Optimal. Leaf size=269 \[ \frac{43 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{96 a^4 c^2 f}+\frac{21 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{64 a^3 c^2 f}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{a^{5/2} c^2 f}-\frac{107 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{64 \sqrt{2} a^{5/2} c^2 f}-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{3/2}}{16 a^4 c^2 f}-\frac{15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{3/2}}{32 a^4 c^2 f} \]
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Rubi [A] time = 0.335235, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{43 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{96 a^4 c^2 f}+\frac{21 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{64 a^3 c^2 f}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{a^{5/2} c^2 f}-\frac{107 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{64 \sqrt{2} a^{5/2} c^2 f}-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{3/2}}{16 a^4 c^2 f}-\frac{15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{3/2}}{32 a^4 c^2 f} \]
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))^2} \, dx &=\frac{\int \frac{\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 )^3} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a^4 c^2 f}\\ &=-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}-\frac{\operatorname{Subst}\left (\int \frac{a-7 a^2 x^2}{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 )}{4 a^5 c^2 f}\\ &=-\frac{15 \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}}{32 a^4 c^2 f}-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}-\frac{\operatorname{Subst}\left (\int \frac{-43 a^2-75 a^3 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 )}{16 a^6 c^2 f}\\ &=\frac{43 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{96 a^4 c^2 f}-\frac{15 \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}}{32 a^4 c^2 f}-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}+\frac{\operatorname{Subst}\left (\int \frac{63 a^3-129 a^4 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 )}{96 a^6 c^2 f}\\ &=\frac{21 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{64 a^3 c^2 f}+\frac{43 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{96 a^4 c^2 f}-\frac{15 \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}}{32 a^4 c^2 f}-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}-\frac{\operatorname{Subst}\left (\int \frac{447 a^4+63 a^5 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 )}{192 a^6 c^2 f}\\ &=\frac{21 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{64 a^3 c^2 f}+\frac{43 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{96 a^4 c^2 f}-\frac{15 \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}}{32 a^4 c^2 f}-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}+\frac{107 \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 )}{64 a^2 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^2 c^2 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^2 f}-\frac{107 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a+a \sec (e+f x)}}\right )}{64 \sqrt{2} a^{5/2} c^2 f}+\frac{21 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{64 a^3 c^2 f}+\frac{43 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{96 a^4 c^2 f}-\frac{15 \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}}{32 a^4 c^2 f}-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}\\ \end{align*}
Mathematica [C] time = 24.0523, size = 5660, normalized size = 21.04 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.326, size = 725, normalized size = 2.7 \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(-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 [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(-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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