Integrand size = 40, antiderivative size = 117 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a^3 (7 B+6 C) x+\frac {a^3 (B+3 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 B \sin (c+d x)}{2 d}+\frac {a B \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(B-2 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{2 d} \]
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Time = 0.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4157, 4102, 4103, 4081, 3855} \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (B+3 C) \text {arctanh}(\sin (c+d x))}{d}-\frac {(B-2 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{2 d}+\frac {5 a^3 B \sin (c+d x)}{2 d}+\frac {1}{2} a^3 x (7 B+6 C)+\frac {a B \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d} \]
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Rule 3855
Rule 4081
Rule 4102
Rule 4103
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \cos ^2(c+d x) (a+a \sec (c+d x))^3 (B+C \sec (c+d x)) \, dx \\ & = \frac {a B \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {1}{2} \int \cos (c+d x) (a+a \sec (c+d x))^2 (2 a (2 B+C)-a (B-2 C) \sec (c+d x)) \, dx \\ & = \frac {a B \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(B-2 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {1}{2} \int \cos (c+d x) (a+a \sec (c+d x)) \left (5 a^2 B+2 a^2 (B+3 C) \sec (c+d x)\right ) \, dx \\ & = \frac {5 a^3 B \sin (c+d x)}{2 d}+\frac {a B \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(B-2 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{2 d}-\frac {1}{2} \int \left (-a^3 (7 B+6 C)-2 a^3 (B+3 C) \sec (c+d x)\right ) \, dx \\ & = \frac {1}{2} a^3 (7 B+6 C) x+\frac {5 a^3 B \sin (c+d x)}{2 d}+\frac {a B \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(B-2 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\left (a^3 (B+3 C)\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} a^3 (7 B+6 C) x+\frac {a^3 (B+3 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 B \sin (c+d x)}{2 d}+\frac {a B \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(B-2 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(272\) vs. \(2(117)=234\).
Time = 2.10 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.32 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{32} a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (2 (7 B+6 C) x-\frac {4 (B+3 C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {4 (B+3 C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {4 (3 B+C) \cos (d x) \sin (c)}{d}+\frac {B \cos (2 d x) \sin (2 c)}{d}+\frac {4 (3 B+C) \cos (c) \sin (d x)}{d}+\frac {B \cos (2 c) \sin (2 d x)}{d}+\frac {4 C \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 C \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {a^{3} \left (-8 \cos \left (d x +c \right ) \left (B +3 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+8 \cos \left (d x +c \right ) \left (B +3 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+4 \left (3 B +C \right ) \sin \left (2 d x +2 c \right )+B \sin \left (3 d x +3 c \right )+28 \left (B +\frac {6 C}{7}\right ) x d \cos \left (d x +c \right )+\sin \left (d x +c \right ) \left (B +8 C \right )\right )}{8 d \cos \left (d x +c \right )}\) | \(122\) |
derivativedivides | \(\frac {B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \tan \left (d x +c \right )+3 B \,a^{3} \left (d x +c \right )+3 a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \sin \left (d x +c \right )+3 a^{3} C \left (d x +c \right )+B \,a^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} C \sin \left (d x +c \right )}{d}\) | \(128\) |
default | \(\frac {B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \tan \left (d x +c \right )+3 B \,a^{3} \left (d x +c \right )+3 a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \sin \left (d x +c \right )+3 a^{3} C \left (d x +c \right )+B \,a^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} C \sin \left (d x +c \right )}{d}\) | \(128\) |
risch | \(\frac {7 a^{3} B x}{2}+3 a^{3} x C -\frac {i B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {3 i B \,a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{3} C}{2 d}+\frac {3 i B \,a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{3} C}{2 d}+\frac {i B \,a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i a^{3} C}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(240\) |
norman | \(\frac {\left (\frac {7}{2} B \,a^{3}+3 a^{3} C \right ) x +\left (-\frac {21}{2} B \,a^{3}-9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {21}{2} B \,a^{3}-9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-\frac {7}{2} B \,a^{3}-3 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {7}{2} B \,a^{3}-3 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {7}{2} B \,a^{3}+3 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {21}{2} B \,a^{3}+9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {21}{2} B \,a^{3}+9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {a^{3} \left (7 B +4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {5 B \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}-\frac {a^{3} \left (B +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {4 a^{3} \left (2 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}+\frac {8 a^{3} \left (3 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {4 a^{3} \left (4 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {a^{3} \left (11 B -4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}+\frac {a^{3} \left (B +3 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{3} \left (B +3 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(445\) |
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Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (7 \, B + 6 \, C\right )} a^{3} d x \cos \left (d x + c\right ) + {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (B a^{3} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, B + C\right )} a^{3} \cos \left (d x + c\right ) + 2 \, C a^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.20 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 12 \, {\left (d x + c\right )} B a^{3} + 12 \, {\left (d x + c\right )} C a^{3} + 2 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, B a^{3} \sin \left (d x + c\right ) + 4 \, C a^{3} \sin \left (d x + c\right ) + 4 \, C a^{3} \tan \left (d x + c\right )}{4 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.64 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {\frac {4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - {\left (7 \, B a^{3} + 6 \, C a^{3}\right )} {\left (d x + c\right )} - 2 \, {\left (B a^{3} + 3 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 2 \, {\left (B a^{3} + 3 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (5 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
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Time = 15.89 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.68 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {7\,B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,C\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \]
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