Optimal. Leaf size=89 \[ \frac{2 (b B-a C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}-\frac{2 a (b B-a C) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}+\frac{2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d} \]
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Rubi [A] time = 0.29845, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3029, 3002, 2639, 2803, 2641, 2805} \[ \frac{2 (b B-a C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}-\frac{2 a (b B-a C) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}+\frac{2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 3002
Rule 2639
Rule 2803
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx &=\int \frac{\sqrt{\cos (c+d x)} (B+C \cos (c+d x))}{a+b \cos (c+d x)} \, dx\\ &=\frac{C \int \sqrt{\cos (c+d x)} \, dx}{b}-\frac{(-b B+a C) \int \frac{\sqrt{\cos (c+d x)}}{a+b \cos (c+d x)} \, dx}{b}\\ &=\frac{2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d}+\frac{(b B-a C) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{b^2}-\frac{(a (b B-a C)) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{b^2}\\ &=\frac{2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d}+\frac{2 (b B-a C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}-\frac{2 a (b B-a C) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 (a+b) d}\\ \end{align*}
Mathematica [A] time = 0.855516, size = 131, normalized size = 1.47 \[ \frac{b B \left (2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-\frac{2 a \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}\right )-\frac{2 C \sin (c+d x) \left (-(a+b) F\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-a \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{\sqrt{\sin ^2(c+d x)}}}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.706, size = 295, normalized size = 3.3 \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 ( a-b \right ){b}^{2}\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} \left ( B{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) ab-B{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){b}^{2}-B{\it EllipticPi} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,-2\,{\frac{b}{a-b}},\sqrt{2} \right ) ab-C{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){a}^{2}+C{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) ab-C{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) ab+C{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){b}^{2}+C{\it EllipticPi} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,-2\,{\frac{b}{a-b}},\sqrt{2} \right ){a}^{2} \right ) } \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{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\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(-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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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{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\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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