Integrand size = 33, antiderivative size = 187 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {3 b^2 B x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {A b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {3 b^2 B \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {b^2 B \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}-\frac {A b^2 \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}} \]
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Time = 0.08 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {17, 2827, 2713, 2715, 8} \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=-\frac {A b^2 \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}+\frac {A b^2 \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}+\frac {3 b^2 B x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {b^2 B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{4 d}+\frac {3 b^2 B \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{8 d} \]
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Rule 8
Rule 17
Rule 2713
Rule 2715
Rule 2827
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) (A+B \cos (c+d x)) \, dx}{\sqrt {\cos (c+d x)}} \\ & = \frac {\left (A b^2 \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{\sqrt {\cos (c+d x)}}+\frac {\left (b^2 B \sqrt {b \cos (c+d x)}\right ) \int \cos ^4(c+d x) \, dx}{\sqrt {\cos (c+d x)}} \\ & = \frac {b^2 B \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {\left (3 b^2 B \sqrt {b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 \sqrt {\cos (c+d x)}}-\frac {\left (A b^2 \sqrt {b \cos (c+d x)}\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {\cos (c+d x)}} \\ & = \frac {A b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {3 b^2 B \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {b^2 B \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}-\frac {A b^2 \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {\left (3 b^2 B \sqrt {b \cos (c+d x)}\right ) \int 1 \, dx}{8 \sqrt {\cos (c+d x)}} \\ & = \frac {3 b^2 B x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {A b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {3 b^2 B \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {b^2 B \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}-\frac {A b^2 \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.43 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {(b \cos (c+d x))^{5/2} (36 B c+36 B d x+72 A \sin (c+d x)+24 B \sin (2 (c+d x))+8 A \sin (3 (c+d x))+3 B \sin (4 (c+d x)))}{96 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Time = 5.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.50
method | result | size |
default | \(\frac {b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (6 B \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+8 A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+9 B \sin \left (d x +c \right ) \cos \left (d x +c \right )+16 A \sin \left (d x +c \right )+9 B \left (d x +c \right )\right )}{24 d \sqrt {\cos \left (d x +c \right )}}\) | \(94\) |
parts | \(\frac {A \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right ) b}\, \sin \left (d x +c \right )}{3 d \sqrt {\cos \left (d x +c \right )}}+\frac {B \,b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (2 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )+3 d x +3 c \right )}{8 d \sqrt {\cos \left (d x +c \right )}}\) | \(110\) |
risch | \(\frac {3 b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{i \left (d x +c \right )} x B}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{5 i \left (d x +c \right )} B}{32 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{4 i \left (d x +c \right )} A}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {3 i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{2 i \left (d x +c \right )} A}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {3 i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) A}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{-i \left (d x +c \right )} B}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{-2 i \left (d x +c \right )} A}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {7 i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) B \cos \left (3 d x +3 c \right )}{32 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {9 b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) B \sin \left (3 d x +3 c \right )}{32 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}\) | \(439\) |
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Time = 0.36 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.49 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\left [\frac {9 \, B \sqrt {-b} b^{2} \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) + 2 \, {\left (6 \, B b^{2} \cos \left (d x + c\right )^{3} + 8 \, A b^{2} \cos \left (d x + c\right )^{2} + 9 \, B b^{2} \cos \left (d x + c\right ) + 16 \, A b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )}, \frac {9 \, B b^{\frac {5}{2}} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right ) \cos \left (d x + c\right ) + {\left (6 \, B b^{2} \cos \left (d x + c\right )^{3} + 8 \, A b^{2} \cos \left (d x + c\right )^{2} + 9 \, B b^{2} \cos \left (d x + c\right ) + 16 \, A b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )}\right ] \]
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Timed out. \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.51 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.59 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {8 \, {\left (b^{2} \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b^{2} \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} A \sqrt {b} + 3 \, {\left (12 \, {\left (d x + c\right )} b^{2} + b^{2} \sin \left (4 \, d x + 4 \, c\right ) + 8 \, b^{2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} B \sqrt {b}}{96 \, d} \]
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Time = 3.70 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.49 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {9 \, B b^{\frac {5}{2}} d x \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 36 \, B b^{\frac {5}{2}} d x \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 48 \, A b^{\frac {5}{2}} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, B b^{\frac {5}{2}} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 54 \, B b^{\frac {5}{2}} d x \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 80 \, A b^{\frac {5}{2}} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B b^{\frac {5}{2}} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, B b^{\frac {5}{2}} d x \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 80 \, A b^{\frac {5}{2}} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 18 \, B b^{\frac {5}{2}} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, B b^{\frac {5}{2}} d x + 48 \, A b^{\frac {5}{2}} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 30 \, B b^{\frac {5}{2}} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{24 \, {\left (d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 4 \, d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6 \, d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4 \, d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + d\right )}} \]
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Time = 15.84 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.58 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (24\,B\,\sin \left (c+d\,x\right )+80\,A\,\sin \left (2\,c+2\,d\,x\right )+8\,A\,\sin \left (4\,c+4\,d\,x\right )+27\,B\,\sin \left (3\,c+3\,d\,x\right )+3\,B\,\sin \left (5\,c+5\,d\,x\right )+72\,B\,d\,x\,\cos \left (c+d\,x\right )\right )}{96\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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