Integrand size = 25, antiderivative size = 121 \[ \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 a^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {6 a^2 \sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {12 a^2 \sin (c+d x)}{5 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \]
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Time = 0.24 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2841, 21, 2851, 2850} \[ \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {6 a^2 \sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {12 a^2 \sin (c+d x)}{5 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}} \]
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Rule 21
Rule 2841
Rule 2850
Rule 2851
Rubi steps \begin{align*} \text {integral}& = \frac {2 a^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}-\frac {1}{5} (2 a) \int \frac {-\frac {9 a}{2}-\frac {9}{2} a \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx \\ & = \frac {2 a^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {1}{5} (9 a) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {6 a^2 \sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {1}{5} (6 a) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {6 a^2 \sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {12 a^2 \sin (c+d x)}{5 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.51 \[ \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 a \sqrt {a (1+\cos (c+d x))} (4+3 \cos (c+d x)+3 \cos (2 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Time = 4.95 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52
method | result | size |
default | \(\frac {2 \sin \left (d x +c \right ) \left (6 \left (\cos ^{2}\left (d x +c \right )\right )+3 \cos \left (d x +c \right )+1\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a}{5 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {5}{2}}}\) | \(63\) |
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Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.60 \[ \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \, {\left (6 \, a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{5 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}} \]
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Timed out. \[ \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (103) = 206\).
Time = 0.35 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.79 \[ \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {4 \, {\left (\frac {5 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {7 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {2 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{5 \, d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}} \]
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Timed out. \[ \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Time = 16.40 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {4\,a\,\sqrt {a\,\left (\cos \left (c+d\,x\right )+1\right )}\,\left (8\,\sin \left (c+d\,x\right )+6\,\sin \left (2\,c+2\,d\,x\right )+11\,\sin \left (3\,c+3\,d\,x\right )+3\,\sin \left (4\,c+4\,d\,x\right )+3\,\sin \left (5\,c+5\,d\,x\right )\right )}{5\,d\,\sqrt {\cos \left (c+d\,x\right )}\,\left (10\,\cos \left (c+d\,x\right )+8\,\cos \left (2\,c+2\,d\,x\right )+5\,\cos \left (3\,c+3\,d\,x\right )+2\,\cos \left (4\,c+4\,d\,x\right )+\cos \left (5\,c+5\,d\,x\right )+6\right )} \]
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