Integrand size = 29, antiderivative size = 193 \[ \int \frac {\cos ^5(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 a \left (3 a^4-4 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{b^7 d}+\frac {\left (5 a^4-6 a^2 b^2+b^4\right ) \sin (c+d x)}{b^6 d}-\frac {2 a \left (a^2-b^2\right ) \sin ^2(c+d x)}{b^5 d}-\frac {\left (2-\frac {3 a^2}{b^2}\right ) \sin ^3(c+d x)}{3 b^2 d}-\frac {a \sin ^4(c+d x)}{2 b^3 d}+\frac {\sin ^5(c+d x)}{5 b^2 d}-\frac {a^2 \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))} \]
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Time = 0.18 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2916, 12, 962} \[ \int \frac {\cos ^5(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\left (2-\frac {3 a^2}{b^2}\right ) \sin ^3(c+d x)}{3 b^2 d}-\frac {a^2 \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))}-\frac {2 a \left (a^2-b^2\right ) \sin ^2(c+d x)}{b^5 d}-\frac {2 a \left (3 a^4-4 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{b^7 d}+\frac {\left (5 a^4-6 a^2 b^2+b^4\right ) \sin (c+d x)}{b^6 d}-\frac {a \sin ^4(c+d x)}{2 b^3 d}+\frac {\sin ^5(c+d x)}{5 b^2 d} \]
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Rule 12
Rule 962
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2 \left (b^2-x^2\right )^2}{b^2 (a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {x^2 \left (b^2-x^2\right )^2}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^7 d} \\ & = \frac {\text {Subst}\left (\int \left (5 a^4 \left (1+\frac {-6 a^2 b^2+b^4}{5 a^4}\right )-4 a \left (a^2-b^2\right ) x+\left (3 a^2-2 b^2\right ) x^2-2 a x^3+x^4+\frac {\left (a^3-a b^2\right )^2}{(a+x)^2}-\frac {2 a \left (3 a^4-4 a^2 b^2+b^4\right )}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d} \\ & = -\frac {2 a \left (3 a^4-4 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{b^7 d}+\frac {\left (5 a^4-6 a^2 b^2+b^4\right ) \sin (c+d x)}{b^6 d}-\frac {2 a \left (a^2-b^2\right ) \sin ^2(c+d x)}{b^5 d}+\frac {\left (3 a^2-2 b^2\right ) \sin ^3(c+d x)}{3 b^4 d}-\frac {a \sin ^4(c+d x)}{2 b^3 d}+\frac {\sin ^5(c+d x)}{5 b^2 d}-\frac {a^2 \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))} \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.17 \[ \int \frac {\cos ^5(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {-30 a^2 \left (a^2-b^2\right ) \left (a^2-b^2+\left (6 a^2-2 b^2\right ) \log (a+b \sin (c+d x))\right )-30 a b \left (a^2-b^2\right ) \left (-5 a^2+b^2+\left (6 a^2-2 b^2\right ) \log (a+b \sin (c+d x))\right ) \sin (c+d x)+30 b^2 \left (3 a^4-4 a^2 b^2+b^4\right ) \sin ^2(c+d x)+\left (-30 a^3 b^3+40 a b^5\right ) \sin ^3(c+d x)+5 b^4 \left (3 a^2-4 b^2\right ) \sin ^4(c+d x)-9 a b^5 \sin ^5(c+d x)+6 b^6 \sin ^6(c+d x)}{30 b^7 d (a+b \sin (c+d x))} \]
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Time = 1.04 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\sin ^{5}\left (d x +c \right )\right ) b^{4}}{5}-\frac {a \left (\sin ^{4}\left (d x +c \right )\right ) b^{3}}{2}+a^{2} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )-\frac {2 b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-2 a^{3} b \left (\sin ^{2}\left (d x +c \right )\right )+2 a \,b^{3} \left (\sin ^{2}\left (d x +c \right )\right )+5 a^{4} \sin \left (d x +c \right )-6 \sin \left (d x +c \right ) a^{2} b^{2}+\sin \left (d x +c \right ) b^{4}}{b^{6}}-\frac {a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}{b^{7} \left (a +b \sin \left (d x +c \right )\right )}-\frac {2 a \left (3 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{7}}}{d}\) | \(198\) |
default | \(\frac {\frac {\frac {\left (\sin ^{5}\left (d x +c \right )\right ) b^{4}}{5}-\frac {a \left (\sin ^{4}\left (d x +c \right )\right ) b^{3}}{2}+a^{2} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )-\frac {2 b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-2 a^{3} b \left (\sin ^{2}\left (d x +c \right )\right )+2 a \,b^{3} \left (\sin ^{2}\left (d x +c \right )\right )+5 a^{4} \sin \left (d x +c \right )-6 \sin \left (d x +c \right ) a^{2} b^{2}+\sin \left (d x +c \right ) b^{4}}{b^{6}}-\frac {a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}{b^{7} \left (a +b \sin \left (d x +c \right )\right )}-\frac {2 a \left (3 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{7}}}{d}\) | \(198\) |
parallelrisch | \(\frac {-2880 \left (a +b \right ) \left (a^{2}-\frac {b^{2}}{3}\right ) \left (a +b \sin \left (d x +c \right )\right ) \left (a -b \right ) a \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )+2880 \left (a +b \right ) \left (a^{2}-\frac {b^{2}}{3}\right ) \left (a +b \sin \left (d x +c \right )\right ) \left (a -b \right ) a \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-720 a^{4} b^{2}+840 a^{2} b^{4}-125 b^{6}\right ) \cos \left (2 d x +2 c \right )+\left (30 a^{2} b^{4}-22 b^{6}\right ) \cos \left (4 d x +4 c \right )+\left (120 a^{3} b^{3}-115 a \,b^{5}\right ) \sin \left (3 d x +3 c \right )-3 b^{6} \cos \left (6 d x +6 c \right )-9 a \,b^{5} \sin \left (5 d x +5 c \right )+\left (2880 a^{5} b -4200 a^{3} b^{3}+1350 a \,b^{5}\right ) \sin \left (d x +c \right )+720 a^{4} b^{2}-870 a^{2} b^{4}+150 b^{6}}{480 b^{7} d \left (a +b \sin \left (d x +c \right )\right )}\) | \(280\) |
risch | \(\frac {4 i a c}{b^{3} d}-\frac {a \cos \left (4 d x +4 c \right )}{16 b^{3} d}+\frac {5 \sin \left (3 d x +3 c \right )}{48 b^{2} d}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 b^{6} d}+\frac {21 i {\mathrm e}^{i \left (d x +c \right )} a^{2}}{8 b^{4} d}-\frac {\sin \left (3 d x +3 c \right ) a^{2}}{4 b^{4} d}+\frac {6 i x \,a^{5}}{b^{7}}-\frac {8 i x \,a^{3}}{b^{5}}+\frac {2 i x a}{b^{3}}+\frac {\sin \left (5 d x +5 c \right )}{80 b^{2} d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 b^{6} d}-\frac {21 i {\mathrm e}^{-i \left (d x +c \right )} a^{2}}{8 b^{4} d}-\frac {2 a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) {\mathrm e}^{i \left (d x +c \right )}}{b^{7} d \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )}+\frac {12 i a^{5} c}{b^{7} d}-\frac {16 i a^{3} c}{b^{5} d}-\frac {3 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{3} d}-\frac {6 a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{7} d}+\frac {8 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{5} d}-\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{3} d}+\frac {a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{2 b^{5} d}-\frac {3 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{3} d}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )}}{16 b^{2} d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )}}{16 b^{2} d}+\frac {a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{2 b^{5} d}\) | \(517\) |
norman | \(\frac {\frac {4 \left (54 a^{4}-66 a^{2} b^{2}+10 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b^{5} d}+\frac {4 \left (54 a^{4}-66 a^{2} b^{2}+10 b^{4}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b^{5} d}+\frac {2 \left (600 a^{4}-680 a^{2} b^{2}+104 b^{4}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 b^{5} d}+\frac {4 \left (675 a^{4}-780 a^{2} b^{2}+113 b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 b^{5} d}+\frac {4 \left (675 a^{4}-780 a^{2} b^{2}+113 b^{4}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 b^{5} d}+\frac {4 \left (3 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5} d}+\frac {4 \left (3 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5} d}+\frac {2 \left (126 a^{6}-180 a^{4} b^{2}+58 a^{2} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \,b^{6} d}+\frac {2 \left (126 a^{6}-180 a^{4} b^{2}+58 a^{2} b^{4}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \,b^{6} d}+\frac {2 \left (1890 a^{6}-2820 a^{4} b^{2}+958 a^{2} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a \,b^{6} d}+\frac {2 \left (1890 a^{6}-2820 a^{4} b^{2}+958 a^{2} b^{4}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a \,b^{6} d}+\frac {2 \left (3150 a^{6}-4800 a^{4} b^{2}+1634 a^{2} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a \,b^{6} d}+\frac {2 \left (3150 a^{6}-4800 a^{4} b^{2}+1634 a^{2} b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a \,b^{6} d}+\frac {2 \left (6 a^{6}-8 a^{4} b^{2}+2 a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{6} d a}+\frac {2 \left (6 a^{6}-8 a^{4} b^{2}+2 a^{2} b^{4}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{6} d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}+\frac {2 a \left (3 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{7} d}-\frac {2 a \left (3 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{b^{7} d}\) | \(754\) |
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Time = 0.43 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.27 \[ \int \frac {\cos ^5(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {48 \, b^{6} \cos \left (d x + c\right )^{6} + 240 \, a^{6} - 1440 \, a^{4} b^{2} + 1275 \, a^{2} b^{4} - 128 \, b^{6} - 8 \, {\left (15 \, a^{2} b^{4} - 2 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (45 \, a^{4} b^{2} - 45 \, a^{2} b^{4} + 4 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 480 \, {\left (3 \, a^{6} - 4 \, a^{4} b^{2} + a^{2} b^{4} + {\left (3 \, a^{5} b - 4 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (72 \, a b^{5} \cos \left (d x + c\right )^{4} - 1200 \, a^{5} b + 1440 \, a^{3} b^{3} - 293 \, a b^{5} - 16 \, {\left (15 \, a^{3} b^{3} - 11 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}} \]
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Timed out. \[ \int \frac {\cos ^5(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^5(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {30 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )}}{b^{8} \sin \left (d x + c\right ) + a b^{7}} - \frac {6 \, b^{4} \sin \left (d x + c\right )^{5} - 15 \, a b^{3} \sin \left (d x + c\right )^{4} + 10 \, {\left (3 \, a^{2} b^{2} - 2 \, b^{4}\right )} \sin \left (d x + c\right )^{3} - 60 \, {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )^{2} + 30 \, {\left (5 \, a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )}{b^{6}} + \frac {60 \, {\left (3 \, a^{5} - 4 \, a^{3} b^{2} + a b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{7}}}{30 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.29 \[ \int \frac {\cos ^5(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {60 \, {\left (3 \, a^{5} - 4 \, a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{7}} - \frac {30 \, {\left (6 \, a^{5} b \sin \left (d x + c\right ) - 8 \, a^{3} b^{3} \sin \left (d x + c\right ) + 2 \, a b^{5} \sin \left (d x + c\right ) + 5 \, a^{6} - 6 \, a^{4} b^{2} + a^{2} b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{7}} - \frac {6 \, b^{8} \sin \left (d x + c\right )^{5} - 15 \, a b^{7} \sin \left (d x + c\right )^{4} + 30 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} - 20 \, b^{8} \sin \left (d x + c\right )^{3} - 60 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} + 60 \, a b^{7} \sin \left (d x + c\right )^{2} + 150 \, a^{4} b^{4} \sin \left (d x + c\right ) - 180 \, a^{2} b^{6} \sin \left (d x + c\right ) + 30 \, b^{8} \sin \left (d x + c\right )}{b^{10}}}{30 \, d} \]
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Time = 13.00 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.32 \[ \int \frac {\cos ^5(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {a^3}{b^5}+\frac {a\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {2}{3\,b^2}-\frac {a^2}{b^4}\right )}{d}+\frac {\sin \left (c+d\,x\right )\,\left (\frac {1}{b^2}+\frac {a^2\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{b^2}-\frac {2\,a\,\left (\frac {2\,a^3}{b^5}+\frac {2\,a\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{b}\right )}{d}+\frac {{\sin \left (c+d\,x\right )}^5}{5\,b^2\,d}-\frac {a\,{\sin \left (c+d\,x\right )}^4}{2\,b^3\,d}-\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (6\,a^5-8\,a^3\,b^2+2\,a\,b^4\right )}{b^7\,d}-\frac {a^6-2\,a^4\,b^2+a^2\,b^4}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^7+a\,b^6\right )} \]
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