Integrand size = 33, antiderivative size = 246 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=4 a b^3 C x+\frac {\left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}-\frac {b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
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Time = 0.99 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3127, 3126, 3110, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{12 d}+\frac {\left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{8 d}+\frac {\left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+4 a b^3 C x \]
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Rule 2814
Rule 3102
Rule 3110
Rule 3126
Rule 3127
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \cos (c+d x))^3 \left (4 A b+a (3 A+4 C) \cos (c+d x)-b (A-4 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \cos (c+d x))^2 \left (3 \left (4 A b^2+a^2 (3 A+4 C)\right )+2 a b (7 A+12 C) \cos (c+d x)-b^2 (7 A-12 C) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{24} \int (a+b \cos (c+d x)) \left (2 \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 b C)\right )+a \left (3 a^2 (3 A+4 C)+2 b^2 (13 A+36 C)\right ) \cos (c+d x)-b \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-3 \left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right )-96 a b^3 C \cos (c+d x)+b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-3 \left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right )-96 a b^3 C \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = 4 a b^3 C x-\frac {b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{8} \left (-8 A b^4-24 a^2 b^2 (A+2 C)-a^4 (3 A+4 C)\right ) \int \sec (c+d x) \, dx \\ & = 4 a b^3 C x+\frac {\left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}-\frac {b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(612\) vs. \(2(246)=492\).
Time = 9.18 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.49 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {4 a b^3 C (c+d x)}{d}+\frac {\left (-3 a^4 A-24 a^2 A b^2-8 A b^4-4 a^4 C-48 a^2 b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {\left (3 a^4 A+24 a^2 A b^2+8 A b^4+4 a^4 C+48 a^2 b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {a^4 A}{16 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {9 a^4 A+16 a^3 A b+72 a^2 A b^2+12 a^4 C}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {a^4 A}{16 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {-9 a^4 A-16 a^3 A b-72 a^2 A b^2-12 a^4 C}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 \left (2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )+3 a A b^3 \sin \left (\frac {1}{2} (c+d x)\right )+3 a^3 b C \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 \left (2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )+3 a A b^3 \sin \left (\frac {1}{2} (c+d x)\right )+3 a^3 b C \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {b^4 C \sin (c+d x)}{d} \]
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Time = 10.31 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.89
method | result | size |
parts | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {\left (A \,b^{4}+6 C \,a^{2} b^{2}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 A a \,b^{3}+4 C \,a^{3} b \right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+C \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\sin \left (d x +c \right ) C \,b^{4}}{d}-\frac {4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {4 C a \,b^{3} \left (d x +c \right )}{d}\) | \(219\) |
derivativedivides | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 C \,a^{3} b \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A a \,b^{3} \tan \left (d x +c \right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \sin \left (d x +c \right ) b^{4}}{d}\) | \(247\) |
default | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 C \,a^{3} b \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A a \,b^{3} \tan \left (d x +c \right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \sin \left (d x +c \right ) b^{4}}{d}\) | \(247\) |
parallelrisch | \(\frac {-36 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (\left (A +\frac {4 C}{3}\right ) a^{4}+8 a^{2} b^{2} \left (A +2 C \right )+\frac {8 A \,b^{4}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+36 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (\left (A +\frac {4 C}{3}\right ) a^{4}+8 a^{2} b^{2} \left (A +2 C \right )+\frac {8 A \,b^{4}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+384 C a \,b^{3} d x \cos \left (2 d x +2 c \right )+96 C a \,b^{3} d x \cos \left (4 d x +4 c \right )+\left (\left (18 A +24 C \right ) a^{4}+144 A \,a^{2} b^{2}+36 C \,b^{4}\right ) \sin \left (3 d x +3 c \right )+256 b \left (a^{2} \left (A +\frac {3 C}{4}\right )+\frac {3 A \,b^{2}}{4}\right ) a \sin \left (2 d x +2 c \right )+64 \left (\left (A +\frac {3 C}{2}\right ) a^{2}+\frac {3 A \,b^{2}}{2}\right ) b a \sin \left (4 d x +4 c \right )+12 C \sin \left (5 d x +5 c \right ) b^{4}+\left (\left (66 A +24 C \right ) a^{4}+144 A \,a^{2} b^{2}+24 C \,b^{4}\right ) \sin \left (d x +c \right )+288 C a \,b^{3} d x}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(350\) |
risch | \(4 a \,b^{3} C x -\frac {i C \,b^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i C \,b^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a \left (9 A \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+72 A a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+12 C \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-96 A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-96 C \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+33 A \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+72 A a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+12 C \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-192 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-288 A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-288 C \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-33 A \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-72 A a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-12 C \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-256 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-288 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-288 C \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-9 A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-72 A a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-12 C \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-64 A \,a^{2} b -96 A \,b^{3}-96 C \,a^{2} b \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{4}}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}-\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,a^{2} b^{2}}{d}+\frac {3 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{4}}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,a^{2} b^{2}}{d}\) | \(629\) |
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Time = 0.29 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.96 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {192 \, C a b^{3} d x \cos \left (d x + c\right )^{4} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, C b^{4} \cos \left (d x + c\right )^{4} + 32 \, A a^{3} b \cos \left (d x + c\right ) + 6 \, A a^{4} + 32 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{3} b + 3 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.24 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} b + 192 \, {\left (d x + c\right )} C a b^{3} - 3 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, C a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C b^{4} \sin \left (d x + c\right ) + 192 \, C a^{3} b \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (234) = 468\).
Time = 0.38 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.40 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {96 \, {\left (d x + c\right )} C a b^{3} + \frac {48 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 3 \, {\left (3 \, A a^{4} + 4 \, C a^{4} + 24 \, A a^{2} b^{2} + 48 \, C a^{2} b^{2} + 8 \, A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, A a^{4} + 4 \, C a^{4} + 24 \, A a^{2} b^{2} + 48 \, C a^{2} b^{2} + 8 \, A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a b^{3} \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 = 4.23 (sec) , antiderivative size = 1988, normalized size of antiderivative = 8.08 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Too large to display} \]
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