Optimal. Leaf size=115 \[ \frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^2 d}+\frac{2 (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{15 b d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^4 d} \]
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Rubi [A] time = 0.0931476, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {16, 3014, 2635, 2640, 2639} \[ \frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^2 d}+\frac{2 (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{15 b d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^4 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3014
Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt{b \cos (c+d x)}} \, dx &=\frac{\int (b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^3}\\ &=\frac{2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^4 d}+\frac{(9 A+7 C) \int (b \cos (c+d x))^{5/2} \, dx}{9 b^3}\\ &=\frac{2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b^2 d}+\frac{2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^4 d}+\frac{(9 A+7 C) \int \sqrt{b \cos (c+d x)} \, dx}{15 b}\\ &=\frac{2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b^2 d}+\frac{2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^4 d}+\frac{\left ((9 A+7 C) \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{15 b \sqrt{\cos (c+d x)}}\\ &=\frac{2 (9 A+7 C) \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 b d \sqrt{\cos (c+d x)}}+\frac{2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b^2 d}+\frac{2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^4 d}\\ \end{align*}
Mathematica [A] time = 0.395381, size = 83, normalized size = 0.72 \[ \frac{\sin (c+d x) \cos ^2(c+d x) (18 A+5 C \cos (2 (c+d x))+19 C)+6 (9 A+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{45 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.267, size = 321, normalized size = 2.8 \begin{align*} -{\frac{2}{45\,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 ( -160\,C\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+320\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}\cos \left ( 1/2\,dx+c/2 \right ) + \left ( -72\,A-296\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( 72\,A+136\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -18\,A-24\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -27\,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 ) -21\,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 EllipticE} \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 \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt{b \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 (\frac{{\left (C \cos \left (d x + c\right )^{4} + A \cos \left (d x + c\right )^{2}\right )} \sqrt{b \cos \left (d x + c\right )}}{b}, 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{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt{b \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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