Optimal. Leaf size=133 \[ \frac {a^2 x}{c^4}-\frac {4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}-\frac {12 a^2 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {59 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac {164 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))} \]
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Rubi [A]
time = 0.33, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {3988, 3862,
4007, 4004, 3879, 3881, 3882} \begin {gather*} -\frac {164 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))}-\frac {59 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac {12 a^2 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}+\frac {a^2 x}{c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 3862
Rule 3879
Rule 3881
Rule 3882
Rule 3988
Rule 4004
Rule 4007
Rubi steps
\begin {align*} \int \frac {(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx &=\frac {\int \left (\frac {a^2}{(1-\sec (e+f x))^4}+\frac {2 a^2 \sec (e+f x)}{(1-\sec (e+f x))^4}+\frac {a^2 \sec ^2(e+f x)}{(1-\sec (e+f x))^4}\right ) \, dx}{c^4}\\ &=\frac {a^2 \int \frac {1}{(1-\sec (e+f x))^4} \, dx}{c^4}+\frac {a^2 \int \frac {\sec ^2(e+f x)}{(1-\sec (e+f x))^4} \, dx}{c^4}+\frac {\left (2 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^4} \, dx}{c^4}\\ &=-\frac {4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}-\frac {a^2 \int \frac {-7-3 \sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{7 c^4}-\frac {\left (4 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{7 c^4}+\frac {\left (6 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{7 c^4}\\ &=-\frac {4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}-\frac {12 a^2 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}+\frac {a^2 \int \frac {35+20 \sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{35 c^4}-\frac {\left (8 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{35 c^4}+\frac {\left (12 a^2\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{35 c^4}\\ &=-\frac {4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}-\frac {12 a^2 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {59 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac {a^2 \int \frac {-105-55 \sec (e+f x)}{1-\sec (e+f x)} \, dx}{105 c^4}-\frac {\left (8 a^2\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{105 c^4}+\frac {\left (4 a^2\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{35 c^4}\\ &=\frac {a^2 x}{c^4}-\frac {4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}-\frac {12 a^2 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {59 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac {4 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))}+\frac {\left (32 a^2\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{21 c^4}\\ &=\frac {a^2 x}{c^4}-\frac {4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}-\frac {12 a^2 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {59 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac {164 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 227, normalized size = 1.71 \begin {gather*} \frac {a^2 \csc \left (\frac {e}{2}\right ) \csc ^7\left (\frac {1}{2} (e+f x)\right ) \left (3675 f x \cos \left (\frac {f x}{2}\right )-3675 f x \cos \left (e+\frac {f x}{2}\right )-2205 f x \cos \left (e+\frac {3 f x}{2}\right )+2205 f x \cos \left (2 e+\frac {3 f x}{2}\right )+735 f x \cos \left (2 e+\frac {5 f x}{2}\right )-735 f x \cos \left (3 e+\frac {5 f x}{2}\right )-105 f x \cos \left (3 e+\frac {7 f x}{2}\right )+105 f x \cos \left (4 e+\frac {7 f x}{2}\right )-11900 \sin \left (\frac {f x}{2}\right )-10430 \sin \left (e+\frac {f x}{2}\right )+8568 \sin \left (e+\frac {3 f x}{2}\right )+4830 \sin \left (2 e+\frac {3 f x}{2}\right )-3206 \sin \left (2 e+\frac {5 f x}{2}\right )-1260 \sin \left (3 e+\frac {5 f x}{2}\right )+638 \sin \left (3 e+\frac {7 f x}{2}\right )\right )}{13440 c^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 77, normalized size = 0.58
method | result | size |
derivativedivides | \(\frac {a^{2} \left (4 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {3}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {4}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}\right )}{2 f \,c^{4}}\) | \(77\) |
default | \(\frac {a^{2} \left (4 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {3}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {4}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}\right )}{2 f \,c^{4}}\) | \(77\) |
risch | \(\frac {a^{2} x}{c^{4}}+\frac {2 i a^{2} \left (630 \,{\mathrm e}^{6 i \left (f x +e \right )}-2415 \,{\mathrm e}^{5 i \left (f x +e \right )}+5215 \,{\mathrm e}^{4 i \left (f x +e \right )}-5950 \,{\mathrm e}^{3 i \left (f x +e \right )}+4284 \,{\mathrm e}^{2 i \left (f x +e \right )}-1603 \,{\mathrm e}^{i \left (f x +e \right )}+319\right )}{105 f \,c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{7}}\) | \(103\) |
norman | \(\frac {\frac {a^{2} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {a^{2}}{14 c f}-\frac {13 a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}+\frac {29 a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{30 c f}-\frac {8 a^{2} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {2 a^{2} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {a^{2} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\) | \(170\) |
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. 320 vs.
\(2 (125) = 250\).
time = 0.50, size = 320, normalized size = 2.41 \begin {gather*} \frac {5 \, a^{2} {\left (\frac {336 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{4}} + \frac {{\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {77 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {315 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 3\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}}\right )} + \frac {a^{2} {\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {105 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 15\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}} + \frac {6 \, a^{2} {\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {35 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 5\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}}}{840 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.96, size = 184, normalized size = 1.38 \begin {gather*} \frac {319 \, a^{2} \cos \left (f x + e\right )^{4} - 327 \, a^{2} \cos \left (f x + e\right )^{3} - 95 \, a^{2} \cos \left (f x + e\right )^{2} + 387 \, a^{2} \cos \left (f x + e\right ) - 164 \, a^{2} + 105 \, {\left (a^{2} f x \cos \left (f x + e\right )^{3} - 3 \, a^{2} f x \cos \left (f x + e\right )^{2} + 3 \, a^{2} f x \cos \left (f x + e\right ) - a^{2} f x\right )} \sin \left (f x + e\right )}{105 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} 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^{2} \left (\int \frac {2 \sec {\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {1}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx\right )}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 88, normalized size = 0.66 \begin {gather*} \frac {\frac {210 \, {\left (f x + e\right )} a^{2}}{c^{4}} + \frac {420 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 140 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 63 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, a^{2}}{c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7}}}{210 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.50, size = 124, normalized size = 0.93 \begin {gather*} \frac {a^2\,x}{c^4}-\frac {\frac {a^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{14}-\frac {3\,a^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{10}+\frac {2\,a^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{3}-2\,a^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6}{c^4\,f\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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