Optimal. Leaf size=35 \[ \frac{2 C \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{d}+\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d} \]
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Rubi [A] time = 0.0613801, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4064, 2748, 2641, 2639} \[ \frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 C F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4064
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \frac{C+B \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=B \int \sqrt{\cos (c+d x)} \, dx+C \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 C F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0637345, size = 35, normalized size = 1. \[ \frac{2 C \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{d}+\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.767, size = 152, normalized size = 4.3 \begin{align*} 2\,{\frac{\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1} \left ( B{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -C{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) }{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) \sec \left (d x + c\right )\right )} \sqrt{\cos \left (d x + c\right )}, x\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{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\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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