Integrand size = 28, antiderivative size = 330 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {3 a^4 \text {arctanh}(\sin (c+d x))}{8 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {4 a^3 b \sec ^5(c+d x)}{5 d}-\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {3 a^4 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {3 a^2 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{d}-\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d} \]
[Out]
Time = 0.38 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3169, 3853, 3855, 2686, 30, 2691, 14} \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {3 a^4 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 a^4 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {4 a^3 b \sec ^5(c+d x)}{5 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{d}-\frac {a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac {3 a^2 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {4 a b^3 \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac {b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac {b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac {3 b^4 \tan (c+d x) \sec (c+d x)}{128 d} \]
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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 ^5(c+d x)+4 a^3 b \sec ^5(c+d x) \tan (c+d x)+6 a^2 b^2 \sec ^5(c+d x) \tan ^2(c+d x)+4 a b^3 \sec ^5(c+d x) \tan ^3(c+d x)+b^4 \sec ^5(c+d x) \tan ^4(c+d x)\right ) \, dx \\ & = a^4 \int \sec ^5(c+d x) \, dx+\left (4 a^3 b\right ) \int \sec ^5(c+d x) \tan (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \sec ^5(c+d x) \tan ^3(c+d x) \, dx+b^4 \int \sec ^5(c+d x) \tan ^4(c+d x) \, dx \\ & = \frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{d}+\frac {b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}+\frac {1}{4} \left (3 a^4\right ) \int \sec ^3(c+d x) \, dx-\left (a^2 b^2\right ) \int \sec ^5(c+d x) \, dx-\frac {1}{8} \left (3 b^4\right ) \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx+\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int x^4 \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (4 a b^3\right ) \text {Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {4 a^3 b \sec ^5(c+d x)}{5 d}+\frac {3 a^4 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{d}-\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}+\frac {1}{8} \left (3 a^4\right ) \int \sec (c+d x) \, dx-\frac {1}{4} \left (3 a^2 b^2\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{16} b^4 \int \sec ^5(c+d x) \, dx+\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {3 a^4 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {4 a^3 b \sec ^5(c+d x)}{5 d}-\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {3 a^4 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {3 a^2 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{d}-\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}-\frac {1}{8} \left (3 a^2 b^2\right ) \int \sec (c+d x) \, dx+\frac {1}{64} \left (3 b^4\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {3 a^4 \text {arctanh}(\sin (c+d x))}{8 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {4 a^3 b \sec ^5(c+d x)}{5 d}-\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {3 a^4 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {3 a^2 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{d}-\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d}+\frac {1}{128} \left (3 b^4\right ) \int \sec (c+d x) \, dx \\ & = \frac {3 a^4 \text {arctanh}(\sin (c+d x))}{8 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {4 a^3 b \sec ^5(c+d x)}{5 d}-\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {3 a^4 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {3 a^2 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{d}-\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1732\) vs. \(2(330)=660\).
Time = 7.98 (sec) , antiderivative size = 1732, normalized size of antiderivative = 5.25 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a b \left (42 a^2-17 b^2\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{140 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {3 \left (16 a^4-16 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}{128 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {3 \left (16 a^4-16 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}{128 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}{128 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^8 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (56 a^2 b^2+16 a b^3-7 b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{448 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 (560 a^4+896 a^3 b-256 a b^3-35 b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{8960 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 (1680 a^4+1344 a^3 b-1680 a^2 b^2-544 a b^3+105 b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{8960 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}{14 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^7 (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}{128 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8 (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}{14 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^7 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (-56 a^2 b^2+16 a b^3+7 b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{448 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 (-560 a^4+896 a^3 b-256 a b^3+35 b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{8960 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 (-1680 a^4+1344 a^3 b+1680 a^2 b^2-544 a b^3-105 b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{8960 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 (42 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )-17 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{140 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 (42 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )-17 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{140 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 (7 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )-2 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{35 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 {\cos ^4(c+d x) \left (-7 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )+2 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{35 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 {\cos ^4(c+d x) \left (-42 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )+17 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{140 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 (-42 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )+17 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{140 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 = 1.95 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.92
method | result | size |
parts | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\sec \left (d x +c \right )^{7}}{7}-\frac {\sec \left (d x +c \right )^{5}}{5}\right )}{d}+\frac {4 a^{3} b \sec \left (d x +c \right )^{5}}{5 d}+\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(304\) |
derivativedivides | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {4 a^{3} b}{5 \cos \left (d x +c \right )^{5}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{35}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}\) | \(363\) |
default | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {4 a^{3} b}{5 \cos \left (d x +c \right )^{5}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{35}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}\) | \(363\) |
parallelrisch | \(\frac {-13440 \left (a^{4}-a^{2} b^{2}+\frac {1}{16} b^{4}\right ) \left (\frac {35}{8}+\frac {\cos \left (8 d x +8 c \right )}{8}+\cos \left (6 d x +6 c \right )+\frac {7 \cos \left (4 d x +4 c \right )}{2}+7 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+13440 \left (a^{4}-a^{2} b^{2}+\frac {1}{16} b^{4}\right ) \left (\frac {35}{8}+\frac {\cos \left (8 d x +8 c \right )}{8}+\cos \left (6 d x +6 c \right )+\frac {7 \cos \left (4 d x +4 c \right )}{2}+7 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (200704 a^{3} b -57344 a \,b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (100352 a^{3} b -28672 a \,b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (28672 a^{3} b -8192 a \,b^{3}\right ) \cos \left (6 d x +6 c \right )+\left (3584 a^{3} b -1024 a \,b^{3}\right ) \cos \left (8 d x +8 c \right )+\left (57120 a^{4}+86240 a^{2} b^{2}-23310 b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (25760 a^{4}-25760 a^{2} b^{2}+1610 b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (3360 a^{4}-3360 a^{2} b^{2}+210 b^{4}\right ) \sin \left (7 d x +7 c \right )+\left (114688 a^{3} b -114688 a \,b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (34720 a^{4}+108640 a^{2} b^{2}+46970 b^{4}\right ) \sin \left (d x +c \right )+344064 b a \left (\left (a^{2}-\frac {b^{2}}{21}\right ) \cos \left (d x +c \right )+\frac {35 a^{2}}{96}-\frac {5 b^{2}}{48}\right )}{4480 d \left (\cos \left (8 d x +8 c \right )+8 \cos \left (6 d x +6 c \right )+28 \cos \left (4 d x +4 c \right )+56 \cos \left (2 d x +2 c \right )+35\right )}\) | \(462\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (-1680 a^{4}-105 b^{4}+1680 a^{2} b^{2}+28560 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}-805 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-23485 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-11655 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-12880 a^{2} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-1680 a^{2} b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-57344 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+57344 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}+43120 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+172032 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+57344 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+172032 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}-8192 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-8192 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-57344 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+17360 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+23485 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-17360 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-28560 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+11655 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-12880 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+54320 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-54320 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-43120 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+12880 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+12880 a^{4} {\mathrm e}^{12 i \left (d x +c \right )}+1680 a^{4} {\mathrm e}^{14 i \left (d x +c \right )}+105 b^{4} {\mathrm e}^{14 i \left (d x +c \right )}+805 b^{4} {\mathrm e}^{12 i \left (d x +c \right )}\right )}{2240 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{4}}{8 d}+\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{8 d}-\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d}+\frac {3 \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{4}}{8 d}-\frac {3 b^{2} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{2}}{8 d}+\frac {3 b^{4} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{128 d}\) | \(684\) |
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Time = 0.27 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.65 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {105 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 5120 \, a b^{3} \cos \left (d x + c\right ) + 7168 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 70 \, {\left (3 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 16 \, b^{4} + 8 \, {\left (16 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{8960 \, d \cos \left (d x + c\right )^{8}} \]
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Timed out. \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.98 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {35 \, b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{7} - 11 \, \sin \left (d x + c\right )^{5} - 11 \, \sin \left (d x + c\right )^{3} + 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 560 \, a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 560 \, a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {7168 \, a^{3} b}{\cos \left (d x + c\right )^{5}} + \frac {1024 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a b^{3}}{\cos \left (d x + c\right )^{7}}}{8960 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (302) = 604\).
Time = 0.47 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.14 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Too large to display} \]
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Time = 26.24 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.72 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {5\,a^4}{4}+\frac {3\,a^2\,b^2}{4}-\frac {3\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-\frac {17\,a^4}{4}+\frac {41\,a^2\,b^2}{4}+\frac {23\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (-\frac {17\,a^4}{4}+\frac {41\,a^2\,b^2}{4}+\frac {23\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {21\,a^4}{4}-\frac {13\,a^2\,b^2}{4}+\frac {333\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {21\,a^4}{4}-\frac {13\,a^2\,b^2}{4}+\frac {333\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {9\,a^4}{4}+\frac {31\,a^2\,b^2}{4}-\frac {671\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {9\,a^4}{4}+\frac {31\,a^2\,b^2}{4}-\frac {671\,b^4}{64}\right )-\frac {16\,a\,b^3}{35}+\frac {8\,a^3\,b}{5}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,a^4}{4}+\frac {3\,a^2\,b^2}{4}-\frac {3\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (16\,a\,b^3-24\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (16\,a\,b^3-56\,a^3\,b\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {104\,a^3\,b}{5}+\frac {16\,a\,b^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {128\,a\,b^3}{35}-\frac {24\,a^3\,b}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {128\,a\,b^3}{5}-\frac {248\,a^3\,b}{5}\right )-40\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^4}{4}-\frac {3\,a^2\,b^2}{4}+\frac {3\,b^4}{64}\right )}{d} \]
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