Optimal. Leaf size=80 \[ -\frac {(a+a \sec (e+f x))^3 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}-\frac {(a+a \sec (e+f x))^3 \tan (e+f x)}{63 c f (c-c \sec (e+f x))^4} \]
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Rubi [A]
time = 0.11, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4036, 4035}
\begin {gather*} -\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{63 c f (c-c \sec (e+f x))^4}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{9 f (c-c \sec (e+f x))^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 4035
Rule 4036
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^5} \, dx &=-\frac {(a+a \sec (e+f x))^3 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}+\frac {\int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx}{9 c}\\ &=-\frac {(a+a \sec (e+f x))^3 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}-\frac {(a+a \sec (e+f x))^3 \tan (e+f x)}{63 c f (c-c \sec (e+f x))^4}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 141, normalized size = 1.76 \begin {gather*} -\frac {a^3 \csc \left (\frac {e}{2}\right ) \csc ^9\left (\frac {1}{2} (e+f x)\right ) \left (693 \sin \left (\frac {f x}{2}\right )+315 \sin \left (e+\frac {f x}{2}\right )-189 \sin \left (e+\frac {3 f x}{2}\right )-483 \sin \left (2 e+\frac {3 f x}{2}\right )+225 \sin \left (2 e+\frac {5 f x}{2}\right )+63 \sin \left (3 e+\frac {5 f x}{2}\right )-9 \sin \left (3 e+\frac {7 f x}{2}\right )-63 \sin \left (4 e+\frac {7 f x}{2}\right )+8 \sin \left (4 e+\frac {9 f x}{2}\right )\right )}{16128 c^5 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 39, normalized size = 0.49
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}\right )}{2 f \,c^{5}}\) | \(39\) |
default | \(\frac {a^{3} \left (-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}\right )}{2 f \,c^{5}}\) | \(39\) |
risch | \(\frac {2 i a^{3} \left (63 \,{\mathrm e}^{8 i \left (f x +e \right )}-63 \,{\mathrm e}^{7 i \left (f x +e \right )}+483 \,{\mathrm e}^{6 i \left (f x +e \right )}-315 \,{\mathrm e}^{5 i \left (f x +e \right )}+693 \,{\mathrm e}^{4 i \left (f x +e \right )}-189 \,{\mathrm e}^{3 i \left (f x +e \right )}+225 \,{\mathrm e}^{2 i \left (f x +e \right )}-9 \,{\mathrm e}^{i \left (f x +e \right )}+8\right )}{63 f \,c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9}}\) | \(116\) |
norman | \(\frac {-\frac {a^{3}}{18 c f}+\frac {5 a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}-\frac {8 a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}+\frac {17 a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 c f}-\frac {a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{14 c f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3} c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 389 vs.
\(2 (84) = 168\).
time = 0.30, size = 389, normalized size = 4.86 \begin {gather*} -\frac {\frac {a^{3} {\left (\frac {180 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {378 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {420 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {315 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} + \frac {15 \, a^{3} {\left (\frac {18 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {42 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 7\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} - \frac {5 \, a^{3} {\left (\frac {18 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {42 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + 7\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} + \frac {21 \, a^{3} {\left (\frac {18 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {45 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 5\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}}}{5040 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.20, size = 150, normalized size = 1.88 \begin {gather*} \frac {8 \, a^{3} \cos \left (f x + e\right )^{5} + 31 \, a^{3} \cos \left (f x + e\right )^{4} + 44 \, a^{3} \cos \left (f x + e\right )^{3} + 26 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) - a^{3}}{63 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} + 6 \, c^{5} f \cos \left (f x + e\right )^{2} - 4 \, c^{5} f \cos \left (f x + e\right ) + c^{5} f\right )} \sin \left (f x + e\right )} \end {gather*}
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 \begin {gather*} - \frac {a^{3} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {3 \sec ^{3}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx\right )}{c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 41, normalized size = 0.51 \begin {gather*} -\frac {9 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7 \, a^{3}}{126 \, c^{5} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.62, size = 37, normalized size = 0.46 \begin {gather*} \frac {a^3\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (7\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-9\right )}{126\,c^5\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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