Optimal. Leaf size=112 \[ -\frac{2 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0885878, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2748, 2636, 2640, 2639, 2642, 2641} \[ -\frac{2 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2636
Rule 2640
Rule 2639
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{3/2}} \, dx &=A \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx+\frac{B \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx}{b}\\ &=\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}-\frac{A \int \sqrt{b \cos (c+d x)} \, dx}{b^2}+\frac{\left (B \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{b \sqrt{b \cos (c+d x)}}\\ &=\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}-\frac{\left (A \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{b^2 \sqrt{\cos (c+d x)}}\\ &=-\frac{2 A \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0664098, size = 76, normalized size = 0.68 \[ \frac{2 \left (A \sin (c+d x)-A \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{b d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.102, size = 215, normalized size = 1.9 \begin{align*} -2\,{\frac{\sqrt{-2\,b \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b} \left ( A\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\,A\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+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 ) \right ) }{b\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) }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 \cos \left (d x + c\right ) + A}{\left (b \cos \left (d x + c\right )\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{{\left (B \cos \left (d x + c\right ) + A\right )} \sqrt{b \cos \left (d x + c\right )}}{b^{2} \cos \left (d x + c\right )^{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 \cos \left (d x + c\right ) + A}{\left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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