Optimal. Leaf size=67 \[ \frac{\cot ^5(e+f x)}{5 a^3 c^3 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac{\cot (e+f x)}{a^3 c^3 f}+\frac{x}{a^3 c^3} \]
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Rubi [A] time = 0.0805624, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3904, 3473, 8} \[ \frac{\cot ^5(e+f x)}{5 a^3 c^3 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac{\cot (e+f x)}{a^3 c^3 f}+\frac{x}{a^3 c^3} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3} \, dx &=-\frac{\int \cot ^6(e+f x) \, dx}{a^3 c^3}\\ &=\frac{\cot ^5(e+f x)}{5 a^3 c^3 f}+\frac{\int \cot ^4(e+f x) \, dx}{a^3 c^3}\\ &=-\frac{\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^3 f}-\frac{\int \cot ^2(e+f x) \, dx}{a^3 c^3}\\ &=\frac{\cot (e+f x)}{a^3 c^3 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^3 f}+\frac{\int 1 \, dx}{a^3 c^3}\\ &=\frac{x}{a^3 c^3}+\frac{\cot (e+f x)}{a^3 c^3 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^3 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^3 f}\\ \end{align*}
Mathematica [C] time = 0.0737104, size = 39, normalized size = 0.58 \[ \frac{\cot ^5(e+f x) \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-\tan ^2(e+f x)\right )}{5 a^3 c^3 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+a\sec \left ( fx+e \right ) \right ) ^{3} \left ( c-c\sec \left ( fx+e \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57528, size = 76, normalized size = 1.13 \begin{align*} \frac{\frac{15 \,{\left (f x + e\right )}}{a^{3} c^{3}} + \frac{15 \, \tan \left (f x + e\right )^{4} - 5 \, \tan \left (f x + e\right )^{2} + 3}{a^{3} c^{3} \tan \left (f x + e\right )^{5}}}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11877, size = 290, normalized size = 4.33 \begin{align*} \frac{23 \, \cos \left (f x + e\right )^{5} - 35 \, \cos \left (f x + e\right )^{3} + 15 \,{\left (f x \cos \left (f x + e\right )^{4} - 2 \, f x \cos \left (f x + e\right )^{2} + f x\right )} \sin \left (f x + e\right ) + 15 \, \cos \left (f x + e\right )}{15 \,{\left (a^{3} c^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{3} f \cos \left (f x + e\right )^{2} + a^{3} c^{3} f\right )} \sin \left (f x + e\right )} \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}{\sec ^{6}{\left (e + f x \right )} - 3 \sec ^{4}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} - 1}\, dx}{a^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.45574, size = 184, normalized size = 2.75 \begin{align*} \frac{\frac{480 \,{\left (f x + e\right )}}{a^{3} c^{3}} + \frac{330 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 35 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3}{a^{3} c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5}} - \frac{3 \, a^{12} c^{12} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 35 \, a^{12} c^{12} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 330 \, a^{12} c^{12} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a^{15} c^{15}}}{480 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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