Optimal. Leaf size=111 \[ \frac{(A+2 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0951999, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {18, 3021, 2748, 3767, 8, 3770} \[ \frac{(A+2 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 18
Rule 3021
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx}{\sqrt{b \cos (c+d x)}}\\ &=\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \int (2 B+(A+2 C) \cos (c+d x)) \sec ^2(c+d x) \, dx}{2 \sqrt{b \cos (c+d x)}}\\ &=\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{\left (B \sqrt{\cos (c+d x)}\right ) \int \sec ^2(c+d x) \, dx}{\sqrt{b \cos (c+d x)}}+\frac{\left ((A+2 C) \sqrt{\cos (c+d x)}\right ) \int \sec (c+d x) \, dx}{2 \sqrt{b \cos (c+d x)}}\\ &=\frac{(A+2 C) \tanh ^{-1}(\sin (c+d x)) \sqrt{\cos (c+d x)}}{2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}-\frac{\left (B \sqrt{\cos (c+d x)}\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d \sqrt{b \cos (c+d x)}}\\ &=\frac{(A+2 C) \tanh ^{-1}(\sin (c+d x)) \sqrt{\cos (c+d x)}}{2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.088331, size = 69, normalized size = 0.62 \[ \frac{\sin (c+d x) (A+2 B \cos (c+d x))+(A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.448, size = 150, normalized size = 1.4 \begin{align*}{\frac{1}{2\,d} \left ( A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) -\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -4\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}{\it Artanh} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +2\,B\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) +A\sin \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{b\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.32784, size = 1060, normalized size = 9.55 \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 = 2.06991, size = 653, normalized size = 5.88 \begin{align*} \left [\frac{{\left (A + 2 \, C\right )} \sqrt{b} \cos \left (d x + c\right )^{3} \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 \, B \cos \left (d x + c\right ) + A\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{4 \, b d \cos \left (d x + c\right )^{3}}, -\frac{{\left (A + 2 \, C\right )} \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 )^{3} -{\left (2 \, B \cos \left (d x + c\right ) + A\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, b d \cos \left (d x + c\right )^{3}}\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{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{\sqrt{b \cos \left (d x + c\right )} \cos \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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