Optimal. Leaf size=120 \[ \frac{b^2 (A+2 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.0969051, antiderivative size = 120, 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 = {17, 3021, 2748, 3767, 8, 3770} \[ \frac{b^2 (A+2 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3021
Rule 2748
Rule 3767
Rule 8
Rule 3770
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 )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{\left (b^2 \sqrt{b \cos (c+d x)}\right ) \int \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{A b^2 \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (b^2 \sqrt{b \cos (c+d x)}\right ) \int (2 B+(A+2 C) \cos (c+d x)) \sec ^2(c+d x) \, dx}{2 \sqrt{\cos (c+d x)}}\\ &=\frac{A b^2 \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (b^2 B \sqrt{b \cos (c+d x)}\right ) \int \sec ^2(c+d x) \, dx}{\sqrt{\cos (c+d x)}}+\frac{\left (b^2 (A+2 C) \sqrt{b \cos (c+d x)}\right ) \int \sec (c+d x) \, dx}{2 \sqrt{\cos (c+d x)}}\\ &=\frac{b^2 (A+2 C) \tanh ^{-1}(\sin (c+d x)) \sqrt{b \cos (c+d x)}}{2 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{\left (b^2 B \sqrt{b \cos (c+d x)}\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d \sqrt{\cos (c+d x)}}\\ &=\frac{b^2 (A+2 C) \tanh ^{-1}(\sin (c+d x)) \sqrt{b \cos (c+d x)}}{2 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b^2 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.138371, size = 69, normalized size = 0.57 \[ \frac{(b \cos (c+d x))^{5/2} \left (\sin (c+d x) (A+2 B \cos (c+d x))+(A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right )}{2 d \cos ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.262, size = 151, normalized size = 1.3 \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 ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{5}{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 = 2.37679, size = 1179, normalized size = 9.82 \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 = 2.02197, size = 675, normalized size = 5.62 \begin{align*} \left [\frac{{\left (A + 2 \, C\right )} b^{\frac{5}{2}} \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 b^{2} \cos \left (d x + c\right ) + A b^{2}\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{3}}, -\frac{{\left (A + 2 \, C\right )} \sqrt{-b} b^{2} \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 b^{2} \cos \left (d x + c\right ) + A b^{2}\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, 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{{\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}}}{\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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