Integrand size = 21, antiderivative size = 194 \[ \int \frac {\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos (c+d x)}{a^6 d}-\frac {2 b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a^5 d}+\frac {\left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^4 d}+\frac {b \cos ^4(c+d x)}{2 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^2}{a^7 d (b+a \cos (c+d x))}+\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (b+a \cos (c+d x))}{a^7 d} \]
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Time = 0.48 (sec) , antiderivative size = 194, 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} \[ \int \frac {\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {b \cos ^4(c+d x)}{2 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^2}{a^7 d (a \cos (c+d x)+b)}-\frac {2 b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a^5 d}+\frac {\left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^4 d}+\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b)}{a^7 d}-\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos (c+d x)}{a^6 d} \]
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Rule 12
Rule 962
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x) \sin ^5(c+d x)}{(-b-a \cos (c+d x))^2} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )^2}{a^2 (-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )^2}{(-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \left (a^4 \left (1+\frac {-6 a^2 b^2+5 b^4}{a^4}\right )+\frac {b^2 \left (a^2-b^2\right )^2}{(b-x)^2}-\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right )}{b-x}+4 b \left (-a^2+b^2\right ) x-\left (2 a^2-3 b^2\right ) x^2+2 b x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = -\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos (c+d x)}{a^6 d}-\frac {2 b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a^5 d}+\frac {\left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^4 d}+\frac {b \cos ^4(c+d x)}{2 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^2}{a^7 d (b+a \cos (c+d x))}+\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (b+a \cos (c+d x))}{a^7 d} \\ \end{align*}
Time = 1.12 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.44 \[ \int \frac {\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {-150 a^6+1740 a^4 b^2-2160 a^2 b^4+480 b^6-5 \left (25 a^6-168 a^4 b^2+144 a^2 b^4\right ) \cos (2 (c+d x))-115 a^5 b \cos (3 (c+d x))+120 a^3 b^3 \cos (3 (c+d x))+22 a^6 \cos (4 (c+d x))-30 a^4 b^2 \cos (4 (c+d x))+9 a^5 b \cos (5 (c+d x))-3 a^6 \cos (6 (c+d x))+960 a^4 b^2 \log (b+a \cos (c+d x))-3840 a^2 b^4 \log (b+a \cos (c+d x))+2880 b^6 \log (b+a \cos (c+d x))+120 a b \cos (c+d x) \left (-4 a^4+23 a^2 b^2-20 b^4+8 \left (a^4-4 a^2 b^2+3 b^4\right ) \log (b+a \cos (c+d x))\right )}{480 a^7 d (b+a \cos (c+d x))} \]
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Time = 1.39 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {\cos \left (d x +c \right )^{5} a^{4}}{5}-\frac {b \cos \left (d x +c \right )^{4} a^{3}}{2}-\frac {2 \cos \left (d x +c \right )^{3} a^{4}}{3}+\cos \left (d x +c \right )^{3} a^{2} b^{2}+2 \cos \left (d x +c \right )^{2} a^{3} b -2 \cos \left (d x +c \right )^{2} a \,b^{3}+\cos \left (d x +c \right ) a^{4}-6 \cos \left (d x +c \right ) a^{2} b^{2}+5 \cos \left (d x +c \right ) b^{4}}{a^{6}}+\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}{a^{7} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 b \left (a^{4}-4 a^{2} b^{2}+3 b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{7}}}{d}\) | \(198\) |
default | \(\frac {-\frac {\frac {\cos \left (d x +c \right )^{5} a^{4}}{5}-\frac {b \cos \left (d x +c \right )^{4} a^{3}}{2}-\frac {2 \cos \left (d x +c \right )^{3} a^{4}}{3}+\cos \left (d x +c \right )^{3} a^{2} b^{2}+2 \cos \left (d x +c \right )^{2} a^{3} b -2 \cos \left (d x +c \right )^{2} a \,b^{3}+\cos \left (d x +c \right ) a^{4}-6 \cos \left (d x +c \right ) a^{2} b^{2}+5 \cos \left (d x +c \right ) b^{4}}{a^{6}}+\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}{a^{7} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 b \left (a^{4}-4 a^{2} b^{2}+3 b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{7}}}{d}\) | \(198\) |
parallelrisch | \(\frac {960 b^{2} \left (a -b \right ) \left (a +b \right ) \left (a^{2}-3 b^{2}\right ) \left (b +a \cos \left (d x +c \right )\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-960 b^{2} \left (a -b \right ) \left (a +b \right ) \left (a^{2}-3 b^{2}\right ) \left (b +a \cos \left (d x +c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (-125 b \,a^{6}+840 a^{4} b^{3}-720 b^{5} a^{2}\right ) \cos \left (2 d x +2 c \right )+\left (-115 a^{5} b^{2}+120 b^{4} a^{3}\right ) \cos \left (3 d x +3 c \right )+\left (22 b \,a^{6}-30 a^{4} b^{3}\right ) \cos \left (4 d x +4 c \right )+9 \cos \left (5 d x +5 c \right ) a^{5} b^{2}-3 b \cos \left (6 d x +6 c \right ) a^{6}+\left (-256 a^{7}-918 a^{5} b^{2}+3912 b^{4} a^{3}-2880 a \,b^{6}\right ) \cos \left (d x +c \right )-406 b \,a^{6}+1302 a^{4} b^{3}-1008 b^{5} a^{2}}{480 d \,a^{7} b \left (b +a \cos \left (d x +c \right )\right )}\) | \(297\) |
risch | \(-\frac {4 i b c}{a^{3} d}+\frac {16 i b^{3} c}{a^{5} d}-\frac {12 i b^{5} c}{a^{7} d}+\frac {5 \cos \left (3 d x +3 c \right )}{48 d \,a^{2}}+\frac {2 b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) {\mathrm e}^{i \left (d x +c \right )}}{a^{7} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{16 a^{2} d}+\frac {8 i b^{3} x}{a^{5}}-\frac {6 i b^{5} x}{a^{7}}-\frac {3 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{2 a^{5} d}+\frac {21 \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{8 a^{4} d}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )} b^{4}}{2 a^{6} d}+\frac {21 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{8 a^{4} d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )} b^{4}}{2 a^{6} d}-\frac {3 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{2 a^{5} d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 a^{2} d}-\frac {\cos \left (5 d x +5 c \right )}{80 d \,a^{2}}+\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{3} d}-\frac {8 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{5} d}+\frac {6 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{7} d}+\frac {b \cos \left (4 d x +4 c \right )}{16 a^{3} d}-\frac {\cos \left (3 d x +3 c \right ) b^{2}}{4 a^{4} d}-\frac {2 i b x}{a^{3}}\) | \(503\) |
norman | \(\frac {\frac {\left (32 a^{5}+32 a^{4} b +96 a^{3} b^{2}-360 a^{2} b^{3}-144 a \,b^{4}+360 b^{5}\right ) \left (a +b \right )}{60 a^{6} b d}-\frac {\left (32 a^{6}-128 a^{5} b +120 a^{3} b^{3}-408 a^{2} b^{4}-72 a \,b^{5}+360 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{12 a^{6} b d}+\frac {\left (32 a^{6}+128 a^{5} b -120 a^{3} b^{3}-408 a^{2} b^{4}+72 a \,b^{5}+360 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{12 a^{6} b d}-\frac {\left (32 a^{6}-64 a^{5} b +128 a^{4} b^{2}+264 a^{3} b^{3}-504 a^{2} b^{4}-216 a \,b^{5}+360 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{60 d \,a^{6} b}-\frac {\left (64 a^{6}-160 a^{5} b +168 a^{4} b^{2}+432 a^{3} b^{3}-888 a^{2} b^{4}-288 a \,b^{5}+720 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{30 a^{6} b d}+\frac {\left (64 a^{6}+160 a^{5} b +168 a^{4} b^{2}-432 a^{3} b^{3}-888 a^{2} b^{4}+288 a \,b^{5}+720 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{30 d \,a^{6} b}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {2 b \left (a^{4}-4 a^{2} b^{2}+3 b^{4}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a^{7} d}+\frac {2 b \left (a^{4}-4 a^{2} b^{2}+3 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{a^{7} d}\) | \(538\) |
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Time = 0.31 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.24 \[ \int \frac {\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {48 \, a^{6} \cos \left (d x + c\right )^{6} - 72 \, a^{5} b \cos \left (d x + c\right )^{5} - 435 \, a^{4} b^{2} + 720 \, a^{2} b^{4} - 240 \, b^{6} - 40 \, {\left (4 \, a^{6} - 3 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{4} + 80 \, {\left (4 \, a^{5} b - 3 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{3} + 240 \, {\left (a^{6} - 4 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, a^{5} b - 80 \, a^{3} b^{3} + 80 \, a b^{5}\right )} \cos \left (d x + c\right ) - 480 \, {\left (a^{4} b^{2} - 4 \, a^{2} b^{4} + 3 \, b^{6} + {\left (a^{5} b - 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{240 \, {\left (a^{8} d \cos \left (d x + c\right ) + a^{7} b d\right )}} \]
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Timed out. \[ \int \frac {\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {30 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )}}{a^{8} \cos \left (d x + c\right ) + a^{7} b} - \frac {6 \, a^{4} \cos \left (d x + c\right )^{5} - 15 \, a^{3} b \cos \left (d x + c\right )^{4} - 10 \, {\left (2 \, a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 60 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )}{a^{6}} + \frac {60 \, {\left (a^{4} b - 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{7}}}{30 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1102 vs. \(2 (188) = 376\).
Time = 0.34 (sec) , antiderivative size = 1102, normalized size of antiderivative = 5.68 \[ \int \frac {\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
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Time = 0.12 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.30 \[ \int \frac {\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {2}{3\,a^2}-\frac {b^2}{a^4}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {b^3}{a^5}+\frac {b\,\left (\frac {2}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {2}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {2}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^2\,d}+\frac {b\,{\cos \left (c+d\,x\right )}^4}{2\,a^3\,d}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^4\,b-8\,a^2\,b^3+6\,b^5\right )}{a^7\,d}+\frac {a^4\,b^2-2\,a^2\,b^4+b^6}{a\,d\,\left (\cos \left (c+d\,x\right )\,a^7+b\,a^6\right )} \]
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