Integrand size = 28, antiderivative size = 408 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {5 a^4 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {15 a^2 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{256 d}+\frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d} \]
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Time = 0.46 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3169, 3853, 3855, 2686, 30, 2691, 14} \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {5 a^4 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac {5 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {5 a^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {15 a^2 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {3 a^2 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac {a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{8 d}-\frac {5 a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{32 d}-\frac {15 a^2 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{256 d}+\frac {b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{10 d}-\frac {3 b^4 \tan (c+d x) \sec ^7(c+d x)}{80 d}+\frac {b^4 \tan (c+d x) \sec ^5(c+d x)}{160 d}+\frac {b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac {3 b^4 \tan (c+d x) \sec (c+d x)}{256 d} \]
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Rule 14
Rule 30
Rule 2686
Rule 2691
Rule 3169
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^4 \sec ^7(c+d x)+4 a^3 b \sec ^7(c+d x) \tan (c+d x)+6 a^2 b^2 \sec ^7(c+d x) \tan ^2(c+d x)+4 a b^3 \sec ^7(c+d x) \tan ^3(c+d x)+b^4 \sec ^7(c+d x) \tan ^4(c+d x)\right ) \, dx \\ & = a^4 \int \sec ^7(c+d x) \, dx+\left (4 a^3 b\right ) \int \sec ^7(c+d x) \tan (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \sec ^7(c+d x) \tan ^3(c+d x) \, dx+b^4 \int \sec ^7(c+d x) \tan ^4(c+d x) \, dx \\ & = \frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac {1}{6} \left (5 a^4\right ) \int \sec ^5(c+d x) \, dx-\frac {1}{4} \left (3 a^2 b^2\right ) \int \sec ^7(c+d x) \, dx-\frac {1}{10} \left (3 b^4\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int x^6 \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (4 a b^3\right ) \text {Subst}\left (\int x^6 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {4 a^3 b \sec ^7(c+d x)}{7 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac {1}{8} \left (5 a^4\right ) \int \sec ^3(c+d x) \, dx-\frac {1}{8} \left (5 a^2 b^2\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{80} \left (3 b^4\right ) \int \sec ^7(c+d x) \, dx+\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac {1}{16} \left (5 a^4\right ) \int \sec (c+d x) \, dx-\frac {1}{32} \left (15 a^2 b^2\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{32} b^4 \int \sec ^5(c+d x) \, dx \\ & = \frac {5 a^4 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}-\frac {1}{64} \left (15 a^2 b^2\right ) \int \sec (c+d x) \, dx+\frac {1}{128} \left (3 b^4\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {5 a^4 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {15 a^2 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d}+\frac {1}{256} \left (3 b^4\right ) \int \sec (c+d x) \, dx \\ & = \frac {5 a^4 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {15 a^2 b^2 \text {arctanh}(\sin (c+d x))}{64 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{256 d}+\frac {4 a^3 b \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {4 a b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^4 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a^2 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac {5 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{32 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{160 d}+\frac {3 a^2 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac {3 b^4 \sec ^7(c+d x) \tan (c+d x)}{80 d}+\frac {b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{10 d} \\ \end{align*}
Time = 3.16 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.59 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {-80640 \left (80 a^4-60 a^2 b^2+3 b^4\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+3 \sec ^{10}(c+d x) \left (983040 a b \left (a^2-b^2\right ) \cos (3 (c+d x))+420 \left (1552 a^4+1908 a^2 b^2-505 b^4\right ) \sin (3 (c+d x))+7 \left (80 a^4-60 a^2 b^2+3 b^4\right ) (628 \sin (5 (c+d x))+145 \sin (7 (c+d x))+15 \sin (9 (c+d x)))\right )+10 \sec ^9(c+d x) \left (32768 a b \left (27 a^2+b^2\right )+189 \left (592 a^4+1604 a^2 b^2+739 b^4\right ) \tan (c+d x)\right )}{20643840 d} \]
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Time = 2.51 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.86
method | result | size |
parts | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{32 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{256 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{256}-\frac {3 \sin \left (d x +c \right )}{256}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{256}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\sec \left (d x +c \right )^{9}}{9}-\frac {\sec \left (d x +c \right )^{7}}{7}\right )}{d}+\frac {4 a^{3} b \sec \left (d x +c \right )^{7}}{7 d}+\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}\) | \(350\) |
derivativedivides | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {4 a^{3} b}{7 \cos \left (d x +c \right )^{7}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{32 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{256 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{256}-\frac {3 \sin \left (d x +c \right )}{256}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{256}\right )}{d}\) | \(427\) |
default | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {4 a^{3} b}{7 \cos \left (d x +c \right )^{7}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{32 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{256 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{256}-\frac {3 \sin \left (d x +c \right )}{256}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{256}\right )}{d}\) | \(427\) |
parallelrisch | \(\frac {-1134000 \left (\frac {\cos \left (10 d x +10 c \right )}{45}+\frac {2 \cos \left (8 d x +8 c \right )}{9}+\cos \left (6 d x +6 c \right )+\frac {8 \cos \left (4 d x +4 c \right )}{3}+\frac {14 \cos \left (2 d x +2 c \right )}{3}+\frac {14}{5}\right ) \left (a^{4}-\frac {3}{4} a^{2} b^{2}+\frac {3}{80} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+1134000 \left (\frac {\cos \left (10 d x +10 c \right )}{45}+\frac {2 \cos \left (8 d x +8 c \right )}{9}+\cos \left (6 d x +6 c \right )+\frac {8 \cos \left (4 d x +4 c \right )}{3}+\frac {14 \cos \left (2 d x +2 c \right )}{3}+\frac {14}{5}\right ) \left (a^{4}-\frac {3}{4} a^{2} b^{2}+\frac {3}{80} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (46080 a^{3} b -10240 a \,b^{3}\right ) \cos \left (10 d x +10 c \right )+\left (9676800 a^{3} b -2150400 a \,b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (5529600 a^{3} b -1228800 a \,b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (2073600 a^{3} b -460800 a \,b^{3}\right ) \cos \left (6 d x +6 c \right )+\left (460800 a^{3} b -102400 a \,b^{3}\right ) \cos \left (8 d x +8 c \right )+\left (3911040 a^{4}+4808160 a^{2} b^{2}-1272600 b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (2110080 a^{4}-1582560 a^{2} b^{2}+79128 b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (487200 a^{4}-365400 a^{2} b^{2}+18270 b^{4}\right ) \sin \left (7 d x +7 c \right )+\left (50400 a^{4}-37800 a^{2} b^{2}+1890 b^{4}\right ) \sin \left (9 d x +9 c \right )+\left (5898240 a^{3} b -5898240 a \,b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (2237760 a^{4}+6063120 a^{2} b^{2}+2793420 b^{4}\right ) \sin \left (d x +c \right )+17694720 b \left (\left (a^{2}+\frac {b^{2}}{27}\right ) \cos \left (d x +c \right )+\frac {21 a^{2}}{64}-\frac {7 b^{2}}{96}\right ) a}{80640 d \left (\cos \left (10 d x +10 c \right )+10 \cos \left (8 d x +8 c \right )+45 \cos \left (6 d x +6 c \right )+120 \cos \left (4 d x +4 c \right )+210 \cos \left (2 d x +2 c \right )+126\right )}\) | \(547\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (-25200 a^{4}-945 b^{4}+18900 a^{2} b^{2}+1118880 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}-9135 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+636300 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+1396710 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+2404080 a^{2} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-791280 a^{2} b^{2} {\mathrm e}^{14 i \left (d x +c \right )}+327680 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+8847360 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}+3031560 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+8847360 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+2949120 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+327680 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-2949120 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-2949120 i a \,b^{3} {\mathrm e}^{12 i \left (d x +c \right )}+2949120 i a^{3} b \,{\mathrm e}^{12 i \left (d x +c \right )}+243600 a^{4} {\mathrm e}^{16 i \left (d x +c \right )}+9135 b^{4} {\mathrm e}^{16 i \left (d x +c \right )}+25200 a^{4} {\mathrm e}^{18 i \left (d x +c \right )}+945 b^{4} {\mathrm e}^{18 i \left (d x +c \right )}+1055040 a^{4} {\mathrm e}^{14 i \left (d x +c \right )}-1118880 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}-1396710 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-1955520 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-1055040 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-39564 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-243600 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-3031560 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-2404080 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+791280 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+182700 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+1955520 a^{4} {\mathrm e}^{12 i \left (d x +c \right )}+39564 b^{4} {\mathrm e}^{14 i \left (d x +c \right )}-636300 b^{4} {\mathrm e}^{12 i \left (d x +c \right )}-182700 a^{2} b^{2} {\mathrm e}^{16 i \left (d x +c \right )}-18900 a^{2} b^{2} {\mathrm e}^{18 i \left (d x +c \right )}\right )}{40320 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{10}}+\frac {5 \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{4}}{16 d}-\frac {15 b^{2} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{2}}{64 d}+\frac {3 b^{4} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{256 d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{4}}{16 d}+\frac {15 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{64 d}-\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d}\) | \(786\) |
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Time = 0.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.62 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {315 \, {\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{10} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{10} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 71680 \, a b^{3} \cos \left (d x + c\right ) + 92160 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 42 \, {\left (15 \, {\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{6} + 8 \, {\left (80 \, a^{4} - 60 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 384 \, b^{4} + 48 \, {\left (60 \, a^{2} b^{2} - 11 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{161280 \, d \cos \left (d x + c\right )^{10}} \]
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Timed out. \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.94 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {63 \, b^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{9} - 70 \, \sin \left (d x + c\right )^{7} + 128 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 1260 \, a^{2} b^{2} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 1680 \, a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {92160 \, a^{3} b}{\cos \left (d x + c\right )^{7}} + \frac {10240 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} a b^{3}}{\cos \left (d x + c\right )^{9}}}{161280 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 880 vs. \(2 (372) = 744\).
Time = 0.50 (sec) , antiderivative size = 880, normalized size of antiderivative = 2.16 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Too large to display} \]
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Time = 27.36 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.72 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Too large to display} \]
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