Optimal. Leaf size=81 \[ \frac{b (2 A+3 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.0476186, antiderivative size = 81, 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 = {17, 3012, 3767, 8} \[ \frac{b (2 A+3 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3012
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{(b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \int \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{A b \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{\left (b (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 b \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{\left (b (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{A b \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{b (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 = 0.142715, size = 52, normalized size = 0.64 \[ \frac{b \sin (c+d x) \sqrt{b \cos (c+d x)} \left (A \tan ^2(c+d x)+3 (A+C)\right )}{3 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.262, size = 54, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+A \right ) \sin \left ( dx+c \right ) }{3\,d} \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.08734, size = 479, normalized size = 5.91 \begin{align*} \frac{2 \,{\left (\frac{3 \, C b^{\frac{3}{2}} \sin \left (2 \, d x + 2 \, c\right )}{\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1} - \frac{2 \,{\left (3 \, b \cos \left (6 \, d x + 6 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, b \cos \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) -{\left (3 \, b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sin \left (6 \, d x + 6 \, c\right ) - 3 \,{\left (3 \, b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sin \left (4 \, d x + 4 \, c\right )\right )} A \sqrt{b}}{2 \,{\left (3 \, \cos \left (4 \, d x + 4 \, c\right ) + 3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (6 \, d x + 6 \, c\right ) + \cos \left (6 \, d x + 6 \, c\right )^{2} + 6 \,{\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (4 \, d x + 4 \, c\right ) + 9 \, \cos \left (4 \, d x + 4 \, c\right )^{2} + 9 \, \cos \left (2 \, d x + 2 \, c\right )^{2} + 6 \,{\left (\sin \left (4 \, d x + 4 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right )\right )} \sin \left (6 \, d x + 6 \, c\right ) + \sin \left (6 \, d x + 6 \, c\right )^{2} + 9 \, \sin \left (4 \, d x + 4 \, c\right )^{2} + 18 \, \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, \sin \left (2 \, d x + 2 \, c\right )^{2} + 6 \, \cos \left (2 \, d x + 2 \, c\right ) + 1}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35158, size = 134, normalized size = 1.65 \begin{align*} \frac{{\left ({\left (2 \, A + 3 \, C\right )} b \cos \left (d x + c\right )^{2} + A b\right )} \sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{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{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}}}{\cos \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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