Integrand size = 35, antiderivative size = 125 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {b^2 (A+C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {b^2 (A+2 C) \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {b^2 C \sqrt {b \cos (c+d x)} \sin ^5(c+d x)}{5 d \sqrt {\cos (c+d x)}} \]
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Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {17, 3092, 380} \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {b^2 (A+2 C) \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}+\frac {b^2 (A+C) \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}+\frac {b^2 C \sin ^5(c+d x) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}} \]
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Rule 17
Rule 380
Rule 3092
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{\sqrt {\cos (c+d x)}} \\ & = -\frac {\left (b^2 \sqrt {b \cos (c+d x)}\right ) \text {Subst}\left (\int \left (1-x^2\right ) \left (A+C-C x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {\cos (c+d x)}} \\ & = -\frac {\left (b^2 \sqrt {b \cos (c+d x)}\right ) \text {Subst}\left (\int \left (A \left (1+\frac {C}{A}\right )-(A+2 C) x^2+C x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {\cos (c+d x)}} \\ & = \frac {b^2 (A+C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {b^2 (A+2 C) \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {b^2 C \sqrt {b \cos (c+d x)} \sin ^5(c+d x)}{5 d \sqrt {\cos (c+d x)}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.56 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {(b \cos (c+d x))^{5/2} (100 A+89 C+4 (5 A+7 C) \cos (2 (c+d x))+3 C \cos (4 (c+d x))) \sin (c+d x)}{120 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Time = 8.43 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.58
method | result | size |
default | \(\frac {b^{2} \left (3 C \left (\cos ^{4}\left (d x +c \right )\right )+5 A \left (\cos ^{2}\left (d x +c \right )\right )+4 C \left (\cos ^{2}\left (d x +c \right )\right )+10 A +8 C \right ) \sin \left (d x +c \right ) \sqrt {\cos \left (d x +c \right ) b}}{15 d \sqrt {\cos \left (d x +c \right )}}\) | \(73\) |
parts | \(\frac {A \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\cos \left (d x +c \right ) b}}{3 d \sqrt {\cos \left (d x +c \right )}}+\frac {C \,b^{2} \left (3 \left (\cos ^{4}\left (d x +c \right )\right )+4 \left (\cos ^{2}\left (d x +c \right )\right )+8\right ) \sin \left (d x +c \right ) \sqrt {\cos \left (d x +c \right ) b}}{15 d \sqrt {\cos \left (d x +c \right )}}\) | \(100\) |
risch | \(-\frac {i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{6 i \left (d x +c \right )} C}{80 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{2 i \left (d x +c \right )} \left (6 A +5 C \right )}{8 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (6 A +5 C \right )}{8 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{-2 i \left (d x +c \right )} \left (4 A +5 C \right )}{48 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {i b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (10 A +11 C \right ) \cos \left (4 d x +4 c \right )}{120 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {b^{2} \sqrt {\cos \left (d x +c \right ) b}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (5 A +7 C \right ) \sin \left (4 d x +4 c \right )}{60 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}\) | \(322\) |
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Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {{\left (3 \, C b^{2} \cos \left (d x + c\right )^{4} + {\left (5 \, A + 4 \, C\right )} b^{2} \cos \left (d x + c\right )^{2} + 2 \, {\left (5 \, A + 4 \, C\right )} b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, d \sqrt {\cos \left (d x + c\right )}} \]
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Timed out. \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.57 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {20 \, {\left (b^{2} \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b^{2} \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} A \sqrt {b} + {\left (3 \, b^{2} \sin \left (5 \, d x + 5 \, c\right ) + 25 \, b^{2} \sin \left (\frac {3}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right ) + 150 \, b^{2} \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right )\right )} C \sqrt {b}}{240 \, d} \]
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Timed out. \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 2.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.80 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (200\,A\,\sin \left (2\,c+2\,d\,x\right )+20\,A\,\sin \left (4\,c+4\,d\,x\right )+175\,C\,\sin \left (2\,c+2\,d\,x\right )+28\,C\,\sin \left (4\,c+4\,d\,x\right )+3\,C\,\sin \left (6\,c+6\,d\,x\right )\right )}{240\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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