Integrand size = 38, antiderivative size = 114 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(4 a B+3 b C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(b B+a C) \tan (c+d x)}{d}+\frac {(4 a B+3 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {(b B+a C) \tan ^3(c+d x)}{3 d} \]
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Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4157, 4082, 3872, 3853, 3855, 3852} \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(4 a B+3 b C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(a C+b B) \tan ^3(c+d x)}{3 d}+\frac {(a C+b B) \tan (c+d x)}{d}+\frac {(4 a B+3 b C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b C \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4082
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \sec ^3(c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx \\ & = \frac {b C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int \sec ^3(c+d x) (4 a B+3 b C+4 (b B+a C) \sec (c+d x)) \, dx \\ & = \frac {b C \sec ^3(c+d x) \tan (c+d x)}{4 d}+(b B+a C) \int \sec ^4(c+d x) \, dx+\frac {1}{4} (4 a B+3 b C) \int \sec ^3(c+d x) \, dx \\ & = \frac {(4 a B+3 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{8} (4 a B+3 b C) \int \sec (c+d x) \, dx-\frac {(b B+a C) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d} \\ & = \frac {(4 a B+3 b C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(b B+a C) \tan (c+d x)}{d}+\frac {(4 a B+3 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {(b B+a C) \tan ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.75 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 (4 a B+3 b C) \text {arctanh}(\sin (c+d x))+\sec (c+d x) \left (12 a B+9 b C+8 (b B+a C) (2+\cos (2 (c+d x))) \sec (c+d x)+6 b C \sec ^2(c+d x)\right ) \tan (c+d x)}{24 d} \]
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Time = 0.85 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04
method | result | size |
parts | \(-\frac {\left (B b +C a \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {a B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {C b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(119\) |
derivativedivides | \(\frac {a B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C a \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-B b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+C b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(131\) |
default | \(\frac {a B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C a \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-B b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+C b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(131\) |
parallelrisch | \(\frac {-24 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a B +\frac {3 C b}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+24 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a B +\frac {3 C b}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+32 \left (B b +C a \right ) \sin \left (2 d x +2 c \right )+3 \left (4 a B +3 C b \right ) \sin \left (3 d x +3 c \right )+8 \left (B b +C a \right ) \sin \left (4 d x +4 c \right )+3 \sin \left (d x +c \right ) \left (4 a B +11 C b \right )}{12 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(192\) |
norman | \(\frac {\frac {\left (4 a B -8 B b -8 C a +5 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (4 a B +8 B b +8 C a +5 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (12 a B -40 B b -40 C a -9 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {\left (12 a B +40 B b +40 C a -9 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {\left (4 a B +3 C b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (4 a B +3 C b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(201\) |
risch | \(-\frac {i \left (12 a B \,{\mathrm e}^{7 i \left (d x +c \right )}+9 C b \,{\mathrm e}^{7 i \left (d x +c \right )}+12 B a \,{\mathrm e}^{5 i \left (d x +c \right )}+33 C b \,{\mathrm e}^{5 i \left (d x +c \right )}-48 B b \,{\mathrm e}^{4 i \left (d x +c \right )}-48 C a \,{\mathrm e}^{4 i \left (d x +c \right )}-12 a B \,{\mathrm e}^{3 i \left (d x +c \right )}-33 C b \,{\mathrm e}^{3 i \left (d x +c \right )}-64 B b \,{\mathrm e}^{2 i \left (d x +c \right )}-64 C a \,{\mathrm e}^{2 i \left (d x +c \right )}-12 a B \,{\mathrm e}^{i \left (d x +c \right )}-9 C b \,{\mathrm e}^{i \left (d x +c \right )}-16 B b -16 C a \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a B}{2 d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a B}{2 d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\) | \(266\) |
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Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.19 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (4 \, B a + 3 \, C b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, B a + 3 \, C b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, B a + 3 \, C b\right )} \cos \left (d x + c\right )^{2} + 6 \, C b + 8 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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\[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.43 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b - 3 \, C b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, B a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (106) = 212\).
Time = 0.32 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.67 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (4 \, B a + 3 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (4 \, B a + 3 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
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Time = 21.01 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.70 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (B\,a-2\,B\,b-2\,C\,a+\frac {5\,C\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {10\,B\,b}{3}-B\,a+\frac {10\,C\,a}{3}+\frac {3\,C\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {3\,C\,b}{4}-\frac {10\,B\,b}{3}-\frac {10\,C\,a}{3}-B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (B\,a+2\,B\,b+2\,C\,a+\frac {5\,C\,b}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (B\,a+\frac {3\,C\,b}{4}\right )}{d} \]
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