Integrand size = 21, antiderivative size = 55 \[ \int \frac {\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\sin ^4(c+d x)}{4 a d} \]
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Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2914, 2644, 30, 2645, 14} \[ \int \frac {\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sin ^4(c+d x)}{4 a d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\cos ^3(c+d x)}{3 a d} \]
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Rule 14
Rule 30
Rule 2644
Rule 2645
Rule 2914
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) \sin ^5(c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = \frac {\int \cos (c+d x) \sin ^3(c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^3(c+d x) \, dx}{a} \\ & = \frac {\text {Subst}\left (\int x^3 \, dx,x,\sin (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\sin ^4(c+d x)}{4 a d}+\frac {\text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\sin ^4(c+d x)}{4 a d} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76 \[ \int \frac {\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {2 (13+21 \cos (c+d x)+6 \cos (2 (c+d x))) \sin ^6\left (\frac {1}{2} (c+d x)\right )}{15 a d} \]
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Time = 0.53 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {-\frac {\cos \left (d x +c \right )^{5}}{5}+\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {\cos \left (d x +c \right )^{2}}{2}}{d a}\) | \(49\) |
default | \(\frac {-\frac {\cos \left (d x +c \right )^{5}}{5}+\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {\cos \left (d x +c \right )^{2}}{2}}{d a}\) | \(49\) |
parallelrisch | \(\frac {-60 \cos \left (2 d x +2 c \right )+109+15 \cos \left (4 d x +4 c \right )+60 \cos \left (d x +c \right )+10 \cos \left (3 d x +3 c \right )-6 \cos \left (5 d x +5 c \right )}{480 d a}\) | \(63\) |
norman | \(\frac {\frac {4}{15 a d}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d a}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d a}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}\) | \(83\) |
risch | \(\frac {\cos \left (d x +c \right )}{8 a d}-\frac {\cos \left (5 d x +5 c \right )}{80 a d}+\frac {\cos \left (4 d x +4 c \right )}{32 a d}+\frac {\cos \left (3 d x +3 c \right )}{48 a d}-\frac {\cos \left (2 d x +2 c \right )}{8 a d}\) | \(84\) |
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {12 \, \cos \left (d x + c\right )^{5} - 15 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right )^{2}}{60 \, a d} \]
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Timed out. \[ \int \frac {\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {12 \, \cos \left (d x + c\right )^{5} - 15 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right )^{2}}{60 \, a d} \]
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Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.76 \[ \int \frac {\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {4 \, {\left (\frac {5 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {10 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {30 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )}}{15 \, a d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}} \]
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Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.05 \[ \int \frac {\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^2}{2\,a}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a}-\frac {{\cos \left (c+d\,x\right )}^4}{4\,a}+\frac {{\cos \left (c+d\,x\right )}^5}{5\,a}}{d} \]
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