Integrand size = 48, antiderivative size = 214 \[ \int (a+b \sec (c+d x))^3 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=a^4 (b B-a C) x+\frac {b \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b^2 \left (34 a^2 b B+4 b^3 B-15 a^3 C+12 a b^2 C\right ) \tan (c+d x)}{6 d}+\frac {b^3 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \]
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Time = 0.55 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4126, 4003, 4141, 4133, 3855, 3852, 8} \[ \int (a+b \sec (c+d x))^3 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=a^4 x (b B-a C)+\frac {b^3 \left (-6 a^2 C+32 a b B+9 b^2 C\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac {b^2 \left (-15 a^3 C+34 a^2 b B+12 a b^2 C+4 b^3 B\right ) \tan (c+d x)}{6 d}+\frac {b \left (-24 a^4 C+32 a^3 b B+8 a^2 b^2 C+16 a b^3 B+3 b^4 C\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b^2 (3 a C+4 b B) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {b^2 C \tan (c+d x) (a+b \sec (c+d x))^3}{4 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 4003
Rule 4126
Rule 4133
Rule 4141
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+b \sec (c+d x))^4 \left (b^2 (b B-a C)+b^3 C \sec (c+d x)\right ) \, dx}{b^2} \\ & = \frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\int (a+b \sec (c+d x))^2 \left (4 a^2 b^2 (b B-a C)+b^3 \left (8 a b B-4 a^2 C+3 b^2 C\right ) \sec (c+d x)+b^4 (4 b B+3 a C) \sec ^2(c+d x)\right ) \, dx}{4 b^2} \\ & = \frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\int (a+b \sec (c+d x)) \left (12 a^3 b^2 (b B-a C)+b^3 \left (36 a^2 b B+8 b^3 B-24 a^3 C+15 a b^2 C\right ) \sec (c+d x)+b^4 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx}{12 b^2} \\ & = \frac {b^3 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\int \left (24 a^4 b^2 (b B-a C)+3 b^3 \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right ) \sec (c+d x)+4 b^4 \left (34 a^2 b B+4 b^3 B-15 a^3 C+12 a b^2 C\right ) \sec ^2(c+d x)\right ) \, dx}{24 b^2} \\ & = a^4 (b B-a C) x+\frac {b^3 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{6} \left (b^2 \left (34 a^2 b B+4 b^3 B-15 a^3 C+12 a b^2 C\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (b \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right )\right ) \int \sec (c+d x) \, dx \\ & = a^4 (b B-a C) x+\frac {b \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b^3 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {\left (b^2 \left (34 a^2 b B+4 b^3 B-15 a^3 C+12 a b^2 C\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d} \\ & = a^4 (b B-a C) x+\frac {b \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b^2 \left (34 a^2 b B+4 b^3 B-15 a^3 C+12 a b^2 C\right ) \tan (c+d x)}{6 d}+\frac {b^3 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \\ \end{align*}
Time = 1.71 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.79 \[ \int (a+b \sec (c+d x))^3 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=\frac {24 a^4 (b B-a C) d x+3 b \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right ) \text {arctanh}(\sin (c+d x))+3 b^2 \left (8 \left (6 a^2 b B+b^3 B-2 a^3 C+3 a b^2 C\right )+b \left (16 a b B+8 a^2 C+3 b^2 C\right ) \sec (c+d x)+2 b^3 C \sec ^3(c+d x)\right ) \tan (c+d x)+8 b^4 (b B+3 a C) \tan ^3(c+d x)}{24 d} \]
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Time = 1.36 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.02
method | result | size |
parts | \(a^{4} \left (B b -C a \right ) x -\frac {\left (B \,b^{5}+3 C a \,b^{4}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a \,b^{4} B +2 C \,a^{2} b^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (6 B \,a^{2} b^{3}-2 C \,a^{3} b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{3} b^{2} B -3 a^{4} b C \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \,b^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(219\) |
derivativedivides | \(\frac {B \,a^{4} b \left (d x +c \right )-a^{5} C \left (d x +c \right )+4 a^{3} b^{2} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-2 C \,a^{3} b^{2} \tan \left (d x +c \right )-3 a^{4} b C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 B \,a^{2} b^{3} \tan \left (d x +c \right )+2 C \,a^{2} b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a \,b^{4} B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 C a \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-B \,b^{5} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+C \,b^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(278\) |
default | \(\frac {B \,a^{4} b \left (d x +c \right )-a^{5} C \left (d x +c \right )+4 a^{3} b^{2} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-2 C \,a^{3} b^{2} \tan \left (d x +c \right )-3 a^{4} b C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 B \,a^{2} b^{3} \tan \left (d x +c \right )+2 C \,a^{2} b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a \,b^{4} B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 C a \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-B \,b^{5} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+C \,b^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(278\) |
parallelrisch | \(\frac {-384 \left (B \,a^{3} b +\frac {1}{2} B a \,b^{3}-\frac {3}{4} a^{4} C +\frac {1}{4} C \,a^{2} b^{2}+\frac {3}{32} C \,b^{4}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+384 \left (B \,a^{3} b +\frac {1}{2} B a \,b^{3}-\frac {3}{4} a^{4} C +\frac {1}{4} C \,a^{2} b^{2}+\frac {3}{32} C \,b^{4}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+96 a^{4} d x \left (B b -C a \right ) \cos \left (2 d x +2 c \right )+24 a^{4} d x \left (B b -C a \right ) \cos \left (4 d x +4 c \right )+\left (288 B \,a^{2} b^{3}+64 B \,b^{5}-96 C \,a^{3} b^{2}+192 C a \,b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (144 B \,a^{2} b^{3}+16 B \,b^{5}-48 C \,a^{3} b^{2}+48 C a \,b^{4}\right ) \sin \left (4 d x +4 c \right )+\left (96 a \,b^{4} B +48 C \,a^{2} b^{3}+18 C \,b^{5}\right ) \sin \left (3 d x +3 c \right )+\left (96 a \,b^{4} B +48 C \,a^{2} b^{3}+66 C \,b^{5}\right ) \sin \left (d x +c \right )+72 a^{4} d x \left (B b -C a \right )}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(385\) |
norman | \(\frac {\left (B \,a^{4} b -a^{5} C \right ) x +\left (-4 B \,a^{4} b +4 a^{5} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-4 B \,a^{4} b +4 a^{5} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (B \,a^{4} b -a^{5} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (6 B \,a^{4} b -6 a^{5} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {b^{2} \left (48 B \,a^{2} b -16 B a \,b^{2}+8 B \,b^{3}-16 a^{3} C -8 a^{2} b C +24 C a \,b^{2}-5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {b^{2} \left (48 B \,a^{2} b +16 B a \,b^{2}+8 B \,b^{3}-16 a^{3} C +8 a^{2} b C +24 C a \,b^{2}+5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {b^{2} \left (432 B \,a^{2} b -48 B a \,b^{2}+40 B \,b^{3}-144 a^{3} C -24 a^{2} b C +120 C a \,b^{2}+9 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {b^{2} \left (432 B \,a^{2} b +48 B a \,b^{2}+40 B \,b^{3}-144 a^{3} C +24 a^{2} b C +120 C a \,b^{2}-9 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {b \left (32 B \,a^{3} b +16 B a \,b^{3}-24 a^{4} C +8 C \,a^{2} b^{2}+3 C \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {b \left (32 B \,a^{3} b +16 B a \,b^{3}-24 a^{4} C +8 C \,a^{2} b^{2}+3 C \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(511\) |
risch | \(B \,a^{4} b x -C \,a^{5} x -\frac {i b^{2} \left (48 a^{3} C -48 C a \,b^{2}-16 B \,b^{3}-144 B \,a^{2} b -48 B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+24 C \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-144 C a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-432 B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-432 B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+24 C \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+48 B a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-144 B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+48 B a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-48 B a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-24 C \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-192 C a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-24 C \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-48 B a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-33 C \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-64 B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-9 C \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+144 C \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+9 C \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+48 C \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+144 C \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+33 C \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {4 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,a^{3}}{d}-\frac {2 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a}{d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{4} C}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,a^{2}}{d}-\frac {3 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}+\frac {4 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,a^{3}}{d}+\frac {2 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a}{d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{4} C}{d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,a^{2}}{d}+\frac {3 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}\) | \(643\) |
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Time = 0.29 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.25 \[ \int (a+b \sec (c+d x))^3 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=-\frac {48 \, {\left (C a^{5} - B a^{4} b\right )} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (24 \, C a^{4} b - 32 \, B a^{3} b^{2} - 8 \, C a^{2} b^{3} - 16 \, B a b^{4} - 3 \, C b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (24 \, C a^{4} b - 32 \, B a^{3} b^{2} - 8 \, C a^{2} b^{3} - 16 \, B a b^{4} - 3 \, C b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (6 \, C b^{5} - 16 \, {\left (3 \, C a^{3} b^{2} - 9 \, B a^{2} b^{3} - 3 \, C a b^{4} - B b^{5}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, C a^{2} b^{3} + 16 \, B a b^{4} + 3 \, C b^{5}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, C a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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\[ \int (a+b \sec (c+d x))^3 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=- \int C a^{5}\, dx - \int \left (- B a^{4} b\right )\, dx - \int \left (- B b^{5} \sec ^{4}{\left (c + d x \right )}\right )\, dx - \int \left (- C b^{5} \sec ^{5}{\left (c + d x \right )}\right )\, dx - \int \left (- 4 B a b^{4} \sec ^{3}{\left (c + d x \right )}\right )\, dx - \int \left (- 6 B a^{2} b^{3} \sec ^{2}{\left (c + d x \right )}\right )\, dx - \int \left (- 4 B a^{3} b^{2} \sec {\left (c + d x \right )}\right )\, dx - \int \left (- 3 C a b^{4} \sec ^{4}{\left (c + d x \right )}\right )\, dx - \int \left (- 2 C a^{2} b^{3} \sec ^{3}{\left (c + d x \right )}\right )\, dx - \int 2 C a^{3} b^{2} \sec ^{2}{\left (c + d x \right )}\, dx - \int 3 C a^{4} b \sec {\left (c + d x \right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.50 \[ \int (a+b \sec (c+d x))^3 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=-\frac {48 \, {\left (d x + c\right )} C a^{5} - 48 \, {\left (d x + c\right )} B a^{4} b - 48 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b^{4} - 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b^{5} + 3 \, C b^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{2} b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, B a b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, C a^{4} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 192 \, B a^{3} b^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 96 \, C a^{3} b^{2} \tan \left (d x + c\right ) - 288 \, B a^{2} b^{3} \tan \left (d x + c\right )}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (205) = 410\).
Time = 0.36 (sec) , antiderivative size = 658, normalized size of antiderivative = 3.07 \[ \int (a+b \sec (c+d x))^3 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=-\frac {24 \, {\left (C a^{5} - B a^{4} b\right )} {\left (d x + c\right )} + 3 \, {\left (24 \, C a^{4} b - 32 \, B a^{3} b^{2} - 8 \, C a^{2} b^{3} - 16 \, B a b^{4} - 3 \, C b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (24 \, C a^{4} b - 32 \, B a^{3} b^{2} - 8 \, C a^{2} b^{3} - 16 \, B a b^{4} - 3 \, C b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (48 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 144 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 48 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 144 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 432 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 48 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 120 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, C b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 144 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 432 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, C b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 144 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C b^{5} \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 = 20.51 (sec) , antiderivative size = 3157, normalized size of antiderivative = 14.75 \[ \int (a+b \sec (c+d x))^3 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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