Optimal. Leaf size=85 \[ \frac{(2 A+3 C) \sin (c+d x)}{3 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0469938, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {18, 3012, 3767, 8} \[ \frac{(2 A+3 C) \sin (c+d x)}{3 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 18
Rule 3012
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{5/2}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx}{b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{A \sin (c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{\left ((2 A+3 C) \sqrt{\cos (c+d x)}\right ) \int \sec ^2(c+d x) \, dx}{3 b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{A \sin (c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}-\frac{\left ((2 A+3 C) \sqrt{\cos (c+d x)}\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 b^2 d \sqrt{b \cos (c+d x)}}\\ &=\frac{A \sin (c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{(2 A+3 C) \sin (c+d x)}{3 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.111845, size = 51, normalized size = 0.6 \[ \frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (A \tan ^2(c+d x)+3 (A+C)\right )}{3 d (b \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.256, size = 54, normalized size = 0.6 \begin{align*}{\frac{\sin \left ( dx+c \right ) \left ( 2\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+A \right ) }{3\,d} \left ( b\cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.93057, size = 556, normalized size = 6.54 \begin{align*} \frac{2 \,{\left (\frac{3 \, C \sqrt{b} \sin \left (2 \, d x + 2 \, c\right )}{b^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} + b^{3} \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, b^{3} \cos \left (2 \, d x + 2 \, c\right ) + b^{3}} + \frac{2 \,{\left ({\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (6 \, d x + 6 \, c\right ) + 3 \,{\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (4 \, d x + 4 \, c\right ) - 3 \, \cos \left (6 \, d x + 6 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) - 9 \, \cos \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right )\right )} A}{{\left (b^{2} \cos \left (6 \, d x + 6 \, c\right )^{2} + 9 \, b^{2} \cos \left (4 \, d x + 4 \, c\right )^{2} + 9 \, b^{2} \cos \left (2 \, d x + 2 \, c\right )^{2} + b^{2} \sin \left (6 \, d x + 6 \, c\right )^{2} + 9 \, b^{2} \sin \left (4 \, d x + 4 \, c\right )^{2} + 18 \, b^{2} \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, b^{2} \sin \left (2 \, d x + 2 \, c\right )^{2} + 6 \, b^{2} \cos \left (2 \, d x + 2 \, c\right ) + b^{2} + 2 \,{\left (3 \, b^{2} \cos \left (4 \, d x + 4 \, c\right ) + 3 \, b^{2} \cos \left (2 \, d x + 2 \, c\right ) + b^{2}\right )} \cos \left (6 \, d x + 6 \, c\right ) + 6 \,{\left (3 \, b^{2} \cos \left (2 \, d x + 2 \, c\right ) + b^{2}\right )} \cos \left (4 \, d x + 4 \, c\right ) + 6 \,{\left (b^{2} \sin \left (4 \, d x + 4 \, c\right ) + b^{2} \sin \left (2 \, d x + 2 \, c\right )\right )} \sin \left (6 \, d x + 6 \, c\right )\right )} \sqrt{b}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34058, size = 134, normalized size = 1.58 \begin{align*} \frac{{\left ({\left (2 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{2} + A\right )} \sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{3 \, b^{3} d \cos \left (d x + c\right )^{\frac{7}{2}}} \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} + A}{\left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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