Optimal. Leaf size=168 \[ \frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}+\frac{6 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^2 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{10 b B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.142352, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {16, 2748, 2635, 2640, 2639, 2642, 2641} \[ \frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}+\frac{6 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^2 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{10 b B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2748
Rule 2635
Rule 2640
Rule 2639
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx &=\frac{\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx}{b^2}\\ &=\frac{A \int (b \cos (c+d x))^{5/2} \, dx}{b^2}+\frac{B \int (b \cos (c+d x))^{7/2} \, dx}{b^3}\\ &=\frac{2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^2 d}+\frac{1}{5} (3 A) \int \sqrt{b \cos (c+d x)} \, dx+\frac{(5 B) \int (b \cos (c+d x))^{3/2} \, dx}{7 b}\\ &=\frac{10 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^2 d}+\frac{1}{21} (5 b B) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx+\frac{\left (3 A \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \sqrt{\cos (c+d x)}}\\ &=\frac{6 A \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)}}+\frac{10 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^2 d}+\frac{\left (5 b B \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 \sqrt{b \cos (c+d x)}}\\ &=\frac{6 A \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)}}+\frac{10 b B \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{10 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^2 d}\\ \end{align*}
Mathematica [A] time = 0.516783, size = 100, normalized size = 0.6 \[ \frac{\sqrt{b \cos (c+d x)} \left (2 \sin (c+d x) \sqrt{\cos (c+d x)} (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+100 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{210 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.447, size = 299, normalized size = 1.8 \begin{align*} -{\frac{2\,b}{105\,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 ( 240\,B\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+ \left ( -168\,A-360\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( 168\,A+280\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -42\,A-80\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -63\,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 ) +25\,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 ){\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 (B \cos \left (d x + c\right ) + A\right )} \sqrt{b \cos \left (d x + c\right )} \cos \left (d x + c\right )^{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 ({\left (B \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )^{2}\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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