Optimal. Leaf size=109 \[ \frac{\cos ^3(e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{5 \cos (e+f x) \sqrt{d \tan (e+f x)}}{6 d f}+\frac{5 \sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{12 f \sqrt{d \tan (e+f x)}} \]
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Rubi [A] time = 0.127342, 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 = {2612, 2614, 2573, 2641} \[ \frac{\cos ^3(e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{5 \cos (e+f x) \sqrt{d \tan (e+f x)}}{6 d f}+\frac{5 \sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{12 f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2612
Rule 2614
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cos ^3(e+f x)}{\sqrt{d \tan (e+f x)}} \, dx &=\frac{\cos ^3(e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{5}{6} \int \frac{\cos (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{5 \cos (e+f x) \sqrt{d \tan (e+f x)}}{6 d f}+\frac{\cos ^3(e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{5}{12} \int \frac{\sec (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{5 \cos (e+f x) \sqrt{d \tan (e+f x)}}{6 d f}+\frac{\cos ^3(e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{\left (5 \sqrt{\sin (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)} \sqrt{\sin (e+f x)}} \, dx}{12 \sqrt{\cos (e+f x)} \sqrt{d \tan (e+f x)}}\\ &=\frac{5 \cos (e+f x) \sqrt{d \tan (e+f x)}}{6 d f}+\frac{\cos ^3(e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{\left (5 \sec (e+f x) \sqrt{\sin (2 e+2 f x)}\right ) \int \frac{1}{\sqrt{\sin (2 e+2 f x)}} \, dx}{12 \sqrt{d \tan (e+f x)}}\\ &=\frac{5 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sec (e+f x) \sqrt{\sin (2 e+2 f x)}}{12 f \sqrt{d \tan (e+f x)}}+\frac{5 \cos (e+f x) \sqrt{d \tan (e+f x)}}{6 d f}+\frac{\cos ^3(e+f x) \sqrt{d \tan (e+f x)}}{3 d f}\\ \end{align*}
Mathematica [C] time = 0.970626, size = 94, normalized size = 0.86 \[ \frac{11 \sin (e+f x)+\sin (3 (e+f x))-10 \sqrt [4]{-1} \cos (e+f x) \sqrt{\tan (e+f x)} \sqrt{\sec ^2(e+f x)} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tan (e+f x)}\right )\right |-1\right )}{12 f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.174, size = 224, normalized size = 2.1 \begin{align*} -{\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) -1 \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{12\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) } \left ( 5\,\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 ) }}}{\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 ) ^{4}\sqrt{2}+2\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\sqrt{2}-5\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sqrt{2}+5\,\cos \left ( fx+e \right ) \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{\cos \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 )} \cos \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(-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{\cos \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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