Optimal. Leaf size=46 \[ -\frac {\cot ^3(e+f x)}{3 a^2 c^2 f}+\frac {\cot (e+f x)}{a^2 c^2 f}+\frac {x}{a^2 c^2} \]
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Rubi [A] time = 0.07, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3904, 3473, 8} \[ -\frac {\cot ^3(e+f x)}{3 a^2 c^2 f}+\frac {\cot (e+f x)}{a^2 c^2 f}+\frac {x}{a^2 c^2} \]
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))^2 (c-c \sec (e+f x))^2} \, dx &=\frac {\int \cot ^4(e+f x) \, dx}{a^2 c^2}\\ &=-\frac {\cot ^3(e+f x)}{3 a^2 c^2 f}-\frac {\int \cot ^2(e+f x) \, dx}{a^2 c^2}\\ &=\frac {\cot (e+f x)}{a^2 c^2 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^2 f}+\frac {\int 1 \, dx}{a^2 c^2}\\ &=\frac {x}{a^2 c^2}+\frac {\cot (e+f x)}{a^2 c^2 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^2 f}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 39, normalized size = 0.85 \[ -\frac {\cot ^3(e+f x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(e+f x)\right )}{3 a^2 c^2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 81, normalized size = 1.76 \[ \frac {4 \, \cos \left (f x + e\right )^{3} + 3 \, {\left (f x \cos \left (f x + e\right )^{2} - f x\right )} \sin \left (f x + e\right ) - 3 \, \cos \left (f x + e\right )}{3 \, {\left (a^{2} c^{2} f \cos \left (f x + e\right )^{2} - a^{2} c^{2} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 100, normalized size = 2.17 \[ \frac {\frac {24 \, {\left (f x + e\right )}}{a^{2} c^{2}} + \frac {15 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1}{a^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}} + \frac {a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6} c^{6}}}{24 \, f} \]
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 \[ \int \frac {1}{\left (a +a \sec \left (f x +e \right )\right )^{2} \left (c -c \sec \left (f x +e \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 46, normalized size = 1.00 \[ \frac {\frac {3 \, {\left (f x + e\right )}}{a^{2} c^{2}} + \frac {3 \, \tan \left (f x + e\right )^{2} - 1}{a^{2} c^{2} \tan \left (f x + e\right )^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 58, normalized size = 1.26 \[ -\frac {\cos \left (3\,e+3\,f\,x\right )+\frac {3\,\sin \left (3\,e+3\,f\,x\right )\,\left (e+f\,x\right )}{4}-\frac {9\,\sin \left (e+f\,x\right )\,\left (e+f\,x\right )}{4}}{3\,a^2\,c^2\,f\,{\sin \left (e+f\,x\right )}^3} \]
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 \[ \frac {\int \frac {1}{\sec ^{4}{\left (e + f x \right )} - 2 \sec ^{2}{\left (e + f x \right )} + 1}\, dx}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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