Optimal. Leaf size=32 \[ \frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0122905, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {18, 3767, 8} \[ \frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 18
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \sec ^2(c+d x) \, dx}{\sqrt{b \cos (c+d x)}}\\ &=-\frac{\sqrt{\cos (c+d x)} \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d \sqrt{b \cos (c+d x)}}\\ &=\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0228144, size = 32, normalized size = 1. \[ \frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.238, size = 29, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d}{\frac{1}{\sqrt{\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{b\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.7695, size = 80, normalized size = 2.5 \begin{align*} \frac{2 \, \sqrt{b} \sin \left (2 \, d x + 2 \, c\right )}{{\left (b \cos \left (2 \, d x + 2 \, c\right )^{2} + b \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81807, size = 81, normalized size = 2.53 \begin{align*} \frac{\sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{b d \cos \left (d x + c\right )^{\frac{3}{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{1}{\sqrt{b \cos \left (d x + c\right )} \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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