Integrand size = 21, antiderivative size = 73 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {x}{8 a}-\frac {\cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\sin ^3(c+d x)}{3 a d} \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2918, 2644, 30, 2648, 2715, 8} \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sin ^3(c+d x)}{3 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac {\sin (c+d x) \cos (c+d x)}{8 a d}-\frac {x}{8 a} \]
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Rule 8
Rule 30
Rule 2644
Rule 2648
Rule 2715
Rule 2918
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) \sin ^4(c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = \frac {\int \cos (c+d x) \sin ^2(c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{a} \\ & = \frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \cos ^2(c+d x) \, dx}{4 a}+\frac {\text {Subst}\left (\int x^2 \, dx,x,\sin (c+d x)\right )}{a d} \\ & = -\frac {\cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\sin ^3(c+d x)}{3 a d}-\frac {\int 1 \, dx}{8 a} \\ & = -\frac {x}{8 a}-\frac {\cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\sin ^3(c+d x)}{3 a d} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (24 \sin (c+d x)-8 \sin (3 (c+d x))+3 \left (4 c-4 d x+\sin (4 (c+d x))-4 \tan \left (\frac {c}{2}\right )\right )\right )}{48 a d (1+\sec (c+d x))} \]
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Time = 0.60 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(\frac {-12 d x +24 \sin \left (d x +c \right )-8 \sin \left (3 d x +3 c \right )+3 \sin \left (4 d x +4 c \right )}{96 d a}\) | \(44\) |
risch | \(-\frac {x}{8 a}+\frac {\sin \left (d x +c \right )}{4 a d}+\frac {\sin \left (4 d x +4 c \right )}{32 d a}-\frac {\sin \left (3 d x +3 c \right )}{12 d a}\) | \(56\) |
derivativedivides | \(\frac {-\frac {16 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{64}-\frac {53 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(90\) |
default | \(\frac {-\frac {16 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{64}-\frac {53 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(90\) |
norman | \(\frac {-\frac {x}{8 a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 a d}+\frac {53 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 a d}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\) | \(166\) |
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Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {3 \, d x - {\left (6 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{24 \, a d} \]
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\[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sin ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (65) = 130\).
Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.68 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {53 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{12 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 53 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]
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Time = 14.00 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \frac {\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sin \left (c+d\,x\right )}{4\,a\,d}-\frac {x}{8\,a}-\frac {\sin \left (3\,c+3\,d\,x\right )}{12\,a\,d}+\frac {\sin \left (4\,c+4\,d\,x\right )}{32\,a\,d} \]
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