Integrand size = 28, antiderivative size = 259 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {5 a^3 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {15 a b^2 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {3 a^2 b \sec ^7(c+d x)}{7 d}-\frac {b^3 \sec ^7(c+d x)}{7 d}+\frac {b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^3 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a b^2 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a b^2 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a b^2 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {3 a b^2 \sec ^7(c+d x) \tan (c+d x)}{8 d} \]
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Time = 0.32 (sec) , antiderivative size = 259, 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} \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {5 a^3 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac {5 a^3 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {5 a^3 \tan (c+d x) \sec (c+d x)}{16 d}+\frac {3 a^2 b \sec ^7(c+d x)}{7 d}-\frac {15 a b^2 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {3 a b^2 \tan (c+d x) \sec ^7(c+d x)}{8 d}-\frac {a b^2 \tan (c+d x) \sec ^5(c+d x)}{16 d}-\frac {5 a b^2 \tan (c+d x) \sec ^3(c+d x)}{64 d}-\frac {15 a b^2 \tan (c+d x) \sec (c+d x)}{128 d}+\frac {b^3 \sec ^9(c+d x)}{9 d}-\frac {b^3 \sec ^7(c+d x)}{7 d} \]
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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^3 \sec ^7(c+d x)+3 a^2 b \sec ^7(c+d x) \tan (c+d x)+3 a b^2 \sec ^7(c+d x) \tan ^2(c+d x)+b^3 \sec ^7(c+d x) \tan ^3(c+d x)\right ) \, dx \\ & = a^3 \int \sec ^7(c+d x) \, dx+\left (3 a^2 b\right ) \int \sec ^7(c+d x) \tan (c+d x) \, dx+\left (3 a b^2\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+b^3 \int \sec ^7(c+d x) \tan ^3(c+d x) \, dx \\ & = \frac {a^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {3 a b^2 \sec ^7(c+d x) \tan (c+d x)}{8 d}+\frac {1}{6} \left (5 a^3\right ) \int \sec ^5(c+d x) \, dx-\frac {1}{8} \left (3 a b^2\right ) \int \sec ^7(c+d x) \, dx+\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int x^6 \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^3 \text {Subst}\left (\int x^6 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {3 a^2 b \sec ^7(c+d x)}{7 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a b^2 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {3 a b^2 \sec ^7(c+d x) \tan (c+d x)}{8 d}+\frac {1}{8} \left (5 a^3\right ) \int \sec ^3(c+d x) \, dx-\frac {1}{16} \left (5 a b^2\right ) \int \sec ^5(c+d x) \, dx+\frac {b^3 \text {Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {3 a^2 b \sec ^7(c+d x)}{7 d}-\frac {b^3 \sec ^7(c+d x)}{7 d}+\frac {b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^3 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a b^2 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a b^2 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {3 a b^2 \sec ^7(c+d x) \tan (c+d x)}{8 d}+\frac {1}{16} \left (5 a^3\right ) \int \sec (c+d x) \, dx-\frac {1}{64} \left (15 a b^2\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {5 a^3 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {3 a^2 b \sec ^7(c+d x)}{7 d}-\frac {b^3 \sec ^7(c+d x)}{7 d}+\frac {b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^3 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a b^2 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a b^2 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a b^2 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {3 a b^2 \sec ^7(c+d x) \tan (c+d x)}{8 d}-\frac {1}{128} \left (15 a b^2\right ) \int \sec (c+d x) \, dx \\ & = \frac {5 a^3 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {15 a b^2 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {3 a^2 b \sec ^7(c+d x)}{7 d}-\frac {b^3 \sec ^7(c+d x)}{7 d}+\frac {b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^3 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a b^2 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a b^2 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a b^2 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {3 a b^2 \sec ^7(c+d x) \tan (c+d x)}{8 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(810\) vs. \(2(259)=518\).
Time = 5.46 (sec) , antiderivative size = 810, normalized size of antiderivative = 3.13 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {\sec ^9(c+d x) \left (442368 a^2 b+81920 b^3+147456 \left (3 a^2 b-b^3\right ) \cos (2 (c+d x))-211680 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+79380 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-90720 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+34020 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-22680 a^3 \cos (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8505 a b^2 \cos (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-2520 a^3 \cos (9 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+945 a b^2 \cos (9 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-39690 a \left (8 a^2-3 b^2\right ) \cos (c+d x) \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 )+211680 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-79380 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+90720 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-34020 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+22680 a^3 \cos (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-8505 a b^2 \cos (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+2520 a^3 \cos (9 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-945 a b^2 \cos (9 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+223776 a^3 \sin (2 (c+d x))+303156 a b^2 \sin (2 (c+d x))+167328 a^3 \sin (4 (c+d x))-62748 a b^2 \sin (4 (c+d x))+43680 a^3 \sin (6 (c+d x))-16380 a b^2 \sin (6 (c+d x))+5040 a^3 \sin (8 (c+d x))-1890 a b^2 \sin (8 (c+d x))\right )}{2064384 d} \]
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Time = 2.14 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.83
method | result | size |
parts | \(\frac {a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {b^{3} \left (\frac {\sec \left (d x +c \right )^{9}}{9}-\frac {\sec \left (d x +c \right )^{7}}{7}\right )}{d}+\frac {3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}+\frac {3 a^{2} b \sec \left (d x +c \right )^{7}}{7 d}\) | \(214\) |
derivativedivides | \(\frac {a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {3 a^{2} b}{7 \cos \left (d x +c \right )^{7}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )}{d}\) | \(294\) |
default | \(\frac {a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {3 a^{2} b}{7 \cos \left (d x +c \right )^{7}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )}{d}\) | \(294\) |
parallelrisch | \(\frac {-22680 \left (\frac {\cos \left (9 d x +9 c \right )}{9}+\cos \left (7 d x +7 c \right )+4 \cos \left (5 d x +5 c \right )+\frac {28 \cos \left (3 d x +3 c \right )}{3}+14 \cos \left (d x +c \right )\right ) \left (a^{2}-\frac {3 b^{2}}{8}\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+22680 \left (\frac {\cos \left (9 d x +9 c \right )}{9}+\cos \left (7 d x +7 c \right )+4 \cos \left (5 d x +5 c \right )+\frac {28 \cos \left (3 d x +3 c \right )}{3}+14 \cos \left (d x +c \right )\right ) \left (a^{2}-\frac {3 b^{2}}{8}\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (290304 a^{2} b -21504 b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (124416 a^{2} b -9216 b^{3}\right ) \cos \left (5 d x +5 c \right )+\left (31104 a^{2} b -2304 b^{3}\right ) \cos \left (7 d x +7 c \right )+\left (3456 a^{2} b -256 b^{3}\right ) \cos \left (9 d x +9 c \right )+\left (442368 a^{2} b -147456 b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (223776 a^{3}+303156 a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (167328 a^{3}-62748 a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (43680 a^{3}-16380 a \,b^{2}\right ) \sin \left (6 d x +6 c \right )+\left (5040 a^{3}-1890 a \,b^{2}\right ) \sin \left (8 d x +8 c \right )+\left (435456 a^{2} b -32256 b^{3}\right ) \cos \left (d x +c \right )+442368 a^{2} b +81920 b^{3}}{8064 d \left (\cos \left (9 d x +9 c \right )+9 \cos \left (7 d x +7 c \right )+36 \cos \left (5 d x +5 c \right )+84 \cos \left (3 d x +3 c \right )+126 \cos \left (d x +c \right )\right )}\) | \(438\) |
risch | \(-\frac {{\mathrm e}^{i \left (d x +c \right )} \left (2520 i a^{3} {\mathrm e}^{16 i \left (d x +c \right )}-2520 i a^{3}-111888 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+21840 i a^{3} {\mathrm e}^{14 i \left (d x +c \right )}-31374 i a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-151578 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+31374 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+83664 i a^{3} {\mathrm e}^{12 i \left (d x +c \right )}-221184 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}+73728 b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-442368 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-81920 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+945 i a \,b^{2}+151578 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-221184 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+73728 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+111888 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}-945 i a \,b^{2} {\mathrm e}^{16 i \left (d x +c \right )}+8190 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-83664 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-8190 i a \,b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-21840 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{4032 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{9}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}+\frac {15 a \,b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d}+\frac {5 a^{3} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{16 d}-\frac {15 a \,b^{2} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{128 d}\) | \(475\) |
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Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.74 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {315 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{9} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{9} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 1792 \, b^{3} + 2304 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 42 \, {\left (15 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} + 10 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 144 \, a b^{2} \cos \left (d x + c\right ) + 8 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{16128 \, d \cos \left (d x + c\right )^{9}} \]
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Timed out. \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.96 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {63 \, a b^{2} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 168 \, a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {6912 \, a^{2} b}{\cos \left (d x + c\right )^{7}} - \frac {256 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} b^{3}}{\cos \left (d x + c\right )^{9}}}{16128 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (235) = 470\).
Time = 0.42 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.31 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {315 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 315 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (5544 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 945 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} - 24192 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} - 15792 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 24066 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 48384 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 16128 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 29232 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 31374 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 145152 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 26880 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 33264 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 54810 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 241920 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 80640 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 193536 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 48384 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 33264 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 54810 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 145152 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48384 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 29232 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 31374 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 76032 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6912 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 15792 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24066 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6912 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2304 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5544 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 945 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3456 \, a^{2} b + 256 \, b^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{9}}}{8064 \, d} \]
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Time = 26.78 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.11 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {15\,a\,b^2}{64}-\frac {5\,a^3}{8}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {11\,a^3}{8}+\frac {15\,a\,b^2}{64}\right )+\frac {6\,a^2\,b}{7}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\left (\frac {11\,a^3}{8}+\frac {15\,a\,b^2}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {191\,a\,b^2}{32}-\frac {47\,a^3}{12}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {191\,a\,b^2}{32}-\frac {47\,a^3}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {29\,a^3}{4}+\frac {249\,a\,b^2}{32}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {29\,a^3}{4}+\frac {249\,a\,b^2}{32}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {435\,a\,b^2}{32}-\frac {33\,a^3}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {435\,a\,b^2}{32}-\frac {33\,a^3}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (12\,a^2\,b-4\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {12\,a^2\,b}{7}-\frac {4\,b^3}{7}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (36\,a^2\,b-12\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (48\,a^2\,b+12\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (36\,a^2\,b+\frac {20\,b^3}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (60\,a^2\,b-20\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {132\,a^2\,b}{7}+\frac {12\,b^3}{7}\right )-\frac {4\,b^3}{63}+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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