Optimal. Leaf size=57 \[ \frac{2 B \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{d}-\frac{2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 C \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.0695075, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4064, 2748, 2636, 2639, 2641} \[ \frac{2 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}-\frac{2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 C \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4064
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\cos (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)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=B \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+C \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-C \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.139659, size = 51, normalized size = 0.89 \[ \frac{2 \left (B \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{C \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.441, size = 148, normalized size = 2.6 \begin{align*} -2\,{\frac{B\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +C\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) C}{\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 )} \sqrt{\cos \left (d x + c\right )}\,{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 \sec \left (d x + c\right )^{2} + B \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (B + C \sec{\left (c + d x \right )}\right ) \sqrt{\cos{\left (c + d x \right )}} \sec{\left (c + d x \right )}\, dx \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 )} \sqrt{\cos \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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