Integrand size = 31, antiderivative size = 163 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {B \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {(6 A-55 B) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(12 A-215 B) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))}+\frac {(A-B) \sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(3 A-10 B) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3} \]
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Time = 0.53 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4104, 4093, 4083, 3855, 3879} \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {(12 A-215 B) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(6 A-55 B) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {B \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {(A-B) \tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {(3 A-10 B) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
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Rule 3855
Rule 3879
Rule 4083
Rule 4093
Rule 4104
Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\sec ^3(c+d x) (3 a (A-B)+7 a B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = \frac {(A-B) \sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(3 A-10 B) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^2(c+d x) \left (2 a^2 (3 A-10 B)+35 a^2 B \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(6 A-55 B) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(3 A-10 B) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {\sec (c+d x) \left (-2 a^3 (6 A-55 B)-105 a^3 B \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(6 A-55 B) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(3 A-10 B) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {(12 A-215 B) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^3}+\frac {B \int \sec (c+d x) \, dx}{a^4} \\ & = \frac {B \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {(6 A-55 B) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(3 A-10 B) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {(12 A-215 B) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}
Time = 1.81 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.80 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (1680 B \text {arctanh}(\sin (c+d x)) \cos ^7\left (\frac {1}{2} (c+d x)\right )+(96 A-1055 B+(87 A-1480 B) \cos (c+d x)+(24 A-535 B) \cos (2 (c+d x))+3 A \cos (3 (c+d x))-80 B \cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{105 a^4 d (1+\sec (c+d x))^4} \]
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Time = 0.86 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {-56 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+56 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+7 \left (\frac {3 A}{5}-B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \left (A -\frac {11 B}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+7 A -105 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{56 a^{4} d}\) | \(112\) |
derivativedivides | \(\frac {-8 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+8 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}}{8 d \,a^{4}}\) | \(146\) |
default | \(\frac {-8 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+8 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}}{8 d \,a^{4}}\) | \(146\) |
risch | \(-\frac {2 i \left (105 B \,{\mathrm e}^{6 i \left (d x +c \right )}+735 B \,{\mathrm e}^{5 i \left (d x +c \right )}+2170 B \,{\mathrm e}^{4 i \left (d x +c \right )}-210 A \,{\mathrm e}^{3 i \left (d x +c \right )}+3430 B \,{\mathrm e}^{3 i \left (d x +c \right )}-126 A \,{\mathrm e}^{2 i \left (d x +c \right )}+2625 B \,{\mathrm e}^{2 i \left (d x +c \right )}-42 \,{\mathrm e}^{i \left (d x +c \right )} A +1015 B \,{\mathrm e}^{i \left (d x +c \right )}-6 A +160 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{4} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{4} d}\) | \(182\) |
norman | \(\frac {\frac {\left (A -15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (A -15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{280 a d}+\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{56 a d}+\frac {\left (3 A -605 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{840 a d}-\frac {\left (9 A -1465 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{280 a d}-\frac {\left (57 A +55 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{840 a d}+\frac {\left (-1145 B +39 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 a d}-\frac {\left (-169 B +9 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 a d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} a^{3}}+\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{4} d}-\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4} d}\) | \(263\) |
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Time = 0.27 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.45 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, {\left (B \cos \left (d x + c\right )^{4} + 4 \, B \cos \left (d x + c\right )^{3} + 6 \, B \cos \left (d x + c\right )^{2} + 4 \, B \cos \left (d x + c\right ) + B\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (B \cos \left (d x + c\right )^{4} + 4 \, B \cos \left (d x + c\right )^{3} + 6 \, B \cos \left (d x + c\right )^{2} + 4 \, B \cos \left (d x + c\right ) + B\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (3 \, A - 80 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (24 \, A - 535 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (39 \, A - 620 \, B\right )} \cos \left (d x + c\right ) + 36 \, A - 260 \, B\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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\[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \sec ^{4}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{5}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Time = 0.22 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.40 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {5 \, B {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.11 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {840 \, B \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {840 \, B \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 14.11 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.21 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}-\frac {11\,B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}\right )+{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}-\frac {B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8}\right )+{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}-\frac {15\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}\right )+\frac {A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56}-\frac {B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}+\frac {2\,B\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a^4\,d} \]
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