Integrand size = 31, antiderivative size = 252 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {\left (12 a^2 A b+3 A b^3+4 a^3 B+9 a b^2 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (15 a^3 A b+60 a A b^3-3 a^4 B+52 a^2 b^2 B+16 b^4 B\right ) \tan (c+d x)}{30 b d}+\frac {\left (30 a^2 A b+45 A b^3-6 a^3 B+71 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (15 a A b-3 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 A b-a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d} \]
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Time = 0.54 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4095, 4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {\left (-3 a^2 B+15 a A b+16 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{60 b d}+\frac {\left (4 a^3 B+12 a^2 A b+9 a b^2 B+3 A b^3\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (-6 a^3 B+30 a^2 A b+71 a b^2 B+45 A b^3\right ) \tan (c+d x) \sec (c+d x)}{120 d}+\frac {\left (-3 a^4 B+15 a^3 A b+52 a^2 b^2 B+60 a A b^3+16 b^4 B\right ) \tan (c+d x)}{30 b d}+\frac {(5 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^3}{20 b d}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rule 4095
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 (4 b B+(5 A b-a B) \sec (c+d x)) \, dx}{5 b} \\ & = \frac {(5 A b-a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (b (15 A b+13 a B)+\left (15 a A b-3 a^2 B+16 b^2 B\right ) \sec (c+d x)\right ) \, dx}{20 b} \\ & = \frac {\left (15 a A b-3 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 A b-a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x)) \left (b \left (75 a A b+33 a^2 B+32 b^2 B\right )+\left (30 a^2 A b+45 A b^3-6 a^3 B+71 a b^2 B\right ) \sec (c+d x)\right ) \, dx}{60 b} \\ & = \frac {\left (30 a^2 A b+45 A b^3-6 a^3 B+71 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (15 a A b-3 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 A b-a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {\int \sec (c+d x) \left (15 b \left (12 a^2 A b+3 A b^3+4 a^3 B+9 a b^2 B\right )+4 \left (15 a^3 A b+60 a A b^3-3 a^4 B+52 a^2 b^2 B+16 b^4 B\right ) \sec (c+d x)\right ) \, dx}{120 b} \\ & = \frac {\left (30 a^2 A b+45 A b^3-6 a^3 B+71 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (15 a A b-3 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 A b-a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {1}{8} \left (12 a^2 A b+3 A b^3+4 a^3 B+9 a b^2 B\right ) \int \sec (c+d x) \, dx+\frac {\left (15 a^3 A b+60 a A b^3-3 a^4 B+52 a^2 b^2 B+16 b^4 B\right ) \int \sec ^2(c+d x) \, dx}{30 b} \\ & = \frac {\left (12 a^2 A b+3 A b^3+4 a^3 B+9 a b^2 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (30 a^2 A b+45 A b^3-6 a^3 B+71 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (15 a A b-3 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 A b-a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}-\frac {\left (15 a^3 A b+60 a A b^3-3 a^4 B+52 a^2 b^2 B+16 b^4 B\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{30 b d} \\ & = \frac {\left (12 a^2 A b+3 A b^3+4 a^3 B+9 a b^2 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (15 a^3 A b+60 a A b^3-3 a^4 B+52 a^2 b^2 B+16 b^4 B\right ) \tan (c+d x)}{30 b d}+\frac {\left (30 a^2 A b+45 A b^3-6 a^3 B+71 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (15 a A b-3 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 A b-a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d} \\ \end{align*}
Time = 2.74 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.72 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \left (12 a^2 A b+3 A b^3+4 a^3 B+9 a b^2 B\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 \left (12 a^2 A b+3 A b^3+4 a^3 B+9 a b^2 B\right ) \sec (c+d x)+30 b^2 (A b+3 a B) \sec ^3(c+d x)+8 \left (15 \left (a^3 A+3 a A b^2+3 a^2 b B+b^3 B\right )+5 b \left (3 a A b+3 a^2 B+2 b^2 B\right ) \tan ^2(c+d x)+3 b^3 B \tan ^4(c+d x)\right )\right )}{120 d} \]
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Time = 6.41 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.79
method | result | size |
parts | \(\frac {\left (A \,b^{3}+3 B a \,b^{2}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (3 A a \,b^{2}+3 B \,a^{2} b \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {B \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{3} A \tan \left (d x +c \right )}{d}\) | \(200\) |
derivativedivides | \(\frac {a^{3} A \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{2} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-3 A a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B a \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+A \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(275\) |
default | \(\frac {a^{3} A \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{2} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-3 A a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B a \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+A \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(275\) |
parallelrisch | \(\frac {-180 \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right ) \left (A \,a^{2} b +\frac {1}{4} A \,b^{3}+\frac {1}{3} B \,a^{3}+\frac {3}{4} B a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+180 \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right ) \left (A \,a^{2} b +\frac {1}{4} A \,b^{3}+\frac {1}{3} B \,a^{3}+\frac {3}{4} B a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (720 A \,a^{2} b +420 A \,b^{3}+240 B \,a^{3}+1260 B a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (360 a^{3} A +1200 A a \,b^{2}+1200 B \,a^{2} b +320 B \,b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (360 A \,a^{2} b +90 A \,b^{3}+120 B \,a^{3}+270 B a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (120 a^{3} A +240 A a \,b^{2}+240 B \,a^{2} b +64 B \,b^{3}\right ) \sin \left (5 d x +5 c \right )+240 \left (a^{3} A +4 A a \,b^{2}+4 B \,a^{2} b +\frac {8}{3} B \,b^{3}\right ) \sin \left (d x +c \right )}{120 d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(358\) |
norman | \(\frac {-\frac {4 \left (45 a^{3} A +75 A a \,b^{2}+75 B \,a^{2} b +29 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}-\frac {\left (8 a^{3} A -12 A \,a^{2} b +24 A a \,b^{2}-5 A \,b^{3}-4 B \,a^{3}+24 B \,a^{2} b -15 B a \,b^{2}+8 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {\left (8 a^{3} A +12 A \,a^{2} b +24 A a \,b^{2}+5 A \,b^{3}+4 B \,a^{3}+24 B \,a^{2} b +15 B a \,b^{2}+8 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (48 a^{3} A -36 A \,a^{2} b +96 A a \,b^{2}-3 A \,b^{3}-12 B \,a^{3}+96 B \,a^{2} b -9 B a \,b^{2}+16 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {\left (48 a^{3} A +36 A \,a^{2} b +96 A a \,b^{2}+3 A \,b^{3}+12 B \,a^{3}+96 B \,a^{2} b +9 B a \,b^{2}+16 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}-\frac {\left (12 A \,a^{2} b +3 A \,b^{3}+4 B \,a^{3}+9 B a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (12 A \,a^{2} b +3 A \,b^{3}+4 B \,a^{3}+9 B a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(424\) |
risch | \(-\frac {i \left (-240 B \,a^{2} b -240 A a \,b^{2}-64 B \,b^{3}-120 a^{3} A -480 A \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-45 A \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-60 B \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+120 B \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-720 A \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-480 a^{3} A \,{\mathrm e}^{2 i \left (d x +c \right )}+45 A \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-640 B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+210 A \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+60 B \,a^{3} {\mathrm e}^{9 i \left (d x +c \right )}-120 A \,a^{3} {\mathrm e}^{8 i \left (d x +c \right )}-120 B \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-210 A \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-320 B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+180 A \,a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}-720 B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-1680 A a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-630 B a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-1200 A a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-1200 B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-180 A \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-135 B a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+135 B a \,b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+360 A \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+630 B a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-720 A a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-1680 B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-360 A \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,a^{2} b}{2 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{3}}{8 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{2}}{8 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,a^{2} b}{2 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{3}}{8 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}+\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{2}}{8 d}\) | \(662\) |
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Time = 0.29 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.99 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (4 \, B a^{3} + 12 \, A a^{2} b + 9 \, B a b^{2} + 3 \, A b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (4 \, B a^{3} + 12 \, A a^{2} b + 9 \, B a b^{2} + 3 \, A b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (15 \, A a^{3} + 30 \, B a^{2} b + 30 \, A a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{4} + 24 \, B b^{3} + 15 \, {\left (4 \, B a^{3} + 12 \, A a^{2} b + 9 \, B a b^{2} + 3 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (15 \, B a^{2} b + 15 \, A a b^{2} + 4 \, B b^{3}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
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\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sec ^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.35 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} b + 240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b^{2} + 16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B b^{3} - 45 \, B a b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 15 \, A b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, A a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, A a^{3} \tan \left (d x + c\right )}{240 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (239) = 478\).
Time = 0.37 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.87 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Too large to display} \]
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Time = 18.01 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.87 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {B\,a^3}{2}+\frac {3\,A\,a^2\,b}{2}+\frac {9\,B\,a\,b^2}{8}+\frac {3\,A\,b^3}{8}\right )}{2\,B\,a^3+6\,A\,a^2\,b+\frac {9\,B\,a\,b^2}{2}+\frac {3\,A\,b^3}{2}}\right )\,\left (B\,a^3+3\,A\,a^2\,b+\frac {9\,B\,a\,b^2}{4}+\frac {3\,A\,b^3}{4}\right )}{d}-\frac {\left (2\,A\,a^3-\frac {5\,A\,b^3}{4}-B\,a^3+2\,B\,b^3+6\,A\,a\,b^2-3\,A\,a^2\,b-\frac {15\,B\,a\,b^2}{4}+6\,B\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {A\,b^3}{2}-8\,A\,a^3+2\,B\,a^3-\frac {8\,B\,b^3}{3}-16\,A\,a\,b^2+6\,A\,a^2\,b+\frac {3\,B\,a\,b^2}{2}-16\,B\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (12\,A\,a^3+20\,B\,a^2\,b+20\,A\,a\,b^2+\frac {116\,B\,b^3}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-8\,A\,a^3-\frac {A\,b^3}{2}-2\,B\,a^3-\frac {8\,B\,b^3}{3}-16\,A\,a\,b^2-6\,A\,a^2\,b-\frac {3\,B\,a\,b^2}{2}-16\,B\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^3+\frac {5\,A\,b^3}{4}+B\,a^3+2\,B\,b^3+6\,A\,a\,b^2+3\,A\,a^2\,b+\frac {15\,B\,a\,b^2}{4}+6\,B\,a^2\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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