Optimal. Leaf size=66 \[ \frac{A b^2 \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b^2 B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.0263802, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {17, 2735, 3770} \[ \frac{A b^2 \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b^2 B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx &=\frac{\left (b^2 \sqrt{b \cos (c+d x)}\right ) \int (A+B \cos (c+d x)) \sec (c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{b^2 B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{\left (A b^2 \sqrt{b \cos (c+d x)}\right ) \int \sec (c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{b^2 B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{A b^2 \tanh ^{-1}(\sin (c+d x)) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.060486, size = 40, normalized size = 0.61 \[ \frac{(b \cos (c+d x))^{5/2} \left (A \tanh ^{-1}(\sin (c+d x))+B d x\right )}{d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.185, size = 54, normalized size = 0.8 \begin{align*} -{\frac{1}{d} \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( 2\,A{\it Artanh} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -B \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.82097, size = 134, normalized size = 2.03 \begin{align*} \frac{4 \, B b^{\frac{5}{2}} \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) +{\left (b^{2} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - b^{2} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )\right )} A \sqrt{b}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08131, size = 599, normalized size = 9.08 \begin{align*} \left [-\frac{2 \, A \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 ) - B \sqrt{-b} b^{2} \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt{b \cos \left (d x + c\right )} \sqrt{-b} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{2 \, d}, \frac{2 \, B b^{\frac{5}{2}} \arctan \left (\frac{\sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt{b} \cos \left (d x + c\right )^{\frac{3}{2}}}\right ) + A b^{\frac{5}{2}} \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 \, d}\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 (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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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