Integrand size = 23, antiderivative size = 82 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=a^2 A x+\frac {a^2 (4 A+3 B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^2 (2 A+3 B) \tan (c+d x)}{2 d}+\frac {B \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{2 d} \]
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Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4002, 3999, 3852, 8, 3855} \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 (4 A+3 B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^2 (2 A+3 B) \tan (c+d x)}{2 d}+a^2 A x+\frac {B \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3999
Rule 4002
Rubi steps \begin{align*} \text {integral}& = \frac {B \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+a \sec (c+d x)) (2 a A+a (2 A+3 B) \sec (c+d x)) \, dx \\ & = a^2 A x+\frac {B \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a^2 (2 A+3 B)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{2} \left (a^2 (4 A+3 B)\right ) \int \sec (c+d x) \, dx \\ & = a^2 A x+\frac {a^2 (4 A+3 B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {B \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{2 d}-\frac {\left (a^2 (2 A+3 B)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 d} \\ & = a^2 A x+\frac {a^2 (4 A+3 B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^2 (2 A+3 B) \tan (c+d x)}{2 d}+\frac {B \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.65 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 (2 A d x+(4 A+3 B) \text {arctanh}(\sin (c+d x))+(2 A+4 B+B \sec (c+d x)) \tan (c+d x))}{2 d} \]
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Time = 2.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.21
method | result | size |
parts | \(a^{2} A x +\frac {\left (A \,a^{2}+2 B \,a^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (2 A \,a^{2}+B \,a^{2}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(99\) |
derivativedivides | \(\frac {A \,a^{2} \tan \left (d x +c \right )+B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 B \,a^{2} \tan \left (d x +c \right )+A \,a^{2} \left (d x +c \right )+B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(114\) |
default | \(\frac {A \,a^{2} \tan \left (d x +c \right )+B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 B \,a^{2} \tan \left (d x +c \right )+A \,a^{2} \left (d x +c \right )+B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(114\) |
parallelrisch | \(-\frac {2 \left (\left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {3 B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {3 B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {A x d \cos \left (2 d x +2 c \right )}{2}+\left (-\frac {A}{2}-B \right ) \sin \left (2 d x +2 c \right )-\frac {A x d}{2}-\frac {B \sin \left (d x +c \right )}{2}\right ) a^{2}}{d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(127\) |
norman | \(\frac {a^{2} A x +a^{2} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {a^{2} \left (2 A +5 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-2 a^{2} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a^{2} \left (2 A +3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {a^{2} \left (4 A +3 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{2} \left (4 A +3 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(163\) |
risch | \(a^{2} A x -\frac {i a^{2} \left (B \,{\mathrm e}^{3 i \left (d x +c \right )}-2 A \,{\mathrm e}^{2 i \left (d x +c \right )}-4 B \,{\mathrm e}^{2 i \left (d x +c \right )}-B \,{\mathrm e}^{i \left (d x +c \right )}-2 A -4 B \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}\) | \(172\) |
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Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.45 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {4 \, A a^{2} d x \cos \left (d x + c\right )^{2} + {\left (4 \, A + 3 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (4 \, A + 3 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right ) + B a^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=a^{2} \left (\int A\, dx + \int 2 A \sec {\left (c + d x \right )}\, dx + \int A \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sec {\left (c + d x \right )}\, dx + \int 2 B \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.56 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {4 \, {\left (d x + c\right )} A a^{2} - B a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, B a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, A a^{2} \tan \left (d x + c\right ) + 8 \, B a^{2} \tan \left (d x + c\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (76) = 152\).
Time = 0.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.88 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {2 \, {\left (d x + c\right )} A a^{2} + {\left (4 \, A a^{2} + 3 \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (4 \, A a^{2} + 3 \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
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Time = 13.83 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.98 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {2\,A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,B\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {2\,B\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^2\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2} \]
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