Integrand size = 28, antiderivative size = 258 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a^4 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{4 d}+\frac {b^4 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^3 b \sec ^3(c+d x)}{3 d}-\frac {4 a b^3 \sec ^3(c+d x)}{3 d}+\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {a^4 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {3 a^2 b^2 \sec (c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {3 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{2 d}-\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^3(c+d x) \tan ^3(c+d x)}{6 d} \]
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Time = 0.34 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3169, 3853, 3855, 2686, 30, 2691, 14} \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a^4 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^4 \tan (c+d x) \sec (c+d x)}{2 d}+\frac {4 a^3 b \sec ^3(c+d x)}{3 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{4 d}+\frac {3 a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}-\frac {3 a^2 b^2 \tan (c+d x) \sec (c+d x)}{4 d}+\frac {4 a b^3 \sec ^5(c+d x)}{5 d}-\frac {4 a b^3 \sec ^3(c+d x)}{3 d}+\frac {b^4 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {b^4 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac {b^4 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac {b^4 \tan (c+d x) \sec (c+d x)}{16 d} \]
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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 ^3(c+d x)+4 a^3 b \sec ^3(c+d x) \tan (c+d x)+6 a^2 b^2 \sec ^3(c+d x) \tan ^2(c+d x)+4 a b^3 \sec ^3(c+d x) \tan ^3(c+d x)+b^4 \sec ^3(c+d x) \tan ^4(c+d x)\right ) \, dx \\ & = a^4 \int \sec ^3(c+d x) \, dx+\left (4 a^3 b\right ) \int \sec ^3(c+d x) \tan (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \sec ^3(c+d x) \tan ^3(c+d x) \, dx+b^4 \int \sec ^3(c+d x) \tan ^4(c+d x) \, dx \\ & = \frac {a^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {3 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{2 d}+\frac {b^4 \sec ^3(c+d x) \tan ^3(c+d x)}{6 d}+\frac {1}{2} a^4 \int \sec (c+d x) \, dx-\frac {1}{2} \left (3 a^2 b^2\right ) \int \sec ^3(c+d x) \, dx-\frac {1}{2} b^4 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx+\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (4 a b^3\right ) \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {a^4 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {4 a^3 b \sec ^3(c+d x)}{3 d}+\frac {a^4 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {3 a^2 b^2 \sec (c+d x) \tan (c+d x)}{4 d}+\frac {3 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{2 d}-\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^3(c+d x) \tan ^3(c+d x)}{6 d}-\frac {1}{4} \left (3 a^2 b^2\right ) \int \sec (c+d x) \, dx+\frac {1}{8} b^4 \int \sec ^3(c+d x) \, dx+\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {a^4 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{4 d}+\frac {4 a^3 b \sec ^3(c+d x)}{3 d}-\frac {4 a b^3 \sec ^3(c+d x)}{3 d}+\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {a^4 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {3 a^2 b^2 \sec (c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {3 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{2 d}-\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^3(c+d x) \tan ^3(c+d x)}{6 d}+\frac {1}{16} b^4 \int \sec (c+d x) \, dx \\ & = \frac {a^4 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{4 d}+\frac {b^4 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^3 b \sec ^3(c+d x)}{3 d}-\frac {4 a b^3 \sec ^3(c+d x)}{3 d}+\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {a^4 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {3 a^2 b^2 \sec (c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {3 a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{2 d}-\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b^4 \sec ^3(c+d x) \tan ^3(c+d x)}{6 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1342\) vs. \(2(258)=516\).
Time = 7.85 (sec) , antiderivative size = 1342, normalized size of antiderivative = 5.20 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a b \left (20 a^2-11 b^2\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{30 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (-8 a^4+12 a^2 b^2-b^4\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{16 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (8 a^4-12 a^2 b^2+b^4\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{16 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {b^4 \cos ^4(c+d x) (a+b \tan (c+d x))^4}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (30 a^2 b^2+8 a b^3-5 b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{80 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (120 a^4+160 a^3 b-180 a^2 b^2-88 a b^3+15 b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{480 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {a b^3 \cos ^4(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{5 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {b^4 \cos ^4(c+d x) (a+b \tan (c+d x))^4}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {a b^3 \cos ^4(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{5 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (-30 a^2 b^2+8 a b^3+5 b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{80 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (-120 a^4+160 a^3 b+180 a^2 b^2-88 a b^3-15 b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{480 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cos ^4(c+d x) \left (20 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )-11 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{30 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cos ^4(c+d x) \left (20 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )-11 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{30 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cos ^4(c+d x) \left (-20 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )+11 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{30 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cos ^4(c+d x) \left (-20 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )+11 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{30 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^4} \]
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Time = 1.86 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.99
method | result | size |
parts | \(\frac {a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{24 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{48 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{48}-\frac {\sin \left (d x +c \right )}{16}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {4 a^{3} b \sec \left (d x +c \right )^{3}}{3 d}+\frac {4 a \,b^{3} \left (\frac {\sec \left (d x +c \right )^{5}}{5}-\frac {\sec \left (d x +c \right )^{3}}{3}\right )}{d}+\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(255\) |
derivativedivides | \(\frac {a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {4 a^{3} b}{3 \cos \left (d x +c \right )^{3}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{15}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{24 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{48 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{48}-\frac {\sin \left (d x +c \right )}{16}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(296\) |
default | \(\frac {a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {4 a^{3} b}{3 \cos \left (d x +c \right )^{3}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{15}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{24 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{48 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{48}-\frac {\sin \left (d x +c \right )}{16}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(296\) |
parallelrisch | \(\frac {-120 \left (a^{4}-\frac {3}{2} a^{2} b^{2}+\frac {1}{8} b^{4}\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+120 \left (a^{4}-\frac {3}{2} a^{2} b^{2}+\frac {1}{8} b^{4}\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (4800 a^{3} b -1920 a \,b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (1920 a^{3} b -768 a \,b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (320 a^{3} b -128 a \,b^{3}\right ) \cos \left (6 d x +6 c \right )+\left (720 a^{4}+1800 a^{2} b^{2}-470 b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (240 a^{4}-360 a^{2} b^{2}+30 b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (2560 a^{3} b -2560 a \,b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (480 a^{4}+2160 a^{2} b^{2}+780 b^{4}\right ) \sin \left (d x +c \right )+7680 b a \left (\left (a^{2}-\frac {b^{2}}{5}\right ) \cos \left (d x +c \right )+\frac {5 a^{2}}{12}-\frac {b^{2}}{6}\right )}{240 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(377\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (-120 a^{4}-15 b^{4}+180 a^{2} b^{2}+120 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}+235 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+390 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+15 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-180 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+1280 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+3840 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+1280 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+3840 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}-1280 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-768 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-1280 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-768 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+360 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}-235 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+240 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-240 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-390 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-360 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+900 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+1080 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-1080 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-900 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{4}}{2 d}+\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{4 d}-\frac {b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}+\frac {\ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{4}}{2 d}-\frac {3 b^{2} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{2}}{4 d}+\frac {b^{4} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{16 d}\) | \(582\) |
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Time = 0.27 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.72 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {15 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 384 \, a b^{3} \cos \left (d x + c\right ) + 640 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (3 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 8 \, b^{4} + 2 \, {\left (36 \, a^{2} b^{2} - 7 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
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Timed out. \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.97 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {5 \, b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} + 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, a^{2} b^{2} {\left (\frac {2 \, {\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 120 \, a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {640 \, a^{3} b}{\cos \left (d x + c\right )^{3}} + \frac {128 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} a b^{3}}{\cos \left (d x + c\right )^{5}}}{480 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (234) = 468\).
Time = 0.45 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.08 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {15 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 180 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 15 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 960 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 360 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 900 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 85 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2880 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1920 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1080 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 570 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3200 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1280 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1080 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 570 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1920 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 360 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 900 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 85 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 960 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 768 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 180 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 320 \, a^{3} b - 128 \, a b^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]
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Time = 26.55 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.62 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^4-\frac {3\,a^2\,b^2}{2}+\frac {b^4}{8}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (2\,a^4-9\,a^2\,b^2+\frac {19\,b^4}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (2\,a^4-9\,a^2\,b^2+\frac {19\,b^4}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-3\,a^4+\frac {15\,a^2\,b^2}{2}+\frac {17\,b^4}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (-3\,a^4+\frac {15\,a^2\,b^2}{2}+\frac {17\,b^4}{24}\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^4+\frac {3\,a^2\,b^2}{2}-\frac {b^4}{8}\right )-\frac {16\,a\,b^3}{15}+\frac {8\,a^3\,b}{3}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (a^4+\frac {3\,a^2\,b^2}{2}-\frac {b^4}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {32\,a\,b^3}{5}-8\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (16\,a\,b^3-24\,a^3\,b\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {32\,a\,b^3}{3}-\frac {80\,a^3\,b}{3}\right )+16\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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