Optimal. Leaf size=147 \[ \frac{2 (11 A+9 C) \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 b^3 d}+\frac{10 (11 A+9 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{231 b d}+\frac{10 (11 A+9 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^5 d} \]
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Rubi [A] time = 0.126337, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {16, 3014, 2635, 2642, 2641} \[ \frac{2 (11 A+9 C) \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 b^3 d}+\frac{10 (11 A+9 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{231 b d}+\frac{10 (11 A+9 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^5 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3014
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cos ^4(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))^{7/2} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^4}\\ &=\frac{2 C (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^5 d}+\frac{(11 A+9 C) \int (b \cos (c+d x))^{7/2} \, dx}{11 b^4}\\ &=\frac{2 (11 A+9 C) (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^3 d}+\frac{2 C (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^5 d}+\frac{(5 (11 A+9 C)) \int (b \cos (c+d x))^{3/2} \, dx}{77 b^2}\\ &=\frac{10 (11 A+9 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{231 b d}+\frac{2 (11 A+9 C) (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^3 d}+\frac{2 C (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^5 d}+\frac{1}{231} (5 (11 A+9 C)) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx\\ &=\frac{10 (11 A+9 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{231 b d}+\frac{2 (11 A+9 C) (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^3 d}+\frac{2 C (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^5 d}+\frac{\left (5 (11 A+9 C) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{231 \sqrt{b \cos (c+d x)}}\\ &=\frac{10 (11 A+9 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{b \cos (c+d x)}}+\frac{10 (11 A+9 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{231 b d}+\frac{2 (11 A+9 C) (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^3 d}+\frac{2 C (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^5 d}\\ \end{align*}
Mathematica [A] time = 0.375043, size = 94, normalized size = 0.64 \[ \frac{\sin (2 (c+d x)) (12 (11 A+16 C) \cos (2 (c+d x))+572 A+21 C \cos (4 (c+d x))+531 C)+80 (11 A+9 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{1848 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.71, size = 349, normalized size = 2.4 \begin{align*} -{\frac{2}{231\,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 ( 1344\,C\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{12}-3360\,C\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+ \left ( 528\,A+3792\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -792\,A-2328\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( 616\,A+924\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -176\,A-186\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +55\,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 ) +45\,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 \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{4}}{\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 )^{5} + A \cos \left (d x + c\right )^{3}\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 )^{4}}{\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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