Optimal. Leaf size=249 \[ -\frac{3 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{20 a^4 c^3 f}-\frac{5 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{24 a^3 c^3 f}+\frac{21 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{16 a^2 c^3 f}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{a^{3/2} c^3 f}-\frac{11 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{16 \sqrt{2} a^{3/2} c^3 f}+\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{5/2}}{4 a^4 c^3 f} \]
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Rubi [A] time = 0.339961, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{3 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{20 a^4 c^3 f}-\frac{5 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{24 a^3 c^3 f}+\frac{21 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{16 a^2 c^3 f}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{a^{3/2} c^3 f}-\frac{11 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{16 \sqrt{2} a^{3/2} c^3 f}+\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{5/2}}{4 a^4 c^3 f} \]
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))^3} \, dx &=-\frac{\int \cot ^6(e+f x) (a+a \sec (e+f x))^{3/2} \, dx}{a^3 c^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^6 \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^4 c^3 f}\\ &=\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}+\frac{\operatorname{Subst}\left (\int \frac{-3 a-7 a^2 x^2}{x^6 \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^5 c^3 f}\\ &=-\frac{3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}-\frac{\operatorname{Subst}\left (\int \frac{25 a^2-15 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 )}{20 a^5 c^3 f}\\ &=-\frac{5 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{24 a^3 c^3 f}-\frac{3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}+\frac{\operatorname{Subst}\left (\int \frac{315 a^3+75 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 )}{120 a^5 c^3 f}\\ &=\frac{21 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{16 a^2 c^3 f}-\frac{5 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{24 a^3 c^3 f}-\frac{3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}-\frac{\operatorname{Subst}\left (\int \frac{795 a^4+315 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 )}{240 a^5 c^3 f}\\ &=\frac{21 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{16 a^2 c^3 f}-\frac{5 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{24 a^3 c^3 f}-\frac{3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}+\frac{11 \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 )}{16 a c^3 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^3 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^3 f}-\frac{11 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a+a \sec (e+f x)}}\right )}{16 \sqrt{2} a^{3/2} c^3 f}+\frac{21 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{16 a^2 c^3 f}-\frac{5 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{24 a^3 c^3 f}-\frac{3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac{\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}\\ \end{align*}
Mathematica [C] time = 23.7951, size = 5639, normalized size = 22.65 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.362, size = 725, normalized size = 2.9 \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{3}{2}}{\left (c \sec \left (f x + e\right ) - c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18013, size = 1867, normalized size = 7.5 \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(-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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