Integrand size = 31, antiderivative size = 108 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {(A-2 B) \text {arctanh}(\sin (c+d x))}{a^2 d}-\frac {(A-4 B) \tan (c+d x)}{3 a^2 d}-\frac {(A-2 B) \tan (c+d x)}{a^2 d (1+\sec (c+d x))}+\frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2} \]
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Time = 0.42 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4104, 4093, 3872, 3855, 3852, 8} \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {(A-2 B) \text {arctanh}(\sin (c+d x))}{a^2 d}-\frac {(A-4 B) \tan (c+d x)}{3 a^2 d}-\frac {(A-2 B) \tan (c+d x)}{a^2 d (\sec (c+d x)+1)}+\frac {(A-B) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4093
Rule 4104
Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {\sec ^2(c+d x) (2 a (A-B)-a (A-4 B) \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2} \\ & = -\frac {(A-2 B) \tan (c+d x)}{a^2 d (1+\sec (c+d x))}+\frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {\int \sec (c+d x) \left (-3 a^2 (A-2 B)+a^2 (A-4 B) \sec (c+d x)\right ) \, dx}{3 a^4} \\ & = -\frac {(A-2 B) \tan (c+d x)}{a^2 d (1+\sec (c+d x))}+\frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {(A-4 B) \int \sec ^2(c+d x) \, dx}{3 a^2}+\frac {(A-2 B) \int \sec (c+d x) \, dx}{a^2} \\ & = \frac {(A-2 B) \text {arctanh}(\sin (c+d x))}{a^2 d}-\frac {(A-2 B) \tan (c+d x)}{a^2 d (1+\sec (c+d x))}+\frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {(A-4 B) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 a^2 d} \\ & = \frac {(A-2 B) \text {arctanh}(\sin (c+d x))}{a^2 d}-\frac {(A-4 B) \tan (c+d x)}{3 a^2 d}-\frac {(A-2 B) \tan (c+d x)}{a^2 d (1+\sec (c+d x))}+\frac {(A-B) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2} \\ \end{align*}
Time = 0.90 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (6 (A-2 B) \text {arctanh}(\sin (c+d x)) \cos ^3\left (\frac {1}{2} (c+d x)\right )+(-2 A+8 B+(-5 A+14 B) \cos (c+d x)+(-2 A+5 B) \cos (2 (c+d x))) \sec (c+d x) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^2 d (1+\sec (c+d x))^2} \]
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Time = 0.77 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.17
method | result | size |
parallelrisch | \(\frac {-6 \cos \left (d x +c \right ) \left (A -2 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+6 \cos \left (d x +c \right ) \left (A -2 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\left (\frac {2 A}{5}-B \right ) \cos \left (2 d x +2 c \right )+\left (A -\frac {14 B}{5}\right ) \cos \left (d x +c \right )+\frac {2 A}{5}-\frac {8 B}{5}\right )}{6 a^{2} d \cos \left (d x +c \right )}\) | \(126\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\left (-4 B +2 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {2 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (-2 A +4 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {2 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{2 d \,a^{2}}\) | \(134\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\left (-4 B +2 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {2 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (-2 A +4 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {2 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{2 d \,a^{2}}\) | \(134\) |
norman | \(\frac {\frac {\left (4 A -9 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d}+\frac {3 \left (A -3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}-\frac {\left (A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{a d}-\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6 a d}-\frac {\left (13 A -34 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} a}+\frac {\left (A -2 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2} d}-\frac {\left (A -2 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2} d}\) | \(192\) |
risch | \(-\frac {2 i \left (3 A \,{\mathrm e}^{4 i \left (d x +c \right )}-6 B \,{\mathrm e}^{4 i \left (d x +c \right )}+9 A \,{\mathrm e}^{3 i \left (d x +c \right )}-18 B \,{\mathrm e}^{3 i \left (d x +c \right )}+7 A \,{\mathrm e}^{2 i \left (d x +c \right )}-22 B \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )} A -24 B \,{\mathrm e}^{i \left (d x +c \right )}+4 A -10 B \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{a^{2} d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{a^{2} d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{2} d}\) | \(227\) |
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Time = 0.29 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.81 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {3 \, {\left ({\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (A - 2 \, B\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (A - 2 \, B\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, {\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (5 \, A - 14 \, B\right )} \cos \left (d x + c\right ) - 3 \, B\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {A \sec ^{3}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{4}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (104) = 208\).
Time = 0.25 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.26 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {B {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - A {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {6 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {6 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )}}{6 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.40 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {6 \, {\left (A - 2 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac {6 \, {\left (A - 2 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} - \frac {12 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{2}} - \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
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Time = 13.48 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.11 \[ \int \frac {\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A-2\,B\right )}{a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A-B\right )}{6\,a^2\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A-B}{a^2}+\frac {A-3\,B}{2\,a^2}\right )}{d}-\frac {2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^2\right )} \]
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