Integrand size = 33, antiderivative size = 207 \[ \int (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^3 A x+\frac {\left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \]
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Time = 0.36 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4141, 4133, 3855, 3852, 8} \[ \int (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^3 A x+\frac {b \tan (c+d x) \sec (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{24 d}+\frac {\left (8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\tan (c+d x) \left (3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 d}+\frac {(3 a C+4 b B) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^3}{4 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 4133
Rule 4141
Rubi steps \begin{align*} \text {integral}& = \frac {C (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 A+(4 A b+4 a B+3 b C) \sec (c+d x)+(4 b B+3 a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (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^2 A+\left (24 a A b+12 a^2 B+8 b^2 B+15 a b C\right ) \sec (c+d x)+\left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{24} \int \left (24 a^3 A+3 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \sec (c+d x)+4 \left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = a^3 A x+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{6} \left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \int \sec (c+d x) \, dx \\ & = a^3 A x+\frac {\left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d} \\ & = a^3 A x+\frac {\left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \\ \end{align*}
Time = 2.96 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.81 \[ \int (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {24 a^3 A d x+3 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))+3 \left (8 \left (3 a^2 b B+b^3 B+a^3 C+3 a b^2 (A+C)\right )+b \left (4 A b^2+3 \left (4 a b B+4 a^2 C+b^2 C\right )\right ) \sec (c+d x)+2 b^3 C \sec ^3(c+d x)\right ) \tan (c+d x)+8 b^2 (b B+3 a C) \tan ^3(c+d x)}{24 d} \]
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Time = 1.37 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.02
method | result | size |
parts | \(a^{3} A x +\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \tan \left (d x +c \right )}{d}+\frac {C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(212\) |
derivativedivides | \(\frac {a^{3} A \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \tan \left (d x +c \right )+3 A \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \tan \left (d x +c \right ) a^{2} b +3 a^{2} b C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \tan \left (d x +c \right ) a \,b^{2}+3 B a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 C a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+A \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(303\) |
default | \(\frac {a^{3} A \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \tan \left (d x +c \right )+3 A \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \tan \left (d x +c \right ) a^{2} b +3 a^{2} b C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \tan \left (d x +c \right ) a \,b^{2}+3 B a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 C a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+A \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(303\) |
parallelrisch | \(\frac {-72 \left (\left (\frac {A}{6}+\frac {C}{8}\right ) b^{3}+\frac {B a \,b^{2}}{2}+a^{2} \left (A +\frac {C}{2}\right ) b +\frac {B \,a^{3}}{3}\right ) \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+72 \left (\left (\frac {A}{6}+\frac {C}{8}\right ) b^{3}+\frac {B a \,b^{2}}{2}+a^{2} \left (A +\frac {C}{2}\right ) b +\frac {B \,a^{3}}{3}\right ) \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+96 a^{3} A x d \cos \left (2 d x +2 c \right )+24 a^{3} A x d \cos \left (4 d x +4 c \right )+\left (64 B \,b^{3}+144 \left (A +\frac {4 C}{3}\right ) a \,b^{2}+144 B \,a^{2} b +48 a^{3} C \right ) \sin \left (2 d x +2 c \right )+\left (16 B \,b^{3}+72 a \left (A +\frac {2 C}{3}\right ) b^{2}+72 B \,a^{2} b +24 a^{3} C \right ) \sin \left (4 d x +4 c \right )+24 b \left (\left (A +\frac {3 C}{4}\right ) b^{2}+3 B a b +3 C \,a^{2}\right ) \sin \left (3 d x +3 c \right )+24 \left (\left (A +\frac {11 C}{4}\right ) b^{2}+3 B a b +3 C \,a^{2}\right ) b \sin \left (d x +c \right )+72 a^{3} A x d}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(360\) |
norman | \(\frac {a^{3} A x +a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {\left (24 a A \,b^{2}-4 A \,b^{3}+24 B \,a^{2} b -12 B a \,b^{2}+8 B \,b^{3}+8 a^{3} C -12 a^{2} b C +24 C a \,b^{2}-5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (24 a A \,b^{2}+4 A \,b^{3}+24 B \,a^{2} b +12 B a \,b^{2}+8 B \,b^{3}+8 a^{3} C +12 a^{2} b C +24 C a \,b^{2}+5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (216 a A \,b^{2}-12 A \,b^{3}+216 B \,a^{2} b -36 B a \,b^{2}+40 B \,b^{3}+72 a^{3} C -36 a^{2} b C +120 C a \,b^{2}+9 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {\left (216 a A \,b^{2}+12 A \,b^{3}+216 B \,a^{2} b +36 B a \,b^{2}+40 B \,b^{3}+72 a^{3} C +36 a^{2} b C +120 C a \,b^{2}-9 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}-4 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {\left (24 A \,a^{2} b +4 A \,b^{3}+8 B \,a^{3}+12 B a \,b^{2}+12 a^{2} b C +3 C \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (24 A \,a^{2} b +4 A \,b^{3}+8 B \,a^{3}+12 B a \,b^{2}+12 a^{2} b C +3 C \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(512\) |
risch | \(a^{3} A x +\frac {i \left (24 a^{3} C +48 C a \,b^{2}+16 B \,b^{3}+72 B \,a^{2} b +72 a A \,b^{2}+33 C \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-12 A \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+48 B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+64 B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+12 A \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+9 C \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+72 C \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-12 A \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-9 C \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+24 C \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+72 C \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+12 A \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-33 C \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+36 C \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+216 A a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+192 C a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+36 C \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-36 C \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+216 A a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+144 C a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+216 B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+216 B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+36 B a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-36 C \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+72 A a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-36 B a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+72 B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-36 B a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+36 B a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,a^{2} b}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{3}}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,a^{3}}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{2}}{2 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} b C}{2 d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,a^{2} b}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{3}}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,a^{3}}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{2}}{2 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} b C}{2 d}+\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}\) | \(781\) |
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Time = 0.30 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.24 \[ \int (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {48 \, A a^{3} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (8 \, B a^{3} + 12 \, {\left (2 \, A + C\right )} a^{2} b + 12 \, B a b^{2} + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, B a^{3} + 12 \, {\left (2 \, A + C\right )} a^{2} b + 12 \, B a b^{2} + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, C b^{3} + 8 \, {\left (3 \, C a^{3} + 9 \, B a^{2} b + 3 \, {\left (3 \, A + 2 \, C\right )} a b^{2} + 2 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (12 \, C a^{2} b + 12 \, B a b^{2} + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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\[ \int (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \]
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Time = 0.24 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.73 \[ \int (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {48 \, {\left (d x + c\right )} A a^{3} + 48 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b^{2} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b^{3} - 3 \, C b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, C a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, B a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, B a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 144 \, A a^{2} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 48 \, C a^{3} \tan \left (d x + c\right ) + 144 \, B a^{2} b \tan \left (d x + c\right ) + 144 \, A a b^{2} \tan \left (d x + c\right )}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (197) = 394\).
Time = 0.38 (sec) , antiderivative size = 759, normalized size of antiderivative = 3.67 \[ \int (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Time = 19.99 (sec) , antiderivative size = 3210, normalized size of antiderivative = 15.51 \[ \int (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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