Integrand size = 28, antiderivative size = 201 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a^4 \tan (c+d x)}{d}+\frac {2 a^3 b \tan ^2(c+d x)}{d}+\frac {2 a^2 \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {a b \left (2 a^2+b^2\right ) \tan ^4(c+d x)}{d}+\frac {\left (a^4+12 a^2 b^2+b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {2 a b \left (a^2+2 b^2\right ) \tan ^6(c+d x)}{3 d}+\frac {2 b^2 \left (3 a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {a b^3 \tan ^8(c+d x)}{2 d}+\frac {b^4 \tan ^9(c+d x)}{9 d} \]
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Time = 0.19 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3167, 962} \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a^4 \tan (c+d x)}{d}+\frac {2 a^3 b \tan ^2(c+d x)}{d}+\frac {2 b^2 \left (3 a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {2 a b \left (a^2+2 b^2\right ) \tan ^6(c+d x)}{3 d}+\frac {a b \left (2 a^2+b^2\right ) \tan ^4(c+d x)}{d}+\frac {2 a^2 \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {\left (a^4+12 a^2 b^2+b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {a b^3 \tan ^8(c+d x)}{2 d}+\frac {b^4 \tan ^9(c+d x)}{9 d} \]
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Rule 962
Rule 3167
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(b+a x)^4 \left (1+x^2\right )^2}{x^{10}} \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {b^4}{x^{10}}+\frac {4 a b^3}{x^9}+\frac {2 \left (3 a^2 b^2+b^4\right )}{x^8}+\frac {4 a b \left (a^2+2 b^2\right )}{x^7}+\frac {a^4+12 a^2 b^2+b^4}{x^6}+\frac {4 a b \left (2 a^2+b^2\right )}{x^5}+\frac {2 \left (a^4+3 a^2 b^2\right )}{x^4}+\frac {4 a^3 b}{x^3}+\frac {a^4}{x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {a^4 \tan (c+d x)}{d}+\frac {2 a^3 b \tan ^2(c+d x)}{d}+\frac {2 a^2 \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {a b \left (2 a^2+b^2\right ) \tan ^4(c+d x)}{d}+\frac {\left (a^4+12 a^2 b^2+b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {2 a b \left (a^2+2 b^2\right ) \tan ^6(c+d x)}{3 d}+\frac {2 b^2 \left (3 a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {a b^3 \tan ^8(c+d x)}{2 d}+\frac {b^4 \tan ^9(c+d x)}{9 d} \\ \end{align*}
Time = 0.90 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.57 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {\frac {1}{5} \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^5-\frac {2}{3} a \left (a^2+b^2\right ) (a+b \tan (c+d x))^6+\frac {2}{7} \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^7-\frac {1}{2} a (a+b \tan (c+d x))^8+\frac {1}{9} (a+b \tan (c+d x))^9}{b^5 d} \]
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Time = 1.90 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.06
method | result | size |
parts | \(-\frac {a^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\sec \left (d x +c \right )^{8}}{8}-\frac {\sec \left (d x +c \right )^{6}}{6}\right )}{d}+\frac {2 a^{3} b \sec \left (d x +c \right )^{6}}{3 d}+\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \sin \left (d x +c \right )^{3}}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{3}}\right )}{d}\) | \(213\) |
derivativedivides | \(\frac {-a^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+\frac {2 a^{3} b}{3 \cos \left (d x +c \right )^{6}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \sin \left (d x +c \right )^{3}}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{3}}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{24 \cos \left (d x +c \right )^{4}}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )}{d}\) | \(236\) |
default | \(\frac {-a^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+\frac {2 a^{3} b}{3 \cos \left (d x +c \right )^{6}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \sin \left (d x +c \right )^{3}}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{3}}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{24 \cos \left (d x +c \right )^{4}}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )}{d}\) | \(236\) |
risch | \(\frac {16 i \left (21 a^{4}+b^{4}-18 a^{2} b^{2}+945 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}+9 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-126 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-315 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-1260 a^{2} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-2520 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}-1890 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-2520 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}-840 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+840 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+840 i a \,b^{3} {\mathrm e}^{12 i \left (d x +c \right )}-840 i a^{3} b \,{\mathrm e}^{12 i \left (d x +c \right )}+1701 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+441 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+1554 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+756 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+36 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+189 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-378 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-252 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-648 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-162 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+210 a^{4} {\mathrm e}^{12 i \left (d x +c \right )}+210 b^{4} {\mathrm e}^{12 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{9}}\) | \(403\) |
parallelrisch | \(-\frac {2 \left (a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15} a^{3} b +8 \left (a^{2} b^{2}-\frac {2}{3} a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+4 \left (3 a^{3} b -2 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+\frac {4 \left (19 a^{4}-12 a^{2} b^{2}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{5}+\frac {4 \left (-19 a^{3} b -2 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3}+\frac {24 \left (\frac {8}{7} b^{4}-6 a^{4}+\frac {31}{7} a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{5}+4 \left (11 a^{3} b -4 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\frac {2 \left (\frac {269}{3} a^{4}+\frac {1744}{63} b^{4}-\frac {688}{7} a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{5}+4 \left (-11 a^{3} b +4 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\frac {24 \left (\frac {8}{7} b^{4}-6 a^{4}+\frac {31}{7} a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5}+\frac {4 \left (19 a^{3} b +2 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3}+\frac {4 \left (19 a^{4}-12 a^{2} b^{2}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5}+4 \left (-3 a^{3} b +2 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+8 \left (a^{2} b^{2}-\frac {2}{3} a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} b +a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{9}}\) | \(449\) |
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Time = 0.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.83 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {315 \, a b^{3} \cos \left (d x + c\right ) + 420 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (8 \, {\left (21 \, a^{4} - 18 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} + 4 \, {\left (21 \, a^{4} - 18 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (21 \, a^{4} - 18 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 35 \, b^{4} + 10 \, {\left (27 \, a^{2} b^{2} - 5 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{630 \, d \cos \left (d x + c\right )^{9}} \]
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Timed out. \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.96 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {42 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{4} + 36 \, {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a^{2} b^{2} + 2 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} b^{4} + \frac {105 \, {\left (4 \, \sin \left (d x + c\right )^{2} - 1\right )} a b^{3}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - \frac {420 \, a^{3} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{630 \, d} \]
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Time = 0.48 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.06 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {70 \, b^{4} \tan \left (d x + c\right )^{9} + 315 \, a b^{3} \tan \left (d x + c\right )^{8} + 540 \, a^{2} b^{2} \tan \left (d x + c\right )^{7} + 180 \, b^{4} \tan \left (d x + c\right )^{7} + 420 \, a^{3} b \tan \left (d x + c\right )^{6} + 840 \, a b^{3} \tan \left (d x + c\right )^{6} + 126 \, a^{4} \tan \left (d x + c\right )^{5} + 1512 \, a^{2} b^{2} \tan \left (d x + c\right )^{5} + 126 \, b^{4} \tan \left (d x + c\right )^{5} + 1260 \, a^{3} b \tan \left (d x + c\right )^{4} + 630 \, a b^{3} \tan \left (d x + c\right )^{4} + 420 \, a^{4} \tan \left (d x + c\right )^{3} + 1260 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} + 1260 \, a^{3} b \tan \left (d x + c\right )^{2} + 630 \, a^{4} \tan \left (d x + c\right )}{630 \, d} \]
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Time = 26.00 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.22 \[ \int \sec ^{10}(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 )}^5\,\left (\frac {152\,a^4}{5}-\frac {96\,a^2\,b^2}{5}+\frac {32\,b^4}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {152\,a^4}{5}-\frac {96\,a^2\,b^2}{5}+\frac {32\,b^4}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (-\frac {288\,a^4}{5}+\frac {1488\,a^2\,b^2}{35}+\frac {384\,b^4}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-\frac {288\,a^4}{5}+\frac {1488\,a^2\,b^2}{35}+\frac {384\,b^4}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {1076\,a^4}{15}-\frac {2752\,a^2\,b^2}{35}+\frac {6976\,b^4}{315}\right )+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {32\,a^4}{3}-16\,a^2\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {32\,a^4}{3}-16\,a^2\,b^2\right )+2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (16\,a\,b^3-24\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (16\,a\,b^3-24\,a^3\,b\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (32\,a\,b^3-88\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (32\,a\,b^3-88\,a^3\,b\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {152\,a^3\,b}{3}+\frac {16\,a\,b^3}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {152\,a^3\,b}{3}+\frac {16\,a\,b^3}{3}\right )+8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-8\,a^3\,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 )}^2-1\right )}^9} \]
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