Integrand size = 25, antiderivative size = 136 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {A x}{a^4}-\frac {(55 A-8 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {8 (20 A-C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))}-\frac {(A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (5 A-2 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3} \]
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Time = 0.31 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {4138, 4007, 4004, 3879} \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=-\frac {8 (20 A-C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(55 A-8 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {A x}{a^4}-\frac {2 (5 A-2 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A+C) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
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Rule 3879
Rule 4004
Rule 4007
Rule 4138
Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {\int \frac {-7 a A+a (3 A-4 C) \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (5 A-2 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {35 a^2 A-4 a^2 (5 A-2 C) \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(55 A-8 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (5 A-2 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {-105 a^3 A+a^3 (55 A-8 C) \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = \frac {A x}{a^4}-\frac {(55 A-8 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (5 A-2 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(8 (20 A-C)) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^3} \\ & = \frac {A x}{a^4}-\frac {(55 A-8 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (5 A-2 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {8 (20 A-C) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(315\) vs. \(2(136)=272\).
Time = 6.22 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.32 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (3675 A d x \cos \left (\frac {d x}{2}\right )+3675 A d x \cos \left (c+\frac {d x}{2}\right )+2205 A d x \cos \left (c+\frac {3 d x}{2}\right )+2205 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+735 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+735 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+105 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+105 A d x \cos \left (4 c+\frac {7 d x}{2}\right )-9940 A \sin \left (\frac {d x}{2}\right )+560 C \sin \left (\frac {d x}{2}\right )+8260 A \sin \left (c+\frac {d x}{2}\right )-350 C \sin \left (c+\frac {d x}{2}\right )-7140 A \sin \left (c+\frac {3 d x}{2}\right )+336 C \sin \left (c+\frac {3 d x}{2}\right )+3780 A \sin \left (2 c+\frac {3 d x}{2}\right )-210 C \sin \left (2 c+\frac {3 d x}{2}\right )-2800 A \sin \left (2 c+\frac {5 d x}{2}\right )+182 C \sin \left (2 c+\frac {5 d x}{2}\right )+840 A \sin \left (3 c+\frac {5 d x}{2}\right )-520 A \sin \left (3 c+\frac {7 d x}{2}\right )+26 C \sin \left (3 c+\frac {7 d x}{2}\right )\right )}{13440 a^4 d} \]
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Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(\frac {15 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+21 \left (-5 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+35 \left (11 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+105 \left (-15 A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+840 d x A}{840 a^{4} d}\) | \(87\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +16 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(130\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +16 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(130\) |
norman | \(\frac {\frac {A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {A x}{a}+\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{56 a d}-\frac {\left (10 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{70 a d}-\frac {\left (14 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 a d}+\frac {\left (15 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (35 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{3}}\) | \(169\) |
risch | \(\frac {A x}{a^{4}}-\frac {2 i \left (420 A \,{\mathrm e}^{6 i \left (d x +c \right )}+1890 A \,{\mathrm e}^{5 i \left (d x +c \right )}-105 C \,{\mathrm e}^{5 i \left (d x +c \right )}+4130 A \,{\mathrm e}^{4 i \left (d x +c \right )}-175 C \,{\mathrm e}^{4 i \left (d x +c \right )}+4970 A \,{\mathrm e}^{3 i \left (d x +c \right )}-280 C \,{\mathrm e}^{3 i \left (d x +c \right )}+3570 A \,{\mathrm e}^{2 i \left (d x +c \right )}-168 C \,{\mathrm e}^{2 i \left (d x +c \right )}+1400 A \,{\mathrm e}^{i \left (d x +c \right )}-91 C \,{\mathrm e}^{i \left (d x +c \right )}+260 A -13 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(169\) |
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Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.34 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, A d x \cos \left (d x + c\right )^{4} + 420 \, A d x \cos \left (d x + c\right )^{3} + 630 \, A d x \cos \left (d x + c\right )^{2} + 420 \, A d x \cos \left (d x + c\right ) + 105 \, A d x - {\left (13 \, {\left (20 \, A - C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (155 \, A - 13 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (535 \, A - 32 \, C\right )} \cos \left (d x + c\right ) + 160 \, A - 8 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.48 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=-\frac {5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - \frac {C {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.13 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {840 \, {\left (d x + c\right )} A}{a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 16.16 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.20 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {A\,x}{a^4}+\frac {\left (\frac {13\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}-\frac {52\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {16\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}+\frac {13\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{210}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {5\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{28}-\frac {11\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{140}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}+\frac {C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
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