Integrand size = 31, antiderivative size = 145 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {1}{2} \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) x+\frac {b^3 B \text {arctanh}(\sin (c+d x))}{d}+\frac {a \left (2 a^2 A+8 A b^2+9 a b B\right ) \sin (c+d x)}{3 d}+\frac {a^2 (5 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d} \]
[Out]
Time = 0.37 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4110, 4159, 4132, 8, 4130, 3855} \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a \left (2 a^2 A+9 a b B+8 A b^2\right ) \sin (c+d x)}{3 d}+\frac {a^2 (3 a B+5 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {1}{2} x \left (a^3 B+3 a^2 A b+6 a b^2 B+2 A b^3\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {b^3 B \text {arctanh}(\sin (c+d x))}{d} \]
[In]
[Out]
Rule 8
Rule 3855
Rule 4110
Rule 4130
Rule 4132
Rule 4159
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}-\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (-a (5 A b+3 a B)-\left (2 a^2 A+3 A b^2+6 a b B\right ) \sec (c+d x)-3 b^2 B \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 (5 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) \left (2 a \left (2 a^2 A+8 A b^2+9 a b B\right )+3 \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) \sec (c+d x)+6 b^3 B \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 (5 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) \left (2 a \left (2 a^2 A+8 A b^2+9 a b B\right )+6 b^3 B \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) \int 1 \, dx \\ & = \frac {1}{2} \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) x+\frac {a \left (2 a^2 A+8 A b^2+9 a b B\right ) \sin (c+d x)}{3 d}+\frac {a^2 (5 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\left (b^3 B\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) x+\frac {b^3 B \text {arctanh}(\sin (c+d x))}{d}+\frac {a \left (2 a^2 A+8 A b^2+9 a b B\right ) \sin (c+d x)}{3 d}+\frac {a^2 (5 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d} \\ \end{align*}
Time = 1.64 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.10 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {6 \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) (c+d x)-12 b^3 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 b^3 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 a \left (a^2 A+4 A b^2+4 a b B\right ) \sin (c+d x)+3 a^2 (3 A b+a B) \sin (2 (c+d x))+a^3 A \sin (3 (c+d x))}{12 d} \]
[In]
[Out]
Time = 1.90 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.94
method | result | size |
parallelrisch | \(\frac {-12 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{3}+12 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{3}+\left (9 A \,a^{2} b +3 B \,a^{3}\right ) \sin \left (2 d x +2 c \right )+a^{3} A \sin \left (3 d x +3 c \right )+9 a \left (A \,a^{2}+4 A \,b^{2}+4 B a b \right ) \sin \left (d x +c \right )+18 \left (A \,a^{2} b +\frac {2}{3} A \,b^{3}+\frac {1}{3} B \,a^{3}+2 B a \,b^{2}\right ) d x}{12 d}\) | \(137\) |
derivativedivides | \(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{2} b \sin \left (d x +c \right )+3 A a \,b^{2} \sin \left (d x +c \right )+3 B a \,b^{2} \left (d x +c \right )+A \,b^{3} \left (d x +c \right )+B \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(151\) |
default | \(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{2} b \sin \left (d x +c \right )+3 A a \,b^{2} \sin \left (d x +c \right )+3 B a \,b^{2} \left (d x +c \right )+A \,b^{3} \left (d x +c \right )+B \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(151\) |
risch | \(\frac {3 A \,a^{2} b x}{2}+x A \,b^{3}+\frac {a^{3} x B}{2}+3 x B a \,b^{2}-\frac {3 i a^{3} A \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A a \,b^{2}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{2} b}{2 d}+\frac {3 i a^{3} A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A a \,b^{2}}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{2} b}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,b^{3}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,b^{3}}{d}+\frac {a^{3} A \sin \left (3 d x +3 c \right )}{12 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{2} b}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{4 d}\) | \(247\) |
norman | \(\frac {\left (-\frac {3}{2} A \,a^{2} b -A \,b^{3}-\frac {1}{2} B \,a^{3}-3 B a \,b^{2}\right ) x +\left (-\frac {9}{2} A \,a^{2} b -3 A \,b^{3}-\frac {3}{2} B \,a^{3}-9 B a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {3}{2} A \,a^{2} b +A \,b^{3}+\frac {1}{2} B \,a^{3}+3 B a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {9}{2} A \,a^{2} b +3 A \,b^{3}+\frac {3}{2} B \,a^{3}+9 B a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {a \left (2 A \,a^{2}-3 A a b +6 A \,b^{2}-B \,a^{2}+6 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}+\frac {2 a \left (2 A \,a^{2}-3 A a b -6 A \,b^{2}-B \,a^{2}-6 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {2 a \left (2 A \,a^{2}+3 A a b -6 A \,b^{2}+B \,a^{2}-6 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {a \left (2 A \,a^{2}+3 A a b +6 A \,b^{2}+B \,a^{2}+6 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {a \left (14 A \,a^{2}-27 A a b +18 A \,b^{2}-9 B \,a^{2}+18 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}+\frac {a \left (14 A \,a^{2}+27 A a b +18 A \,b^{2}+9 B \,a^{2}+18 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {B \,b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {B \,b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(494\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {3 \, B b^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, B b^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} d x + {\left (2 \, A a^{3} \cos \left (d x + c\right )^{2} + 4 \, A a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d} \]
[In]
[Out]
\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3} \cos ^{3}{\left (c + d x \right )}\, dx \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.05 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b - 36 \, {\left (d x + c\right )} B a b^{2} - 12 \, {\left (d x + c\right )} A b^{3} - 6 \, B b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, B a^{2} b \sin \left (d x + c\right ) - 36 \, A a b^{2} \sin \left (d x + c\right )}{12 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (137) = 274\).
Time = 0.35 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.17 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {6 \, B b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, B b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
[In]
[Out]
Time = 15.86 (sec) , antiderivative size = 1924, normalized size of antiderivative = 13.27 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Too large to display} \]
[In]
[Out]