Optimal. Leaf size=109 \[ \frac{2 \sec ^3(e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{4 \sec (e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{4 \sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{7 f \sqrt{d \tan (e+f x)}} \]
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Rubi [A] time = 0.136933, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2613, 2614, 2573, 2641} \[ \frac{2 \sec ^3(e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{4 \sec (e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{4 \sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{7 f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2613
Rule 2614
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^5(e+f x)}{\sqrt{d \tan (e+f x)}} \, dx &=\frac{2 \sec ^3(e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{6}{7} \int \frac{\sec ^3(e+f x)}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{4 \sec (e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{2 \sec ^3(e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{4}{7} \int \frac{\sec (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{4 \sec (e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{2 \sec ^3(e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{\left (4 \sqrt{\sin (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)} \sqrt{\sin (e+f x)}} \, dx}{7 \sqrt{\cos (e+f x)} \sqrt{d \tan (e+f x)}}\\ &=\frac{4 \sec (e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{2 \sec ^3(e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{\left (4 \sec (e+f x) \sqrt{\sin (2 e+2 f x)}\right ) \int \frac{1}{\sqrt{\sin (2 e+2 f x)}} \, dx}{7 \sqrt{d \tan (e+f x)}}\\ &=\frac{4 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sec (e+f x) \sqrt{\sin (2 e+2 f x)}}{7 f \sqrt{d \tan (e+f x)}}+\frac{4 \sec (e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{2 \sec ^3(e+f x) \sqrt{d \tan (e+f x)}}{7 d f}\\ \end{align*}
Mathematica [C] time = 0.541142, size = 79, normalized size = 0.72 \[ \frac{2 \sin (e+f x) \left (4 \sqrt{\sec ^2(e+f x)} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(e+f x)\right )+(\cos (2 (e+f x))+2) \sec ^4(e+f x)\right )}{7 f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.196, size = 222, normalized size = 2. \begin{align*} -{\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) -1 \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{7\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3} \left ( \cos \left ( fx+e \right ) \right ) ^{4}} \left ( 4\,\sin \left ( fx+e \right ) \sqrt{{\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) -1+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}} \left ( \cos \left ( fx+e \right ) \right ) ^{3}{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},1/2\,\sqrt{2} \right ) -2\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\sqrt{2}+2\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sqrt{2}-\cos \left ( fx+e \right ) \sqrt{2}+\sqrt{2} \right ){\frac{1}{\sqrt{{\frac{d\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}}}} \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{\sec \left (f x + e\right )^{5}}{\sqrt{d \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \tan \left (f x + e\right )} \sec \left (f x + e\right )^{5}}{d \tan \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{5}{\left (e + f x \right )}}{\sqrt{d \tan{\left (e + f x \right )}}}\, dx \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{\sec \left (f x + e\right )^{5}}{\sqrt{d \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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