Integrand size = 29, antiderivative size = 138 \[ \int \frac {\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(3 A+4 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac {2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )} \]
[Out]
Time = 0.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {4085, 3881, 3879} \[ \int \frac {\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^4 \sec (c+d x)+a^4\right )}+\frac {2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac {(3 A+4 B) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}+\frac {(A-B) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
[In]
[Out]
Rule 3879
Rule 3881
Rule 4085
Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(3 A+4 B) \int \frac {\sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{7 a} \\ & = \frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(3 A+4 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {(2 (3 A+4 B)) \int \frac {\sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{35 a^2} \\ & = \frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(3 A+4 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac {(2 (3 A+4 B)) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^3} \\ & = \frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(3 A+4 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac {2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}
Time = 3.33 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.40 \[ \int \frac {\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (70 (9 A+4 B) \sin \left (\frac {d x}{2}\right )-35 (18 A+5 B) \sin \left (c+\frac {d x}{2}\right )+441 A \sin \left (c+\frac {3 d x}{2}\right )+168 B \sin \left (c+\frac {3 d x}{2}\right )-315 A \sin \left (2 c+\frac {3 d x}{2}\right )-105 B \sin \left (2 c+\frac {3 d x}{2}\right )+147 A \sin \left (2 c+\frac {5 d x}{2}\right )+91 B \sin \left (2 c+\frac {5 d x}{2}\right )-105 A \sin \left (3 c+\frac {5 d x}{2}\right )+36 A \sin \left (3 c+\frac {7 d x}{2}\right )+13 B \sin \left (3 c+\frac {7 d x}{2}\right )\right )}{420 a^4 d (1+\cos (c+d x))^4} \]
[In]
[Out]
Time = 0.82 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57
method | result | size |
parallelrisch | \(-\frac {\left (\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {7 \left (-3 A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5}+7 \left (A +\frac {B}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7 A -7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{56 a^{4} d}\) | \(78\) |
derivativedivides | \(\frac {\frac {\left (-A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {\left (3 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {\left (-3 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B}{8 d \,a^{4}}\) | \(90\) |
default | \(\frac {\frac {\left (-A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {\left (3 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {\left (-3 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B}{8 d \,a^{4}}\) | \(90\) |
norman | \(\frac {-\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{56 a d}-\frac {\left (A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (3 A +2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 a d}-\frac {\left (12 A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 a d}+\frac {\left (13 A -6 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{140 a d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) a^{3}}\) | \(141\) |
risch | \(\frac {2 i \left (105 A \,{\mathrm e}^{6 i \left (d x +c \right )}+315 A \,{\mathrm e}^{5 i \left (d x +c \right )}+105 B \,{\mathrm e}^{5 i \left (d x +c \right )}+630 A \,{\mathrm e}^{4 i \left (d x +c \right )}+175 B \,{\mathrm e}^{4 i \left (d x +c \right )}+630 A \,{\mathrm e}^{3 i \left (d x +c \right )}+280 B \,{\mathrm e}^{3 i \left (d x +c \right )}+441 A \,{\mathrm e}^{2 i \left (d x +c \right )}+168 B \,{\mathrm e}^{2 i \left (d x +c \right )}+147 \,{\mathrm e}^{i \left (d x +c \right )} A +91 B \,{\mathrm e}^{i \left (d x +c \right )}+36 A +13 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(162\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.90 \[ \int \frac {\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {{\left ({\left (36 \, A + 13 \, B\right )} \cos \left (d x + c\right )^{3} + 13 \, {\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right ) + 6 \, A + 8 \, B\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
[In]
[Out]
\[ \int \frac {\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \sec {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.27 \[ \int \frac {\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {B {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int \frac {\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{840 \, a^{4} d} \]
[In]
[Out]
Time = 13.83 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.64 \[ \int \frac {\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,A+B\right )}{24\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A+B\right )}{8\,a^4}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,A-B\right )}{40\,a^4}}{d} \]
[In]
[Out]