Integrand size = 21, antiderivative size = 102 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {4 \cos (c+d x)}{a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {4 \log (1+\cos (c+d x))}{a^3 d} \]
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Time = 0.23 (sec) , antiderivative size = 102, 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, 2915, 12, 90} \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {4 \log (\cos (c+d x)+1)}{a^3 d} \]
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos ^3(c+d x) \sin ^5(c+d x)}{(-a-a \cos (c+d x))^3} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^2 x^3}{a^3 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^2 x^3}{-a+x} \, dx,x,-a \cos (c+d x)\right )}{a^8 d} \\ & = \frac {\text {Subst}\left (\int \left (4 a^4-\frac {4 a^5}{a-x}+4 a^3 x+4 a^2 x^2+3 a x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^8 d} \\ & = -\frac {4 \cos (c+d x)}{a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {4 \log (1+\cos (c+d x))}{a^3 d} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3857-4920 \cos (c+d x)+1320 \cos (2 (c+d x))-380 \cos (3 (c+d x))+90 \cos (4 (c+d x))-12 \cos (5 (c+d x))+7680 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{960 a^3 d} \]
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Time = 1.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {-\frac {\cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4}}{4}-\frac {4 \cos \left (d x +c \right )^{3}}{3}+2 \cos \left (d x +c \right )^{2}-4 \cos \left (d x +c \right )+4 \ln \left (\cos \left (d x +c \right )+1\right )}{d \,a^{3}}\) | \(68\) |
default | \(\frac {-\frac {\cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4}}{4}-\frac {4 \cos \left (d x +c \right )^{3}}{3}+2 \cos \left (d x +c \right )^{2}-4 \cos \left (d x +c \right )+4 \ln \left (\cos \left (d x +c \right )+1\right )}{d \,a^{3}}\) | \(68\) |
parallelrisch | \(\frac {-1920 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-3361-6 \cos \left (5 d x +5 c \right )+45 \cos \left (4 d x +4 c \right )-190 \cos \left (3 d x +3 c \right )+660 \cos \left (2 d x +2 c \right )-2460 \cos \left (d x +c \right )}{480 a^{3} d}\) | \(77\) |
norman | \(\frac {-\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d a}-\frac {166}{15 a d}-\frac {78 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d a}-\frac {154 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d a}-\frac {278 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} a^{2}}-\frac {4 \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a^{3} d}\) | \(128\) |
risch | \(-\frac {4 i x}{a^{3}}-\frac {41 \,{\mathrm e}^{i \left (d x +c \right )}}{16 a^{3} d}-\frac {41 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 a^{3} d}-\frac {8 i c}{a^{3} d}+\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{3} d}-\frac {\cos \left (5 d x +5 c \right )}{80 d \,a^{3}}+\frac {3 \cos \left (4 d x +4 c \right )}{32 d \,a^{3}}-\frac {19 \cos \left (3 d x +3 c \right )}{48 d \,a^{3}}+\frac {11 \cos \left (2 d x +2 c \right )}{8 d \,a^{3}}\) | \(141\) |
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Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {12 \, \cos \left (d x + c\right )^{5} - 45 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} - 120 \, \cos \left (d x + c\right )^{2} + 240 \, \cos \left (d x + c\right ) - 240 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{60 \, a^{3} d} \]
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Timed out. \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {12 \, \cos \left (d x + c\right )^{5} - 45 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} - 120 \, \cos \left (d x + c\right )^{2} + 240 \, \cos \left (d x + c\right )}{a^{3}} - \frac {240 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}}}{60 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.69 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {60 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac {\frac {85 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {20 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {200 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {205 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {137 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 29}{a^{3} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{15 \, d} \]
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Time = 13.36 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {4\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{a^3}-\frac {4\,\cos \left (c+d\,x\right )}{a^3}+\frac {2\,{\cos \left (c+d\,x\right )}^2}{a^3}-\frac {4\,{\cos \left (c+d\,x\right )}^3}{3\,a^3}+\frac {3\,{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3}}{d} \]
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