Optimal. Leaf size=214 \[ \frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{12 a^3 c^2 f}+\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{8 a^2 c^2 f}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{a^{3/2} c^2 f}-\frac{9 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{8 \sqrt{2} a^{3/2} c^2 f}-\frac{\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{3/2}}{4 a^3 c^2 f} \]
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Rubi [A] time = 0.278458, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3904, 3887, 472, 583, 522, 203} \[ \frac{\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{12 a^3 c^2 f}+\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{8 a^2 c^2 f}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{a^{3/2} c^2 f}-\frac{9 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{8 \sqrt{2} a^{3/2} c^2 f}-\frac{\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{3/2}}{4 a^3 c^2 f} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3887
Rule 472
Rule 583
Rule 522
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^2} \, dx &=\frac{\int \cot ^4(e+f x) \sqrt{a+a \sec (e+f x)} \, dx}{a^2 c^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a^3 c^2 f}\\ &=-\frac{\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f}-\frac{\operatorname{Subst}\left (\int \frac{-a-5 a^2 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{2 a^4 c^2 f}\\ &=\frac{\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{12 a^3 c^2 f}-\frac{\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f}+\frac{\operatorname{Subst}\left (\int \frac{21 a^2-3 a^3 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{12 a^4 c^2 f}\\ &=\frac{7 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{8 a^2 c^2 f}+\frac{\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{12 a^3 c^2 f}-\frac{\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f}-\frac{\operatorname{Subst}\left (\int \frac{69 a^3+21 a^4 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{24 a^4 c^2 f}\\ &=\frac{7 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{8 a^2 c^2 f}+\frac{\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{12 a^3 c^2 f}-\frac{\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{2+a x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{8 a c^2 f}-\frac{2 \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 )}{a c^2 f}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a^{3/2} c^2 f}-\frac{9 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a+a \sec (e+f x)}}\right )}{8 \sqrt{2} a^{3/2} c^2 f}+\frac{7 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{8 a^2 c^2 f}+\frac{\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{12 a^3 c^2 f}-\frac{\cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{4 a^3 c^2 f}\\ \end{align*}
Mathematica [C] time = 23.7926, size = 5622, normalized size = 26.27 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.293, size = 387, normalized size = 1.8 \begin{align*} -{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}-1}{48\,f{c}^{2}{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{5}}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ( 48\, \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 ) +27\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\ln \left ({\frac{1}{\sin \left ( fx+e \right ) } \left ( \sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) +1 \right ) } \right ) -48\,\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 ) }}}-27\,\sin \left ( fx+e \right ) \sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\ln \left ({\frac{1}{\sin \left ( fx+e \right ) } \left ( \sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) +1 \right ) } \right ) -62\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}+4\, \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}}{\left (c \sec \left (f x + e\right ) - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00765, size = 1478, normalized size = 6.91 \begin{align*} \text{result too large to display} \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*} \frac{\int \frac{1}{a \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{3}{\left (e + f x \right )} - a \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} - a \sqrt{a \sec{\left (e + f x \right )} + a} \sec{\left (e + f x \right )} + a \sqrt{a \sec{\left (e + f x \right )} + a}}\, dx}{c^{2}} \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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