Optimal. Leaf size=199 \[ \frac {b^2 x (4 A+3 C) \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {b^2 (4 A+3 C) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{8 d}-\frac {b^2 B \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}+\frac {b^2 B \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}+\frac {b^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{4 d} \]
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Rubi [A] time = 0.11, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {17, 3023, 2748, 2635, 8, 2633} \[ \frac {b^2 x (4 A+3 C) \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {b^2 (4 A+3 C) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{8 d}-\frac {b^2 B \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}+\frac {b^2 B \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}+\frac {b^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 2633
Rule 2635
Rule 2748
Rule 3023
Rubi steps
\begin {align*} \int \frac {(b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx &=\frac {\left (b^2 \sqrt {b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{\sqrt {\cos (c+d x)}}\\ &=\frac {b^2 C \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {\left (b^2 \sqrt {b \cos (c+d x)}\right ) \int \cos ^2(c+d x) (4 A+3 C+4 B \cos (c+d x)) \, dx}{4 \sqrt {\cos (c+d x)}}\\ &=\frac {b^2 C \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\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 {\left (b^2 (4 A+3 C) \sqrt {b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 \sqrt {\cos (c+d x)}}\\ &=\frac {b^2 (4 A+3 C) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {b^2 C \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {\left (b^2 (4 A+3 C) \sqrt {b \cos (c+d x)}\right ) \int 1 \, dx}{8 \sqrt {\cos (c+d x)}}-\frac {\left (b^2 B \sqrt {b \cos (c+d x)}\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {\cos (c+d x)}}\\ &=\frac {b^2 (4 A+3 C) x \sqrt {b \cos (c+d x)}}{8 \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 {b^2 (4 A+3 C) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {b^2 C \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 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*}
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Mathematica [A] time = 0.30, size = 92, normalized size = 0.46 \[ \frac {(b \cos (c+d x))^{5/2} (24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x+72 B \sin (c+d x)+8 B \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 c C+36 C d x)}{96 d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.84, size = 303, normalized size = 1.52 \[ \left [\frac {3 \, {\left (4 \, A + 3 \, C\right )} \sqrt {-b} b^{2} \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) + 2 \, {\left (6 \, C b^{2} \cos \left (d x + c\right )^{3} + 8 \, B b^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + 3 \, C\right )} b^{2} \cos \left (d x + c\right ) + 16 \, B b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )}, \frac {3 \, {\left (4 \, A + 3 \, C\right )} b^{\frac {5}{2}} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right ) \cos \left (d x + c\right ) + {\left (6 \, C b^{2} \cos \left (d x + c\right )^{3} + 8 \, B b^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + 3 \, C\right )} b^{2} \cos \left (d x + c\right ) + 16 \, B b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}}{\sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 114, normalized size = 0.57 \[ \frac {\left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \left (6 C \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+8 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+12 A \cos \left (d x +c \right ) \sin \left (d x +c \right )+9 C \sin \left (d x +c \right ) \cos \left (d x +c \right )+12 A \left (d x +c \right )+16 B \sin \left (d x +c \right )+9 C \left (d x +c \right )\right )}{24 d \cos \left (d x +c \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 140, normalized size = 0.70 \[ \frac {24 \, {\left (2 \, {\left (d x + c\right )} b^{2} + b^{2} \sin \left (2 \, d x + 2 \, c\right )\right )} A \sqrt {b} + 8 \, {\left (b^{2} \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b^{2} \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} B \sqrt {b} + 3 \, {\left (12 \, {\left (d x + c\right )} b^{2} + b^{2} \sin \left (4 \, d x + 4 \, c\right ) + 8 \, b^{2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} C \sqrt {b}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 94, normalized size = 0.47 \[ \frac {b^2\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (72\,B\,\sin \left (c+d\,x\right )+24\,A\,\sin \left (2\,c+2\,d\,x\right )+8\,B\,\sin \left (3\,c+3\,d\,x\right )+24\,C\,\sin \left (2\,c+2\,d\,x\right )+3\,C\,\sin \left (4\,c+4\,d\,x\right )+48\,A\,d\,x+36\,C\,d\,x\right )}{96\,d\,\sqrt {\cos \left (c+d\,x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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