Integrand size = 33, antiderivative size = 148 \[ \int \frac {(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\frac {A b^2 x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {b^2 B \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {A b^2 \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d}-\frac {b^2 B \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}} \]
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Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {17, 2827, 2715, 8, 2713} \[ \int \frac {(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\frac {A b^2 x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {A b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{2 d}-\frac {b^2 B \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}+\frac {b^2 B \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}} \]
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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 ^2(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 ^2(c+d x) \, dx}{\sqrt {\cos (c+d x)}}+\frac {\left (b^2 B \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{\sqrt {\cos (c+d x)}} \\ & = \frac {A b^2 \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac {\left (A b^2 \sqrt {b \cos (c+d x)}\right ) \int 1 \, dx}{2 \sqrt {\cos (c+d x)}}-\frac {\left (b^2 B \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 x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {b^2 B \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {A b^2 \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d}-\frac {b^2 B \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}} \\ \end{align*}
Time = 0.99 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.47 \[ \int \frac {(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\frac {(b \cos (c+d x))^{5/2} (6 A c+6 A d x+9 B \sin (c+d x)+3 A \sin (2 (c+d x))+B \sin (3 (c+d x)))}{12 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Time = 5.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.52
method | result | size |
default | \(\frac {b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (2 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+3 A \sin \left (d x +c \right ) \cos \left (d x +c \right )+3 A \left (d x +c \right )+4 B \sin \left (d x +c \right )\right )}{6 d \sqrt {\cos \left (d x +c \right )}}\) | \(77\) |
parts | \(\frac {A \,b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+d x +c \right )}{2 d \sqrt {\cos \left (d x +c \right )}}+\frac {B \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right ) b}\, \sin \left (d x +c \right )}{3 d \sqrt {\cos \left (d x +c \right )}}\) | \(90\) |
risch | \(\frac {A \,b^{2} x \sqrt {\cos \left (d x +c \right ) b}}{2 \sqrt {\cos \left (d x +c \right )}}+\frac {3 b^{2} B \sin \left (d x +c \right ) \sqrt {\cos \left (d x +c \right ) b}}{4 d \sqrt {\cos \left (d x +c \right )}}+\frac {b^{2} \sqrt {\cos \left (d x +c \right ) b}\, B \sin \left (3 d x +3 c \right )}{12 \sqrt {\cos \left (d x +c \right )}\, d}+\frac {b^{2} \sqrt {\cos \left (d x +c \right ) b}\, A \sin \left (2 d x +2 c \right )}{4 \sqrt {\cos \left (d x +c \right )}\, d}\) | \(132\) |
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Time = 0.35 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.70 \[ \int \frac {(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\left [\frac {3 \, A \sqrt {-b} b^{2} \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) + 2 \, {\left (2 \, B b^{2} \cos \left (d x + c\right )^{2} + 3 \, A b^{2} \cos \left (d x + c\right ) + 4 \, B b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )}, \frac {3 \, A b^{\frac {5}{2}} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right ) \cos \left (d x + c\right ) + {\left (2 \, B b^{2} \cos \left (d x + c\right )^{2} + 3 \, A b^{2} \cos \left (d x + c\right ) + 4 \, B b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )}\right ] \]
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Timed out. \[ \int \frac {(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \]
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Time = 0.48 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.55 \[ \int \frac {(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\frac {3 \, {\left (2 \, {\left (d x + c\right )} b^{2} + b^{2} \sin \left (2 \, d x + 2 \, c\right )\right )} A \sqrt {b} + {\left (b^{2} \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b^{2} \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} B \sqrt {b}}{12 \, d} \]
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\[ \int \frac {(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \]
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Time = 0.76 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.43 \[ \int \frac {(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\frac {b^2\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (9\,B\,\sin \left (c+d\,x\right )+3\,A\,\sin \left (2\,c+2\,d\,x\right )+B\,\sin \left (3\,c+3\,d\,x\right )+6\,A\,d\,x\right )}{12\,d\,\sqrt {\cos \left (c+d\,x\right )}} \]
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