Integrand size = 29, antiderivative size = 125 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {5 a^3 (4 A+3 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 (4 A+3 B) \tan (c+d x)}{d}+\frac {3 a^3 (4 A+3 B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {a^3 (4 A+3 B) \tan ^3(c+d x)}{12 d} \]
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Time = 0.16 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4086, 3876, 3855, 3852, 8, 3853} \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {5 a^3 (4 A+3 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 (4 A+3 B) \tan ^3(c+d x)}{12 d}+\frac {a^3 (4 A+3 B) \tan (c+d x)}{d}+\frac {3 a^3 (4 A+3 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d} \]
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3876
Rule 4086
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{4} (4 A+3 B) \int \sec (c+d x) (a+a \sec (c+d x))^3 \, dx \\ & = \frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{4} (4 A+3 B) \int \left (a^3 \sec (c+d x)+3 a^3 \sec ^2(c+d x)+3 a^3 \sec ^3(c+d x)+a^3 \sec ^4(c+d x)\right ) \, dx \\ & = \frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{4} \left (a^3 (4 A+3 B)\right ) \int \sec (c+d x) \, dx+\frac {1}{4} \left (a^3 (4 A+3 B)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{4} \left (3 a^3 (4 A+3 B)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{4} \left (3 a^3 (4 A+3 B)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {a^3 (4 A+3 B) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {3 a^3 (4 A+3 B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{8} \left (3 a^3 (4 A+3 B)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^3 (4 A+3 B)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{4 d}-\frac {\left (3 a^3 (4 A+3 B)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{4 d} \\ & = \frac {5 a^3 (4 A+3 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 (4 A+3 B) \tan (c+d x)}{d}+\frac {3 a^3 (4 A+3 B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {a^3 (4 A+3 B) \tan ^3(c+d x)}{12 d} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.38 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {5 a^3 A \text {arctanh}(\sin (c+d x))}{2 d}+\frac {15 a^3 B \text {arctanh}(\sin (c+d x))}{8 d}+\frac {4 a^3 A \tan (c+d x)}{d}+\frac {4 a^3 B \tan (c+d x)}{d}+\frac {3 a^3 A \sec (c+d x) \tan (c+d x)}{2 d}+\frac {15 a^3 B \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^3 B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a^3 A \tan ^3(c+d x)}{3 d}+\frac {a^3 B \tan ^3(c+d x)}{d} \]
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Time = 4.39 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.40
method | result | size |
norman | \(\frac {-\frac {73 a^{3} \left (4 A +3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {55 a^{3} \left (4 A +3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {5 a^{3} \left (4 A +3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {a^{3} \left (44 A +49 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {5 a^{3} \left (4 A +3 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {5 a^{3} \left (4 A +3 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(175\) |
parallelrisch | \(\frac {26 \left (-\frac {15 \left (A +\frac {3 B}{4}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{13}+\frac {15 \left (A +\frac {3 B}{4}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{13}+\left (A +\frac {15 B}{13}\right ) \sin \left (2 d x +2 c \right )+\frac {9 \left (A +\frac {5 B}{4}\right ) \sin \left (3 d x +3 c \right )}{26}+\frac {\left (11 A +9 B \right ) \sin \left (4 d x +4 c \right )}{26}+\frac {9 \left (A +\frac {23 B}{12}\right ) \sin \left (d x +c \right )}{26}\right ) a^{3}}{3 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(178\) |
parts | \(-\frac {\left (a^{3} A +3 B \,a^{3}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 a^{3} A +B \,a^{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 a^{3} A +3 B \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{3}}{d}\) | \(181\) |
derivativedivides | \(\frac {-a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \tan \left (d x +c \right )+3 B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \tan \left (d x +c \right )}{d}\) | \(219\) |
default | \(\frac {-a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \tan \left (d x +c \right )+3 B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \tan \left (d x +c \right )}{d}\) | \(219\) |
risch | \(-\frac {i a^{3} \left (36 A \,{\mathrm e}^{7 i \left (d x +c \right )}+45 B \,{\mathrm e}^{7 i \left (d x +c \right )}-72 A \,{\mathrm e}^{6 i \left (d x +c \right )}-24 B \,{\mathrm e}^{6 i \left (d x +c \right )}+36 A \,{\mathrm e}^{5 i \left (d x +c \right )}+69 B \,{\mathrm e}^{5 i \left (d x +c \right )}-264 A \,{\mathrm e}^{4 i \left (d x +c \right )}-216 B \,{\mathrm e}^{4 i \left (d x +c \right )}-36 A \,{\mathrm e}^{3 i \left (d x +c \right )}-69 B \,{\mathrm e}^{3 i \left (d x +c \right )}-280 A \,{\mathrm e}^{2 i \left (d x +c \right )}-264 B \,{\mathrm e}^{2 i \left (d x +c \right )}-36 \,{\mathrm e}^{i \left (d x +c \right )} A -45 B \,{\mathrm e}^{i \left (d x +c \right )}-88 A -72 B \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{8 d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{8 d}\) | \(287\) |
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Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.16 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (4 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (4 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (11 \, A + 9 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 9 \, {\left (4 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 8 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) + 6 \, B a^{3}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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\[ \int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=a^{3} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 3 A \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{3}{\left (c + d x \right )}\, dx + \int A \sec ^{4}{\left (c + d x \right )}\, dx + \int B \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{4}{\left (c + d x \right )}\, dx + \int B \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (117) = 234\).
Time = 0.23 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.10 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 48 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} - 3 \, B a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 144 \, A a^{3} \tan \left (d x + c\right ) + 48 \, B a^{3} \tan \left (d x + c\right )}{48 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.70 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (4 \, A a^{3} + 3 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (4 \, A a^{3} + 3 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (60 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 45 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 220 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 165 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 292 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 219 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 132 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 147 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
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Time = 15.92 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.48 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {\left (-5\,A\,a^3-\frac {15\,B\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {55\,A\,a^3}{3}+\frac {55\,B\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {73\,A\,a^3}{3}-\frac {73\,B\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A\,a^3+\frac {49\,B\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {5\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,A+3\,B\right )}{4\,d} \]
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