Integrand size = 21, antiderivative size = 170 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=-\frac {a \left (a^2-6 b^2\right ) \cos (c+d x)}{d}+\frac {b \left (6 a^2-b^2\right ) \cos ^2(c+d x)}{2 d}+\frac {a \left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 d}-\frac {3 a^2 b \cos ^4(c+d x)}{4 d}-\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {b \left (3 a^2-2 b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
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Time = 0.30 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2916, 12, 962} \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=-\frac {a^3 \cos ^5(c+d x)}{5 d}+\frac {a \left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 d}+\frac {b \left (6 a^2-b^2\right ) \cos ^2(c+d x)}{2 d}-\frac {a \left (a^2-6 b^2\right ) \cos (c+d x)}{d}-\frac {b \left (3 a^2-2 b^2\right ) \log (\cos (c+d x))}{d}-\frac {3 a^2 b \cos ^4(c+d x)}{4 d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
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Rule 12
Rule 962
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x))^3 \sin ^2(c+d x) \tan ^3(c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a^3 (-b+x)^3 \left (a^2-x^2\right )^2}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(-b+x)^3 \left (a^2-x^2\right )^2}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^2 d} \\ & = \frac {\text {Subst}\left (\int \left (a^4 \left (1-\frac {6 b^2}{a^2}\right )-\frac {a^4 b^3}{x^3}+\frac {3 a^4 b^2}{x^2}+\frac {-3 a^4 b+2 a^2 b^3}{x}-b \left (-6 a^2+b^2\right ) x-\left (2 a^2-3 b^2\right ) x^2-3 b x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d} \\ & = -\frac {a \left (a^2-6 b^2\right ) \cos (c+d x)}{d}+\frac {b \left (6 a^2-b^2\right ) \cos ^2(c+d x)}{2 d}+\frac {a \left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 d}-\frac {3 a^2 b \cos ^4(c+d x)}{4 d}-\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {b \left (3 a^2-2 b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \\ \end{align*}
Time = 1.39 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=\frac {-60 a \left (5 a^2-42 b^2\right ) \cos (c+d x)+60 \left (9 a^2 b-2 b^3\right ) \cos (2 (c+d x))+50 a^3 \cos (3 (c+d x))-120 a b^2 \cos (3 (c+d x))-45 a^2 b \cos (4 (c+d x))-6 a^3 \cos (5 (c+d x))-1440 a^2 b \log (\cos (c+d x))+960 b^3 \log (\cos (c+d x))+1440 a b^2 \sec (c+d x)+240 b^3 \sec ^2(c+d x)}{480 d} \]
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Time = 2.45 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {-\frac {a^{3} \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5}+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(174\) |
default | \(\frac {-\frac {a^{3} \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5}+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(174\) |
parts | \(-\frac {a^{3} \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5 d}+\frac {b^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )}{d}+\frac {3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(182\) |
parallelrisch | \(\frac {2880 \left (a^{2}-\frac {2 b^{2}}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) b \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-2880 \left (a^{2}-\frac {2 b^{2}}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-2880 \left (a^{2}-\frac {2 b^{2}}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-512 a^{3}+45 a^{2} b +7680 a \,b^{2}-480 b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (-206 a^{3}+2280 a \,b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (450 a^{2} b -120 b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (38 a^{3}-120 a \,b^{2}\right ) \cos \left (5 d x +5 c \right )-45 a^{2} b \cos \left (6 d x +6 c \right )-6 a^{3} \cos \left (7 d x +7 c \right )+\left (-850 a^{3}+13200 a \,b^{2}\right ) \cos \left (d x +c \right )-512 a^{3}-450 a^{2} b +7680 a \,b^{2}+600 b^{3}}{960 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(294\) |
norman | \(\frac {\frac {\left (16 a^{3}+24 a^{2} b -48 a \,b^{2}+16 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {16 a^{3}-240 a \,b^{2}}{15 d}-\frac {\left (6 a^{2} b -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}-\frac {\left (18 a^{2} b -12 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {\left (16 a^{3}+30 a^{2} b -240 a \,b^{2}-20 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d}-\frac {\left (16 a^{3}+270 a^{2} b -240 a \,b^{2}-180 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{15 d}-\frac {\left (32 a^{3}-72 a^{2} b +96 a \,b^{2}-48 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {b \left (3 a^{2}-2 b^{2}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}-\frac {b \left (3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \left (3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(350\) |
risch | \(\frac {6 i b \,a^{2} c}{d}-\frac {4 i b^{3} c}{d}+\frac {5 \,{\mathrm e}^{3 i \left (d x +c \right )} a^{3}}{96 d}-\frac {{\mathrm e}^{3 i \left (d x +c \right )} a \,b^{2}}{8 d}+\frac {9 \,{\mathrm e}^{2 i \left (d x +c \right )} a^{2} b}{16 d}-\frac {{\mathrm e}^{2 i \left (d x +c \right )} b^{3}}{8 d}-\frac {5 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {21 \,{\mathrm e}^{i \left (d x +c \right )} a \,b^{2}}{8 d}-\frac {5 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {21 \,{\mathrm e}^{-i \left (d x +c \right )} a \,b^{2}}{8 d}+\frac {9 \,{\mathrm e}^{-2 i \left (d x +c \right )} a^{2} b}{16 d}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )} b^{3}}{8 d}+\frac {5 \,{\mathrm e}^{-3 i \left (d x +c \right )} a^{3}}{96 d}-\frac {{\mathrm e}^{-3 i \left (d x +c \right )} a \,b^{2}}{8 d}+3 i a^{2} b x -2 i b^{3} x +\frac {2 b^{2} \left (3 a \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )} a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}+\frac {2 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {a^{3} \cos \left (5 d x +5 c \right )}{80 d}-\frac {3 a^{2} b \cos \left (4 d x +4 c \right )}{32 d}\) | \(381\) |
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Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.03 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=-\frac {96 \, a^{3} \cos \left (d x + c\right )^{7} + 360 \, a^{2} b \cos \left (d x + c\right )^{6} - 160 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 240 \, {\left (6 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{4} - 1440 \, a b^{2} \cos \left (d x + c\right ) + 480 \, {\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 480 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 240 \, b^{3} + 15 \, {\left (39 \, a^{2} b - 8 \, b^{3}\right )} \cos \left (d x + c\right )^{2}}{480 \, d \cos \left (d x + c\right )^{2}} \]
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Timed out. \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.84 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=-\frac {12 \, a^{3} \cos \left (d x + c\right )^{5} + 45 \, a^{2} b \cos \left (d x + c\right )^{4} - 20 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (6 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 60 \, {\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right ) + 60 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \log \left (\cos \left (d x + c\right )\right ) - \frac {30 \, {\left (6 \, a b^{2} \cos \left (d x + c\right ) + b^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{60 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (160) = 320\).
Time = 0.44 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.09 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=\text {Too large to display} \]
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Time = 13.58 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.84 \[ \int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx=-\frac {{\cos \left (c+d\,x\right )}^3\,\left (a\,b^2-\frac {2\,a^3}{3}\right )-{\cos \left (c+d\,x\right )}^2\,\left (3\,a^2\,b-\frac {b^3}{2}\right )+\ln \left (\cos \left (c+d\,x\right )\right )\,\left (3\,a^2\,b-2\,b^3\right )-\frac {\frac {b^3}{2}+3\,a\,\cos \left (c+d\,x\right )\,b^2}{{\cos \left (c+d\,x\right )}^2}-\cos \left (c+d\,x\right )\,\left (6\,a\,b^2-a^3\right )+\frac {a^3\,{\cos \left (c+d\,x\right )}^5}{5}+\frac {3\,a^2\,b\,{\cos \left (c+d\,x\right )}^4}{4}}{d} \]
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