Optimal. Leaf size=230 \[ -\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{32 a^3 c f}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{a^{5/2} c f}-\frac{71 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{32 \sqrt{2} a^{5/2} c f}+\frac{\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{a \sec (e+f x)+a}}{16 a^3 c f}+\frac{13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{a \sec (e+f x)+a}}{32 a^3 c f} \]
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Rubi [A] time = 0.306451, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3904, 3887, 472, 579, 583, 522, 203} \[ -\frac{7 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{32 a^3 c f}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{a^{5/2} c f}-\frac{71 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{32 \sqrt{2} a^{5/2} c f}+\frac{\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{a \sec (e+f x)+a}}{16 a^3 c f}+\frac{13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{a \sec (e+f x)+a}}{32 a^3 c f} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3887
Rule 472
Rule 579
Rule 583
Rule 522
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))} \, dx &=-\frac{\int \frac{\cot ^2(e+f x)}{(a+a \sec (e+f x))^{3/2}} \, dx}{a c}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a^3 c f}\\ &=\frac{\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{a+a \sec (e+f x)}}{16 a^3 c f}+\frac{\operatorname{Subst}\left (\int \frac{3 a-5 a^2 x^2}{x^2 \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 )}{4 a^4 c f}\\ &=\frac{13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{a+a \sec (e+f x)}}{32 a^3 c f}+\frac{\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{a+a \sec (e+f x)}}{16 a^3 c f}+\frac{\operatorname{Subst}\left (\int \frac{-7 a^2-39 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 )}{16 a^5 c f}\\ &=-\frac{7 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{32 a^3 c f}+\frac{13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{a+a \sec (e+f x)}}{32 a^3 c f}+\frac{\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{a+a \sec (e+f x)}}{16 a^3 c f}-\frac{\operatorname{Subst}\left (\int \frac{57 a^3-7 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 )}{32 a^5 c f}\\ &=-\frac{7 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{32 a^3 c f}+\frac{13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{a+a \sec (e+f x)}}{32 a^3 c f}+\frac{\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{a+a \sec (e+f x)}}{16 a^3 c 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^2 c f}+\frac{71 \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 )}{32 a^2 c f}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a^{5/2} c f}-\frac{71 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a+a \sec (e+f x)}}\right )}{32 \sqrt{2} a^{5/2} c f}-\frac{7 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{32 a^3 c f}+\frac{13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{a+a \sec (e+f x)}}{32 a^3 c f}+\frac{\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{a+a \sec (e+f x)}}{16 a^3 c f}\\ \end{align*}
Mathematica [A] time = 1.44271, size = 158, normalized size = 0.69 \[ \frac{\tan ^3\left (\frac{1}{2} (e+f x)\right ) \left (24 \cos (e+f x)+27 \cos (2 (e+f x))+512 \cos ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left (\sqrt{\sec (e+f x)-1}\right )-284 \sqrt{2} \cos ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left (\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right )+13\right )}{64 a^2 c f (\cos (e+f x)-1)^2 \sqrt{a (\sec (e+f x)+1)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.265, size = 545, normalized size = 2.4 \begin{align*} \text{result too large to display} \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{5}{2}}{\left (c \sec \left (f x + e\right ) - c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{3}{\left (e + f x \right )} + a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} - a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec{\left (e + f x \right )} - a^{2} \sqrt{a \sec{\left (e + f x \right )} + a}}\, dx}{c} \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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