Integrand size = 23, antiderivative size = 200 \[ \int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=a^4 A x+\frac {\left (32 a^3 A b+16 a A b^3+8 a^4 B+24 a^2 b^2 B+3 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \left (34 a^2 A b+4 A b^3+19 a^3 B+16 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac {b^2 \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \]
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Time = 0.35 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4003, 4141, 4133, 3855, 3852, 8} \[ \int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=a^4 A x+\frac {b^2 \left (26 a^2 B+32 a A b+9 b^2 B\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac {b \left (19 a^3 B+34 a^2 A b+16 a b^2 B+4 A b^3\right ) \tan (c+d x)}{6 d}+\frac {\left (8 a^4 B+32 a^3 A b+24 a^2 b^2 B+16 a A b^3+3 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b (7 a B+4 A b) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {b B \tan (c+d x) (a+b \sec (c+d x))^3}{4 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 4003
Rule 4133
Rule 4141
Rubi steps \begin{align*} \text {integral}& = \frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \sec (c+d x))^2 \left (4 a^2 A+\left (8 a A b+4 a^2 B+3 b^2 B\right ) \sec (c+d x)+b (4 A b+7 a B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \sec (c+d x)) \left (12 a^3 A+\left (36 a^2 A b+8 A b^3+12 a^3 B+23 a b^2 B\right ) \sec (c+d x)+b \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {b^2 \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{24} \int \left (24 a^4 A+3 \left (32 a^3 A b+16 a A b^3+8 a^4 B+24 a^2 b^2 B+3 b^4 B\right ) \sec (c+d x)+4 b \left (34 a^2 A b+4 A b^3+19 a^3 B+16 a b^2 B\right ) \sec ^2(c+d x)\right ) \, dx \\ & = a^4 A x+\frac {b^2 \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{6} \left (b \left (34 a^2 A b+4 A b^3+19 a^3 B+16 a b^2 B\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (32 a^3 A b+16 a A b^3+8 a^4 B+24 a^2 b^2 B+3 b^4 B\right ) \int \sec (c+d x) \, dx \\ & = a^4 A x+\frac {\left (32 a^3 A b+16 a A b^3+8 a^4 B+24 a^2 b^2 B+3 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b^2 \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {\left (b \left (34 a^2 A b+4 A b^3+19 a^3 B+16 a b^2 B\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d} \\ & = a^4 A x+\frac {\left (32 a^3 A b+16 a A b^3+8 a^4 B+24 a^2 b^2 B+3 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \left (34 a^2 A b+4 A b^3+19 a^3 B+16 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac {b^2 \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.80 \[ \int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {24 a^4 A d x+3 \left (32 a^3 A b+16 a A b^3+8 a^4 B+24 a^2 b^2 B+3 b^4 B\right ) \text {arctanh}(\sin (c+d x))+3 b \left (8 \left (6 a^2 A b+A b^3+4 a^3 B+4 a b^2 B\right )+b \left (16 a A b+24 a^2 B+3 b^2 B\right ) \sec (c+d x)+2 b^3 B \sec ^3(c+d x)\right ) \tan (c+d x)+8 b^3 (A b+4 a B) \tan ^3(c+d x)}{24 d} \]
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Time = 5.31 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.03
method | result | size |
parts | \(a^{4} A x -\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(206\) |
derivativedivides | \(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \tan \left (d x +c \right ) a^{3} b +6 A \tan \left (d x +c \right ) a^{2} b^{2}+6 B \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 A a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-A \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(260\) |
default | \(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \tan \left (d x +c \right ) a^{3} b +6 A \tan \left (d x +c \right ) a^{2} b^{2}+6 B \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 A a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-A \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(260\) |
parallelrisch | \(\frac {-384 \left (A \,a^{3} b +\frac {1}{2} A a \,b^{3}+\frac {1}{4} B \,a^{4}+\frac {3}{4} B \,a^{2} b^{2}+\frac {3}{32} B \,b^{4}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+384 \left (A \,a^{3} b +\frac {1}{2} A a \,b^{3}+\frac {1}{4} B \,a^{4}+\frac {3}{4} B \,a^{2} b^{2}+\frac {3}{32} B \,b^{4}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+96 a^{4} A x d \cos \left (2 d x +2 c \right )+24 a^{4} A x d \cos \left (4 d x +4 c \right )+\left (288 A \,a^{2} b^{2}+64 A \,b^{4}+192 B \,a^{3} b +256 B a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (144 A \,a^{2} b^{2}+16 A \,b^{4}+96 B \,a^{3} b +64 B a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+\left (96 A a \,b^{3}+144 B \,a^{2} b^{2}+18 B \,b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (96 A a \,b^{3}+144 B \,a^{2} b^{2}+66 B \,b^{4}\right ) \sin \left (d x +c \right )+72 a^{4} A x d}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(358\) |
norman | \(\frac {a^{4} A x +a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-4 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {b \left (48 A \,a^{2} b -16 A a \,b^{2}+8 A \,b^{3}+32 B \,a^{3}-24 B \,a^{2} b +32 B a \,b^{2}-5 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {b \left (48 A \,a^{2} b +16 A a \,b^{2}+8 A \,b^{3}+32 B \,a^{3}+24 B \,a^{2} b +32 B a \,b^{2}+5 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {b \left (432 A \,a^{2} b -48 A a \,b^{2}+40 A \,b^{3}+288 B \,a^{3}-72 B \,a^{2} b +160 B a \,b^{2}+9 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {b \left (432 A \,a^{2} b +48 A a \,b^{2}+40 A \,b^{3}+288 B \,a^{3}+72 B \,a^{2} b +160 B a \,b^{2}-9 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {\left (32 A \,a^{3} b +16 A a \,b^{3}+8 B \,a^{4}+24 B \,a^{2} b^{2}+3 B \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (32 A \,a^{3} b +16 A a \,b^{3}+8 B \,a^{4}+24 B \,a^{2} b^{2}+3 B \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(456\) |
risch | \(a^{4} A x -\frac {i b \left (-144 A \,a^{2} b -64 B a \,b^{2}+33 B \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-96 B \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-288 B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-9 b^{3} B \,{\mathrm e}^{i \left (d x +c \right )}-288 B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-33 B \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+9 B \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-64 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-48 A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-96 B \,a^{3}-48 A a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-72 B \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-432 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-256 B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+48 A a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-72 B \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+72 B \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-432 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-192 B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-48 A a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+72 B \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-144 A \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+48 A a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-16 A \,b^{3}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,a^{3} b}{d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A a \,b^{3}}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,a^{2} b^{2}}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,b^{4}}{8 d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,a^{3} b}{d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A a \,b^{3}}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,a^{2} b^{2}}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,b^{4}}{8 d}\) | \(627\) |
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Time = 0.30 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.25 \[ \int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {48 \, A a^{4} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, B b^{4} + 16 \, {\left (6 \, B a^{3} b + 9 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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\[ \int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{4}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.52 \[ \int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {48 \, {\left (d x + c\right )} A a^{4} + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a b^{3} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{4} - 3 \, B b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, B a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 48 \, A a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, B a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 192 \, A a^{3} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 192 \, B a^{3} b \tan \left (d x + c\right ) + 288 \, A a^{2} b^{2} \tan \left (d x + c\right )}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (190) = 380\).
Time = 0.40 (sec) , antiderivative size = 635, normalized size of antiderivative = 3.18 \[ \int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {24 \, {\left (d x + c\right )} A a^{4} + 3 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 3 \, B b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 3 \, B b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (96 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 144 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 288 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 432 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 432 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 144 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B b^{4} \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 = 17.31 (sec) , antiderivative size = 1969, normalized size of antiderivative = 9.84 \[ \int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Too large to display} \]
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