Optimal. Leaf size=152 \[ \frac{(2 A+3 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.11567, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {17, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{(2 A+3 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sqrt{b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{\sqrt{b \cos (c+d x)} \int \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{A \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{\sqrt{b \cos (c+d x)} \int (3 B+(2 A+3 C) \cos (c+d x)) \sec ^3(c+d x) \, dx}{3 \sqrt{\cos (c+d x)}}\\ &=\frac{A \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{\left (B \sqrt{b \cos (c+d x)}\right ) \int \sec ^3(c+d x) \, dx}{\sqrt{\cos (c+d x)}}+\frac{\left ((2 A+3 C) \sqrt{b \cos (c+d x)}\right ) \int \sec ^2(c+d x) \, dx}{3 \sqrt{\cos (c+d x)}}\\ &=\frac{A \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (B \sqrt{b \cos (c+d x)}\right ) \int \sec (c+d x) \, dx}{2 \sqrt{\cos (c+d x)}}-\frac{\left ((2 A+3 C) \sqrt{b \cos (c+d x)}\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d \sqrt{\cos (c+d x)}}\\ &=\frac{B \tanh ^{-1}(\sin (c+d x)) \sqrt{b \cos (c+d x)}}{2 d \sqrt{\cos (c+d x)}}+\frac{A \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{(2 A+3 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.413408, size = 87, normalized size = 0.57 \[ \frac{\sqrt{b \cos (c+d x)} \left (\tan (c+d x) ((2 A+3 C) \cos (2 (c+d x))+4 A+3 B \cos (c+d x)+3 C)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right )}{6 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.444, size = 157, normalized size = 1. \begin{align*}{\frac{1}{6\,d} \left ( -3\,B\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3\,B\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) -\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+4\,A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+6\,C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,B\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) +2\,A\sin \left ( dx+c \right ) \right ) \sqrt{b\cos \left ( dx+c \right ) } \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.3862, size = 1362, normalized size = 8.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97205, size = 722, normalized size = 4.75 \begin{align*} \left [\frac{3 \, B \sqrt{b} \cos \left (d x + c\right )^{4} \log \left (-\frac{b \cos \left (d x + c\right )^{3} - 2 \, \sqrt{b \cos \left (d x + c\right )} \sqrt{b} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{3}}\right ) + 2 \,{\left (2 \,{\left (2 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 2 \, A\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 )^{4}}, -\frac{3 \, B \sqrt{-b} \arctan \left (\frac{\sqrt{b \cos \left (d x + c\right )} \sqrt{-b} \sin \left (d x + c\right )}{b \sqrt{\cos \left (d x + c\right )}}\right ) \cos \left (d x + c\right )^{4} -{\left (2 \,{\left (2 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 2 \, A\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 )^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt{b \cos \left (d x + c\right )}}{\cos \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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