Optimal. Leaf size=108 \[ \frac{2 (a B+A b) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 (5 a A+3 b B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d} \]
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Rubi [A] time = 0.168331, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2968, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 (a B+A b) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 (5 a A+3 b B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3023
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx &=\int \sqrt{\cos (c+d x)} \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2}{5} \int \sqrt{\cos (c+d x)} \left (\frac{1}{2} (5 a A+3 b B)+\frac{5}{2} (A b+a B) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+(A b+a B) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{5} (5 a A+3 b B) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 (5 a A+3 b B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 (A b+a B) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{3} (A b+a B) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (5 a A+3 b B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 (A b+a B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 (A b+a B) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.384445, size = 86, normalized size = 0.8 \[ \frac{2 \left (5 (a B+A b) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+3 (5 a A+3 b B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} (5 a B+5 A b+3 b B \cos (c+d x))\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.474, size = 371, normalized size = 3.4 \begin{align*} -{\frac{2}{15\,d}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -24\,Bb\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+ \left ( 20\,Ab+20\,aB+24\,Bb \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -10\,Ab-10\,aB-6\,Bb \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +5\,Ab\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 ) -15\,A{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \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}a+5\,aB\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 ) -9\,B{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \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}b \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \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 (B \cos \left (d x + c\right ) + A\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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b \cos \left (d x + c\right )^{2} + A a +{\left (B a + A b\right )} \cos \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(-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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