Integrand size = 39, antiderivative size = 286 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (12 a^2 b B+3 b^3 B+4 a^3 (2 A+C)+3 a b^2 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (15 a^3 b B+60 a b^3 B-3 a^4 C+4 b^4 (5 A+4 C)+4 a^2 b^2 (20 A+13 C)\right ) \tan (c+d x)}{30 b d}+\frac {\left (30 a^2 b B+45 b^3 B-6 a^3 C+a b^2 (100 A+71 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (4 b^2 (5 A+4 C)+3 a (5 b B-a C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 b B-a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d} \]
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Time = 0.67 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {4167, 4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\tan (c+d x) \sec (c+d x) \left (-6 a^3 C+30 a^2 b B+a b^2 (100 A+71 C)+45 b^3 B\right )}{120 d}+\frac {\tan (c+d x) \left (-3 a^4 C+15 a^3 b B+4 a^2 b^2 (20 A+13 C)+60 a b^3 B+4 b^4 (5 A+4 C)\right )}{30 b d}+\frac {\tan (c+d x) \left (3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2}{60 b d}+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{20 b d}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rule 4167
Rubi steps \begin{align*} \text {integral}& = \frac {C (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 (b (5 A+4 C)+(5 b B-a C) \sec (c+d x)) \, dx}{5 b} \\ & = \frac {(5 b B-a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (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 (20 a A+15 b B+13 a C)+\left (4 b^2 (5 A+4 C)+3 a (5 b B-a C)\right ) \sec (c+d x)\right ) \, dx}{20 b} \\ & = \frac {\left (4 b^2 (5 A+4 C)+3 a (5 b B-a C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 b B-a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (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 b B+8 b^2 (5 A+4 C)+a^2 (60 A+33 C)\right )+\left (30 a^2 b B+45 b^3 B-6 a^3 C+a b^2 (100 A+71 C)\right ) \sec (c+d x)\right ) \, dx}{60 b} \\ & = \frac {\left (30 a^2 b B+45 b^3 B-6 a^3 C+a b^2 (100 A+71 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (4 b^2 (5 A+4 C)+3 a (5 b B-a C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 b B-a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (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 b B+3 b^3 B+4 a^3 (2 A+C)+3 a b^2 (4 A+3 C)\right )+4 \left (15 a^3 b B+60 a b^3 B-3 a^4 C+4 b^4 (5 A+4 C)+4 a^2 b^2 (20 A+13 C)\right ) \sec (c+d x)\right ) \, dx}{120 b} \\ & = \frac {\left (30 a^2 b B+45 b^3 B-6 a^3 C+a b^2 (100 A+71 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (4 b^2 (5 A+4 C)+3 a (5 b B-a C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 b B-a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {1}{8} \left (12 a^2 b B+3 b^3 B+4 a^3 (2 A+C)+3 a b^2 (4 A+3 C)\right ) \int \sec (c+d x) \, dx+\frac {\left (15 a^3 b B+60 a b^3 B-3 a^4 C+4 b^4 (5 A+4 C)+4 a^2 b^2 (20 A+13 C)\right ) \int \sec ^2(c+d x) \, dx}{30 b} \\ & = \frac {\left (12 a^2 b B+3 b^3 B+4 a^3 (2 A+C)+3 a b^2 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (30 a^2 b B+45 b^3 B-6 a^3 C+a b^2 (100 A+71 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (4 b^2 (5 A+4 C)+3 a (5 b B-a C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 b B-a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}-\frac {\left (15 a^3 b B+60 a b^3 B-3 a^4 C+4 b^4 (5 A+4 C)+4 a^2 b^2 (20 A+13 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{30 b d} \\ & = \frac {\left (12 a^2 b B+3 b^3 B+4 a^3 (2 A+C)+3 a b^2 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (15 a^3 b B+60 a b^3 B-3 a^4 C+4 b^4 (5 A+4 C)+4 a^2 b^2 (20 A+13 C)\right ) \tan (c+d x)}{30 b d}+\frac {\left (30 a^2 b B+45 b^3 B-6 a^3 C+a b^2 (100 A+71 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (4 b^2 (5 A+4 C)+3 a (5 b B-a C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 b B-a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d} \\ \end{align*}
Time = 5.78 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.72 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \left (12 a^2 b B+3 b^3 B+4 a^3 (2 A+C)+3 a b^2 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 \left (12 a^2 b B+3 b^3 B+4 a^3 C+3 a b^2 (4 A+3 C)\right ) \sec (c+d x)+30 b^2 (b B+3 a C) \sec ^3(c+d x)+8 \left (15 \left (a^3 B+3 a b^2 B+3 a^2 b (A+C)+b^3 (A+C)\right )+5 b \left (A b^2+3 a b B+3 a^2 C+2 b^2 C\right ) \tan ^2(c+d x)+3 b^3 C \tan ^4(c+d x)\right )\right )}{120 d} \]
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Time = 1.40 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.85
method | result | size |
parts | \(\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {C \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{3}}{d}\) | \(243\) |
derivativedivides | \(\frac {a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \tan \left (d x +c \right )+a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{2} b \tan \left (d x +c \right )+3 B \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 a^{2} b C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 C a \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-A \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-C \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(369\) |
default | \(\frac {a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \tan \left (d x +c \right )+a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{2} b \tan \left (d x +c \right )+3 B \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 a^{2} b C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 C a \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-A \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-C \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(369\) |
parallelrisch | \(\frac {-600 \left (\left (A +\frac {C}{2}\right ) a^{3}+\frac {3 B \,a^{2} b}{2}+\frac {3 \left (A +\frac {3 C}{4}\right ) b^{2} a}{2}+\frac {3 B \,b^{3}}{8}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+600 \left (\left (A +\frac {C}{2}\right ) a^{3}+\frac {3 B \,a^{2} b}{2}+\frac {3 \left (A +\frac {3 C}{4}\right ) b^{2} a}{2}+\frac {3 B \,b^{3}}{8}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (360 B \,a^{3}+1080 \left (A +\frac {10 C}{9}\right ) b \,a^{2}+1200 B a \,b^{2}+400 b^{3} \left (A +\frac {4 C}{5}\right )\right ) \sin \left (3 d x +3 c \right )+\left (120 B \,a^{3}+360 b \left (A +\frac {2 C}{3}\right ) a^{2}+240 B a \,b^{2}+80 b^{3} \left (A +\frac {4 C}{5}\right )\right ) \sin \left (5 d x +5 c \right )+\left (240 a^{3} C +720 B \,a^{2} b +720 b^{2} \left (A +\frac {7 C}{4}\right ) a +420 B \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (120 a^{3} C +360 B \,a^{2} b +360 \left (A +\frac {3 C}{4}\right ) b^{2} a +90 B \,b^{3}\right ) \sin \left (4 d x +4 c \right )+720 \left (\frac {B \,a^{3}}{3}+b \left (A +\frac {4 C}{3}\right ) a^{2}+\frac {4 B a \,b^{2}}{3}+\frac {4 b^{3} \left (A +2 C \right )}{9}\right ) \sin \left (d x +c \right )}{600 d \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right )}\) | \(406\) |
norman | \(\frac {-\frac {4 \left (135 A \,a^{2} b +25 A \,b^{3}+45 B \,a^{3}+75 B a \,b^{2}+75 a^{2} b C +29 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}-\frac {\left (24 A \,a^{2} b -12 a A \,b^{2}+8 A \,b^{3}+8 B \,a^{3}-12 B \,a^{2} b +24 B a \,b^{2}-5 B \,b^{3}-4 a^{3} C +24 a^{2} b C -15 C a \,b^{2}+8 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {\left (24 A \,a^{2} b +12 a A \,b^{2}+8 A \,b^{3}+8 B \,a^{3}+12 B \,a^{2} b +24 B a \,b^{2}+5 B \,b^{3}+4 a^{3} C +24 a^{2} b C +15 C a \,b^{2}+8 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (144 A \,a^{2} b -36 a A \,b^{2}+32 A \,b^{3}+48 B \,a^{3}-36 B \,a^{2} b +96 B a \,b^{2}-3 B \,b^{3}-12 a^{3} C +96 a^{2} b C -9 C a \,b^{2}+16 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {\left (144 A \,a^{2} b +36 a A \,b^{2}+32 A \,b^{3}+48 B \,a^{3}+36 B \,a^{2} b +96 B a \,b^{2}+3 B \,b^{3}+12 a^{3} C +96 a^{2} b C +9 C a \,b^{2}+16 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {\left (8 a^{3} A +12 a A \,b^{2}+12 B \,a^{2} b +3 B \,b^{3}+4 a^{3} C +9 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (8 a^{3} A +12 a A \,b^{2}+12 B \,a^{2} b +3 B \,b^{3}+4 a^{3} C +9 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(543\) |
risch | \(\text {Expression too large to display}\) | \(937\) |
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Time = 0.29 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.02 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (4 \, {\left (2 \, A + C\right )} a^{3} + 12 \, B a^{2} b + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2} + 3 \, B b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (4 \, {\left (2 \, A + C\right )} a^{3} + 12 \, B a^{2} b + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2} + 3 \, B b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (15 \, B a^{3} + 15 \, {\left (3 \, A + 2 \, C\right )} a^{2} b + 30 \, B a b^{2} + 2 \, {\left (5 \, A + 4 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 24 \, C b^{3} + 15 \, {\left (4 \, C a^{3} + 12 \, B a^{2} b + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2} + 3 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (15 \, C a^{2} b + 15 \, B a b^{2} + {\left (5 \, A + 4 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
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\[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.56 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} b + 240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a b^{2} + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{3} + 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 )} C b^{3} - 45 \, C 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 \, B 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 \, C 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 \, B 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 )} - 180 \, A a 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 )} + 240 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 240 \, B a^{3} \tan \left (d x + c\right ) + 720 \, A a^{2} b \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. 989 vs. \(2 (273) = 546\).
Time = 0.37 (sec) , antiderivative size = 989, normalized size of antiderivative = 3.46 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Time = 19.04 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.10 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A\,a^3+\frac {3\,B\,b^3}{8}+\frac {C\,a^3}{2}+\frac {3\,A\,a\,b^2}{2}+\frac {3\,B\,a^2\,b}{2}+\frac {9\,C\,a\,b^2}{8}\right )}{4\,A\,a^3+\frac {3\,B\,b^3}{2}+2\,C\,a^3+6\,A\,a\,b^2+6\,B\,a^2\,b+\frac {9\,C\,a\,b^2}{2}}\right )\,\left (2\,A\,a^3+\frac {3\,B\,b^3}{4}+C\,a^3+3\,A\,a\,b^2+3\,B\,a^2\,b+\frac {9\,C\,a\,b^2}{4}\right )}{d}-\frac {\left (2\,A\,b^3+2\,B\,a^3-\frac {5\,B\,b^3}{4}-C\,a^3+2\,C\,b^3-3\,A\,a\,b^2+6\,A\,a^2\,b+6\,B\,a\,b^2-3\,B\,a^2\,b-\frac {15\,C\,a\,b^2}{4}+6\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {B\,b^3}{2}-8\,B\,a^3-\frac {16\,A\,b^3}{3}+2\,C\,a^3-\frac {8\,C\,b^3}{3}+6\,A\,a\,b^2-24\,A\,a^2\,b-16\,B\,a\,b^2+6\,B\,a^2\,b+\frac {3\,C\,a\,b^2}{2}-16\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {20\,A\,b^3}{3}+12\,B\,a^3+\frac {116\,C\,b^3}{15}+36\,A\,a^2\,b+20\,B\,a\,b^2+20\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {16\,A\,b^3}{3}-8\,B\,a^3-\frac {B\,b^3}{2}-2\,C\,a^3-\frac {8\,C\,b^3}{3}-6\,A\,a\,b^2-24\,A\,a^2\,b-16\,B\,a\,b^2-6\,B\,a^2\,b-\frac {3\,C\,a\,b^2}{2}-16\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,b^3+2\,B\,a^3+\frac {5\,B\,b^3}{4}+C\,a^3+2\,C\,b^3+3\,A\,a\,b^2+6\,A\,a^2\,b+6\,B\,a\,b^2+3\,B\,a^2\,b+\frac {15\,C\,a\,b^2}{4}+6\,C\,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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