, dx}{45 d^4}\\ &=-\frac{2 b \sqrt{b \tan (e+f x)}}{9 f (d \sec (e+f x))^{9/2}}+\frac{2 b \sqrt{b \tan (e+f x)}}{45 d^2 f (d \sec (e+f x))^{5/2}}+\frac{8 b \sqrt{b \tan (e+f x)}}{45 d^4 f \sqrt{d \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.3127, size = 158, normalized size = 1.53 \[ -\frac{b \sqrt{\sec (e+f x)} \sqrt{b \tan (e+f x)} \left (-21 \sqrt{\sec (e+f x)+1} \sec ^2\left (\frac{1}{2} (e+f x)\right )+\sqrt{\frac{1}{\cos (e+f x)+1}} (21 \cos (3 (e+f x))+5 \cos (5 (e+f x))) \sec ^{\frac{3}{2}}(e+f x)+16 \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{\sec (e+f x)}\right )}{360 d^3 f \sqrt{\frac{1}{\cos (e+f x)+1}} (d \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.162, size = 62, normalized size = 0.6 \begin{align*}{\frac{ \left ( 10\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+8 \right ) \sin \left ( fx+e \right ) }{45\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ({\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{9}{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 (b \tan \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (d \sec \left (f x + e\right )\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81583, size = 174, normalized size = 1.69 \begin{align*} -\frac{2 \,{\left (5 \, b \cos \left (f x + e\right )^{5} - b \cos \left (f x + e\right )^{3} - 4 \, b \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 )}}}{45 \, d^{5} 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{\left (b \tan \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (d \sec \left (f x + e\right )\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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