Optimal. Leaf size=140 \[ \frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}+\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{c^4 f}-\frac{8 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a c^4 f}-\frac{2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^4 f} \]
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Rubi [A] time = 0.184025, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3904, 3887, 461, 203} \[ \frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^4 f}+\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{c^4 f}-\frac{8 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a c^4 f}-\frac{2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^4 f} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3887
Rule 461
Rule 203
Rubi steps
\begin{align*} \int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx &=\frac{\int \cot ^8(e+f x) (a+a \sec (e+f x))^{13/2} \, dx}{a^4 c^4}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{\left (2+a x^2\right )^2}{x^8 \left (1+a x^2\right )} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a c^4 f}\\ &=-\frac{2 \operatorname{Subst}\left (\int \left (\frac{4}{x^8}+\frac{a^2}{x^4}-\frac{a^3}{x^2}+\frac{a^4}{1+a x^2}\right ) \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a c^4 f}\\ &=\frac{2 a^2 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{c^4 f}-\frac{2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^4 f}-\frac{8 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a c^4 f}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{c^4 f}\\ &=\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{c^4 f}+\frac{2 a^2 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{c^4 f}-\frac{2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^4 f}-\frac{8 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a c^4 f}\\ \end{align*}
Mathematica [C] time = 7.93817, size = 361, normalized size = 2.58 \[ -\frac{\sin ^8\left (\frac{e}{2}+\frac{f x}{2}\right ) \sqrt{\frac{1}{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )}} \sqrt{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )} \csc ^7\left (\frac{1}{2} (e+f x)\right ) \sec ^5\left (\frac{1}{2} (e+f x)\right ) \sec ^{\frac{3}{2}}(e+f x) (a (\sec (e+f x)+1))^{5/2} \left (336 \sin ^2\left (\frac{1}{2} (e+f x)\right ) \left (5 \sin ^4\left (\frac{1}{2} (e+f x)\right )-8 \sin ^2\left (\frac{1}{2} (e+f x)\right )+3\right ) \text{Hypergeometric2F1}\left (-\frac{5}{2},-\frac{1}{2},\frac{1}{2},2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )+4 \left (35 \sin ^4\left (\frac{1}{2} (e+f x)\right )-42 \sin ^2\left (\frac{1}{2} (e+f x)\right )+15\right ) \text{Hypergeometric2F1}\left (-\frac{7}{2},-\frac{3}{2},-\frac{1}{2},2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )-105 \left (2 \left (5-4 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )} \sin ^4\left (\frac{1}{2} (e+f x)\right )+3 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sqrt{\sin ^2\left (\frac{1}{2} (e+f x)\right )}\right ) \sin ^2\left (\frac{1}{2} (e+f x)\right )^{3/2}\right ) \cos ^4\left (\frac{1}{2} (e+f x)\right )\right )}{210 f (c-c \sec (e+f x))^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.295, size = 395, normalized size = 2.8 \begin{align*} -{\frac{{a}^{2}}{21\,f{c}^{4}\sin \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) ^{3}}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ( 21\,\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{3}\sin \left ( fx+e \right ) \sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}} \right ) -63\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) \sqrt{2}\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}} \right ) +63\,\sqrt{2}\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) \sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}} \right ) -21\,\sqrt{2}\sin \left ( fx+e \right ){\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}} \right ) \sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}-80\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+154\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-140\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+42\,\cos \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [B] time = 1.61115, size = 1310, normalized size = 9.36 \begin{align*} \left [\frac{21 \,{\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sqrt{-a} \log \left (-\frac{8 \, a \cos \left (f x + e\right )^{3} - 4 \,{\left (2 \, \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 4 \,{\left (40 \, a^{2} \cos \left (f x + e\right )^{4} - 77 \, a^{2} \cos \left (f x + e\right )^{3} + 70 \, a^{2} \cos \left (f x + e\right )^{2} - 21 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{42 \,{\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )}, \frac{21 \,{\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sqrt{a} \arctan \left (\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - a}\right ) \sin \left (f x + e\right ) + 2 \,{\left (40 \, a^{2} \cos \left (f x + e\right )^{4} - 77 \, a^{2} \cos \left (f x + e\right )^{3} + 70 \, a^{2} \cos \left (f x + e\right )^{2} - 21 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{21 \,{\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )}\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(-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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