Optimal. Leaf size=110 \[ \frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{2 b (7 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^2 d} \]
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Rubi [A] time = 0.0919654, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {16, 3014, 2635, 2642, 2641} \[ \frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{2 b (7 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^2 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3014
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \cos (c+d x) \sqrt{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{\int (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b}\\ &=\frac{2 C (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^2 d}+\frac{(7 A+5 C) \int (b \cos (c+d x))^{3/2} \, dx}{7 b}\\ &=\frac{2 (7 A+5 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 C (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^2 d}+\frac{1}{21} (b (7 A+5 C)) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx\\ &=\frac{2 (7 A+5 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 C (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^2 d}+\frac{\left (b (7 A+5 C) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 \sqrt{b \cos (c+d x)}}\\ &=\frac{2 b (7 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 (7 A+5 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 C (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^2 d}\\ \end{align*}
Mathematica [A] time = 0.332497, size = 89, normalized size = 0.81 \[ \frac{(b \cos (c+d x))^{3/2} \left (4 (7 A+5 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \sqrt{\cos (c+d x)} (14 A+3 C \cos (2 (c+d x))+13 C)\right )}{42 b d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.734, size = 294, normalized size = 2.7 \begin{align*} -{\frac{2\,b}{21\,d}\sqrt{b \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 ( 48\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}\cos \left ( 1/2\,dx+c/2 \right ) -72\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) + \left ( 28\,A+56\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -14\,A-16\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +7\,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 EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +5\,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 EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \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{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt{b \cos \left (d x + c\right )} \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 \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )\right )} \sqrt{b \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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