Optimal. Leaf size=66 \[ \frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0266115, 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} \[ \frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int (A+B \cos (c+d x)) \sec (c+d x) \, dx}{b \sqrt{b \cos (c+d x)}}\\ &=\frac{B x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{\left (A \sqrt{\cos (c+d x)}\right ) \int \sec (c+d x) \, dx}{b \sqrt{b \cos (c+d x)}}\\ &=\frac{B x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{A \tanh ^{-1}(\sin (c+d x)) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0381425, size = 40, normalized size = 0.61 \[ \frac{\cos ^{\frac{3}{2}}(c+d x) \left (A \tanh ^{-1}(\sin (c+d x))+B d x\right )}{d (b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.309, size = 54, normalized size = 0.8 \begin{align*} -{\frac{1}{d} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{3}{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 ( b\cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.87296, size = 124, normalized size = 1.88 \begin{align*} \frac{\frac{A{\left (\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 ) - \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 )}}{b^{\frac{3}{2}}} + \frac{4 \, B \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{b^{\frac{3}{2}}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89476, size = 605, normalized size = 9.17 \begin{align*} \left [-\frac{2 \, A \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 ) + B \sqrt{-b} \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 \, b^{2} d}, \frac{2 \, B \sqrt{b} \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 \sqrt{b} \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 \, b^{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 )} \sqrt{\cos \left (d x + c\right )}}{\left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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