Integrand size = 21, antiderivative size = 96 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {7 a^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {2 a^2 \tan (c+d x)}{d}+\frac {7 a^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {2 a^2 \tan ^3(c+d x)}{3 d} \]
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Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3873, 3852, 4131, 3853, 3855} \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {7 a^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {2 a^2 \tan ^3(c+d x)}{3 d}+\frac {2 a^2 \tan (c+d x)}{d}+\frac {a^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {7 a^2 \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rule 3852
Rule 3853
Rule 3855
Rule 3873
Rule 4131
Rubi steps \begin{align*} \text {integral}& = \left (2 a^2\right ) \int \sec ^4(c+d x) \, dx+\int \sec ^3(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \left (7 a^2\right ) \int \sec ^3(c+d x) \, dx-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d} \\ & = \frac {2 a^2 \tan (c+d x)}{d}+\frac {7 a^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {2 a^2 \tan ^3(c+d x)}{3 d}+\frac {1}{8} \left (7 a^2\right ) \int \sec (c+d x) \, dx \\ & = \frac {7 a^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {2 a^2 \tan (c+d x)}{d}+\frac {7 a^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {2 a^2 \tan ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.60 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {a^2 \left (21 \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (21 \sec (c+d x)+6 \sec ^3(c+d x)+16 \left (3+\tan ^2(c+d x)\right )\right )\right )}{24 d} \]
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Time = 0.68 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-2 a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(112\) |
default | \(\frac {a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-2 a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(112\) |
parts | \(\frac {a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {2 a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(117\) |
norman | \(\frac {\frac {25 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {83 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {77 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {7 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {7 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {7 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(133\) |
risch | \(-\frac {i a^{2} \left (21 \,{\mathrm e}^{7 i \left (d x +c \right )}+45 \,{\mathrm e}^{5 i \left (d x +c \right )}-96 \,{\mathrm e}^{4 i \left (d x +c \right )}-45 \,{\mathrm e}^{3 i \left (d x +c \right )}-128 \,{\mathrm e}^{2 i \left (d x +c \right )}-21 \,{\mathrm e}^{i \left (d x +c \right )}-32\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}+\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}\) | \(134\) |
parallelrisch | \(\frac {a^{2} \left (21 \left (-\cos \left (4 d x +4 c \right )-4 \cos \left (2 d x +2 c \right )-3\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+21 \left (3+\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+32 \sin \left (4 d x +4 c \right )+90 \sin \left (d x +c \right )+128 \sin \left (2 d x +2 c \right )+42 \sin \left (3 d x +3 c \right )\right )}{24 d \left (3+\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )\right )}\) | \(149\) |
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Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.16 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {21 \, a^{2} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 21 \, a^{2} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (32 \, a^{2} \cos \left (d x + c\right )^{3} + 21 \, a^{2} \cos \left (d x + c\right )^{2} + 16 \, a^{2} \cos \left (d x + c\right ) + 6 \, a^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \, dx=a^{2} \left (\int \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 \sec ^{4}{\left (c + d x \right )}\, dx + \int \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.51 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {32 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} - 3 \, a^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.27 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {21 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 21 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 77 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 83 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 75 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
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Time = 17.01 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.47 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {7\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {\frac {7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}-\frac {77\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {83\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12}-\frac {25\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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