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 = 0.08, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 8
Rule 3473
Rule 3904
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*}
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Mathematica [C] time = 0.07, size = 39, normalized size = 0.58 \[ \frac {\cot ^5(e+f x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(e+f x)\right )}{5 a^3 c^3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 118, normalized size = 1.76 \[ \frac {23 \, \cos \left (f x + e\right )^{5} - 35 \, \cos \left (f x + e\right )^{3} + 15 \, {\left (f x \cos \left (f x + e\right )^{4} - 2 \, f x \cos \left (f x + e\right )^{2} + f x\right )} \sin \left (f x + e\right ) + 15 \, \cos \left (f x + e\right )}{15 \, {\left (a^{3} c^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{3} f \cos \left (f x + e\right )^{2} + a^{3} c^{3} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 136, normalized size = 2.03 \[ \frac {\frac {480 \, {\left (f x + e\right )}}{a^{3} c^{3}} + \frac {330 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 35 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3}{a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}} - \frac {3 \, a^{12} c^{12} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 35 \, a^{12} c^{12} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 330 \, a^{12} c^{12} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15} c^{15}}}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +a \sec \left (f x +e \right )\right )^{3} \left (c -c \sec \left (f x +e \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 56, normalized size = 0.84 \[ \frac {\frac {15 \, {\left (f x + e\right )}}{a^{3} c^{3}} + \frac {15 \, \tan \left (f x + e\right )^{4} - 5 \, \tan \left (f x + e\right )^{2} + 3}{a^{3} c^{3} \tan \left (f x + e\right )^{5}}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.57, size = 94, normalized size = 1.40 \[ \frac {\frac {5\,\cos \left (e+f\,x\right )}{24}-\frac {5\,\cos \left (3\,e+3\,f\,x\right )}{48}+\frac {23\,\cos \left (5\,e+5\,f\,x\right )}{240}-\frac {5\,\sin \left (3\,e+3\,f\,x\right )\,\left (e+f\,x\right )}{16}+\frac {\sin \left (5\,e+5\,f\,x\right )\,\left (e+f\,x\right )}{16}+\frac {5\,\sin \left (e+f\,x\right )\,\left (e+f\,x\right )}{8}}{a^3\,c^3\,f\,{\sin \left (e+f\,x\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {1}{\sec ^{6}{\left (e + f x \right )} - 3 \sec ^{4}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} - 1}\, dx}{a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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