Optimal. Leaf size=40 \[ \frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d} \]
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Rubi [A] time = 0.0216924, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {21, 2636, 2639} \[ \frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx &=B \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-B \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0589035, size = 40, normalized size = 1. \[ B \left (\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 3.253, size = 102, normalized size = 2.6 \begin{align*} -2\,{\frac{B \left ( \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 ) \right ) }{\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 \frac{B b \cos \left (d x + c\right ) + B a}{{\left (b \cos \left (d x + c\right ) + a\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 (\frac{B}{\cos \left (d x + c\right )^{\frac{3}{2}}}, 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 \frac{B b \cos \left (d x + c\right ) + B a}{{\left (b \cos \left (d x + c\right ) + a\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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