Integrand size = 28, antiderivative size = 213 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {a^3 \tan (c+d x)}{d}+\frac {3 a^2 b \tan ^2(c+d x)}{2 d}+\frac {a \left (a^2+b^2\right ) \tan ^3(c+d x)}{d}+\frac {b \left (9 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac {3 a \left (a^2+3 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {b \left (3 a^2+b^2\right ) \tan ^6(c+d x)}{2 d}+\frac {a \left (a^2+9 b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {3 b \left (a^2+b^2\right ) \tan ^8(c+d x)}{8 d}+\frac {a b^2 \tan ^9(c+d x)}{3 d}+\frac {b^3 \tan ^{10}(c+d x)}{10 d} \]
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Time = 0.20 (sec) , antiderivative size = 213, 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} \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {a^3 \tan (c+d x)}{d}+\frac {3 b \left (a^2+b^2\right ) \tan ^8(c+d x)}{8 d}+\frac {a \left (a^2+9 b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {b \left (3 a^2+b^2\right ) \tan ^6(c+d x)}{2 d}+\frac {3 a \left (a^2+3 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {b \left (9 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac {a \left (a^2+b^2\right ) \tan ^3(c+d x)}{d}+\frac {3 a^2 b \tan ^2(c+d x)}{2 d}+\frac {a b^2 \tan ^9(c+d x)}{3 d}+\frac {b^3 \tan ^{10}(c+d x)}{10 d} \]
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Rule 962
Rule 3167
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(b+a x)^3 \left (1+x^2\right )^3}{x^{11}} \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {b^3}{x^{11}}+\frac {3 a b^2}{x^{10}}+\frac {3 b \left (a^2+b^2\right )}{x^9}+\frac {a^3+9 a b^2}{x^8}+\frac {3 \left (3 a^2 b+b^3\right )}{x^7}+\frac {3 \left (a^3+3 a b^2\right )}{x^6}+\frac {9 a^2 b+b^3}{x^5}+\frac {3 a \left (a^2+b^2\right )}{x^4}+\frac {3 a^2 b}{x^3}+\frac {a^3}{x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {a^3 \tan (c+d x)}{d}+\frac {3 a^2 b \tan ^2(c+d x)}{2 d}+\frac {a \left (a^2+b^2\right ) \tan ^3(c+d x)}{d}+\frac {b \left (9 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac {3 a \left (a^2+3 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {b \left (3 a^2+b^2\right ) \tan ^6(c+d x)}{2 d}+\frac {a \left (a^2+9 b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {3 b \left (a^2+b^2\right ) \tan ^8(c+d x)}{8 d}+\frac {a b^2 \tan ^9(c+d x)}{3 d}+\frac {b^3 \tan ^{10}(c+d x)}{10 d} \\ \end{align*}
Time = 2.17 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.83 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {\frac {1}{4} \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^4-\frac {6}{5} a \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^5+\frac {1}{2} \left (a^2+b^2\right ) \left (5 a^2+b^2\right ) (a+b \tan (c+d x))^6-\frac {4}{7} a \left (5 a^2+3 b^2\right ) (a+b \tan (c+d x))^7+\frac {3}{8} \left (5 a^2+b^2\right ) (a+b \tan (c+d x))^8-\frac {2}{3} a (a+b \tan (c+d x))^9+\frac {1}{10} (a+b \tan (c+d x))^{10}}{b^7 d} \]
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Time = 1.85 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.82
method | result | size |
parts | \(-\frac {a^{3} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{3} \left (\frac {\sec \left (d x +c \right )^{10}}{10}-\frac {\sec \left (d x +c \right )^{8}}{8}\right )}{d}+\frac {3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )}{d}+\frac {3 a^{2} b \sec \left (d x +c \right )^{8}}{8 d}\) | \(175\) |
derivativedivides | \(\frac {-a^{3} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{8 \cos \left (d x +c \right )^{8}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(219\) |
default | \(\frac {-a^{3} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{8 \cos \left (d x +c \right )^{8}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(219\) |
risch | \(-\frac {32 \left (-3 i a^{3}-105 i a^{3} {\mathrm e}^{12 i \left (d x +c \right )}-315 a^{2} b \,{\mathrm e}^{12 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{12 i \left (d x +c \right )}-360 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+126 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-630 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}-126 b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+315 i a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+120 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-315 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+45 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-378 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}+i a \,b^{2}-105 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+10 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-525 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}-135 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-30 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{10}}\) | \(306\) |
parallelrisch | \(-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17} a^{2} b +420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16} a \,b^{2}-105 a^{3}-84 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a \,b^{2}-1176 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} b^{3}-1470 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b^{3}-420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{3}-210 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{3}+5730 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} a^{3}-5730 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a^{3}-4410 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} a^{2} b +4410 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a^{2} b -2520 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{2} b +630 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} b -315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b +105 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{18}-525 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16} a^{3}-210 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15} b^{3}-420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13} b^{3}-2520 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13} a^{2} b +2340 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} a \,b^{2}+4410 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} a^{2} b -1420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} a \,b^{2}+1420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a \,b^{2}-2340 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a \,b^{2}+4080 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{3}-1848 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}+525 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}-420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{2}+630 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15} a^{2} b +84 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14} a \,b^{2}+1848 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}-4080 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} a^{3}-1470 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} b^{3}\right )}{105 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{10}}\) | \(579\) |
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Time = 0.28 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.70 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {84 \, b^{3} + 105 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (16 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{9} + 8 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} + 6 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{5} + 35 \, a b^{2} \cos \left (d x + c\right ) + 5 \, {\left (3 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{10}} \]
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Timed out. \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.86 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {24 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{3} + 8 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} a b^{2} - \frac {21 \, {\left (5 \, \sin \left (d x + c\right )^{2} - 1\right )} b^{3}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} + \frac {315 \, a^{2} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{4}}}{840 \, d} \]
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Time = 0.42 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.03 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {84 \, b^{3} \tan \left (d x + c\right )^{10} + 280 \, a b^{2} \tan \left (d x + c\right )^{9} + 315 \, a^{2} b \tan \left (d x + c\right )^{8} + 315 \, b^{3} \tan \left (d x + c\right )^{8} + 120 \, a^{3} \tan \left (d x + c\right )^{7} + 1080 \, a b^{2} \tan \left (d x + c\right )^{7} + 1260 \, a^{2} b \tan \left (d x + c\right )^{6} + 420 \, b^{3} \tan \left (d x + c\right )^{6} + 504 \, a^{3} \tan \left (d x + c\right )^{5} + 1512 \, a b^{2} \tan \left (d x + c\right )^{5} + 1890 \, a^{2} b \tan \left (d x + c\right )^{4} + 210 \, b^{3} \tan \left (d x + c\right )^{4} + 840 \, a^{3} \tan \left (d x + c\right )^{3} + 840 \, a b^{2} \tan \left (d x + c\right )^{3} + 1260 \, a^{2} b \tan \left (d x + c\right )^{2} + 840 \, a^{3} \tan \left (d x + c\right )}{840 \, d} \]
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Time = 23.92 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.89 \[ \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {a^3\,\sin \left (c+d\,x\right )}{7}-\frac {a\,b^2\,\sin \left (c+d\,x\right )}{21}\right )+{\cos \left (c+d\,x\right )}^5\,\left (\frac {6\,a^3\,\sin \left (c+d\,x\right )}{35}-\frac {2\,a\,b^2\,\sin \left (c+d\,x\right )}{35}\right )+{\cos \left (c+d\,x\right )}^7\,\left (\frac {8\,a^3\,\sin \left (c+d\,x\right )}{35}-\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )}{105}\right )+{\cos \left (c+d\,x\right )}^9\,\left (\frac {16\,a^3\,\sin \left (c+d\,x\right )}{35}-\frac {16\,a\,b^2\,\sin \left (c+d\,x\right )}{105}\right )+{\cos \left (c+d\,x\right )}^2\,\left (\frac {3\,a^2\,b}{8}-\frac {b^3}{8}\right )+\frac {b^3}{10}+\frac {a\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{3}}{d\,{\cos \left (c+d\,x\right )}^{10}} \]
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