Optimal. Leaf size=33 \[ \frac{b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.012162, antiderivative size = 33, 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 = {17, 3767, 8} \[ \frac{b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{(b \cos (c+d x))^{3/2}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx &=\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \int \sec ^2(c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=-\frac{\left (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 \sqrt{b \cos (c+d x)} \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.0234374, size = 32, normalized size = 0.97 \[ \frac{\sin (c+d x) (b \cos (c+d x))^{3/2}}{d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.80648, size = 73, normalized size = 2.21 \begin{align*} \frac{2 \, b^{\frac{3}{2}} \sin \left (2 \, d x + 2 \, c\right )}{{\left (\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\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61215, size = 81, normalized size = 2.45 \begin{align*} \frac{\sqrt{b \cos \left (d x + c\right )} b \sin \left (d x + c\right )}{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{\left (b \cos \left (d x + c\right )\right )^{\frac{3}{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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