Optimal. Leaf size=79 \[ \frac{2 \sec (e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{2 \sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{3 f \sqrt{d \tan (e+f x)}} \]
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Rubi [A] time = 0.0959896, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2613, 2614, 2573, 2641} \[ \frac{2 \sec (e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{2 \sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{3 f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2613
Rule 2614
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^3(e+f x)}{\sqrt{d \tan (e+f x)}} \, dx &=\frac{2 \sec (e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{2}{3} \int \frac{\sec (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{2 \sec (e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{\left (2 \sqrt{\sin (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)} \sqrt{\sin (e+f x)}} \, dx}{3 \sqrt{\cos (e+f x)} \sqrt{d \tan (e+f x)}}\\ &=\frac{2 \sec (e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{\left (2 \sec (e+f x) \sqrt{\sin (2 e+2 f x)}\right ) \int \frac{1}{\sqrt{\sin (2 e+2 f x)}} \, dx}{3 \sqrt{d \tan (e+f x)}}\\ &=\frac{2 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sec (e+f x) \sqrt{\sin (2 e+2 f x)}}{3 f \sqrt{d \tan (e+f x)}}+\frac{2 \sec (e+f x) \sqrt{d \tan (e+f x)}}{3 d f}\\ \end{align*}
Mathematica [C] time = 0.264323, size = 68, normalized size = 0.86 \[ \frac{2 \sin (e+f x) \left (2 \sqrt{\sec ^2(e+f x)} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(e+f x)\right )+\sec ^2(e+f x)\right )}{3 f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.171, size = 194, normalized size = 2.5 \begin{align*} -{\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) -1 \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{3\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3} \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( 2\,{\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 ) \sin \left ( fx+e \right ) \cos \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 ) }}}-\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 )^{3}}{\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 )^{3}}{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 ^{3}{\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 )^{3}}{\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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