Optimal. Leaf size=163 \[ -\frac {2 \tan (e+f x) (a \sec (e+f x)+a)^2}{1155 f \left (c^2-c^2 \sec (e+f x)\right )^3}-\frac {2 \tan (e+f x) (a \sec (e+f x)+a)^2}{231 c^2 f (c-c \sec (e+f x))^4}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{33 c f (c-c \sec (e+f x))^5}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6} \]
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Rubi [A] time = 0.31, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3951, 3950} \[ -\frac {2 \tan (e+f x) (a \sec (e+f x)+a)^2}{1155 f \left (c^2-c^2 \sec (e+f x)\right )^3}-\frac {2 \tan (e+f x) (a \sec (e+f x)+a)^2}{231 c^2 f (c-c \sec (e+f x))^4}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{33 c f (c-c \sec (e+f x))^5}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6} \]
Antiderivative was successfully verified.
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Rule 3950
Rule 3951
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^6} \, dx &=-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}+\frac {3 \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^5} \, dx}{11 c}\\ &=-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{33 c f (c-c \sec (e+f x))^5}+\frac {2 \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx}{33 c^2}\\ &=-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{33 c f (c-c \sec (e+f x))^5}-\frac {2 (a+a \sec (e+f x))^2 \tan (e+f x)}{231 c^2 f (c-c \sec (e+f x))^4}+\frac {2 \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^3} \, dx}{231 c^3}\\ &=-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{33 c f (c-c \sec (e+f x))^5}-\frac {2 (a+a \sec (e+f x))^2 \tan (e+f x)}{231 c^2 f (c-c \sec (e+f x))^4}-\frac {2 (a+a \sec (e+f x))^2 \tan (e+f x)}{1155 f \left (c^2-c^2 \sec (e+f x)\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 167, normalized size = 1.02 \[ -\frac {a^2 \csc \left (\frac {e}{2}\right ) \left (37422 \sin \left (e+\frac {f x}{2}\right )-27060 \sin \left (e+\frac {3 f x}{2}\right )-23100 \sin \left (2 e+\frac {3 f x}{2}\right )+11220 \sin \left (2 e+\frac {5 f x}{2}\right )+13860 \sin \left (3 e+\frac {5 f x}{2}\right )-4895 \sin \left (3 e+\frac {7 f x}{2}\right )-3465 \sin \left (4 e+\frac {7 f x}{2}\right )+517 \sin \left (4 e+\frac {9 f x}{2}\right )+1155 \sin \left (5 e+\frac {9 f x}{2}\right )-152 \sin \left (5 e+\frac {11 f x}{2}\right )+32802 \sin \left (\frac {f x}{2}\right )\right ) \csc ^{11}\left (\frac {1}{2} (e+f x)\right )}{1182720 c^6 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 168, normalized size = 1.03 \[ \frac {152 \, a^{2} \cos \left (f x + e\right )^{6} + 395 \, a^{2} \cos \left (f x + e\right )^{5} + 289 \, a^{2} \cos \left (f x + e\right )^{4} + 15 \, a^{2} \cos \left (f x + e\right )^{3} - 19 \, a^{2} \cos \left (f x + e\right )^{2} + 10 \, a^{2} \cos \left (f x + e\right ) - 2 \, a^{2}}{1155 \, {\left (c^{6} f \cos \left (f x + e\right )^{5} - 5 \, c^{6} f \cos \left (f x + e\right )^{4} + 10 \, c^{6} f \cos \left (f x + e\right )^{3} - 10 \, c^{6} f \cos \left (f x + e\right )^{2} + 5 \, c^{6} f \cos \left (f x + e\right ) - c^{6} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 77, normalized size = 0.47 \[ \frac {231 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 495 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 385 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 105 \, a^{2}}{9240 \, c^{6} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.90, size = 65, normalized size = 0.40 \[ \frac {a^{2} \left (-\frac {1}{11 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{11}}-\frac {3}{7 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{7}}+\frac {1}{3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{9}}+\frac {1}{5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}\right )}{8 f \,c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 389, normalized size = 2.39 \[ \frac {\frac {a^{2} {\left (\frac {385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {990 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {1386 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {1155 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {3465 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 315\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{c^{6} \sin \left (f x + e\right )^{11}} + \frac {6 \, a^{2} {\left (\frac {385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {330 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {462 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {1155 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {1155 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 105\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{c^{6} \sin \left (f x + e\right )^{11}} + \frac {5 \, a^{2} {\left (\frac {385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {990 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {1386 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {1155 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {693 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 63\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{c^{6} \sin \left (f x + e\right )^{11}}}{110880 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.69, size = 108, normalized size = 0.66 \[ -\frac {a^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (105\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-385\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+495\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-231\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{9240\,c^6\,f\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}} \]
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 \[ \frac {a^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec {\left (e + f x \right )} + 1}\, dx\right )}{c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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