Optimal. Leaf size=188 \[ \frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^2 d}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}+\frac{6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^4 d} \]
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Rubi [A] time = 0.201367, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195, Rules used = {16, 3023, 2748, 2635, 2642, 2641, 2640, 2639} \[ \frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^2 d}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}+\frac{6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^4 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3023
Rule 2748
Rule 2635
Rule 2642
Rule 2641
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{3/2}} \, dx &=\frac{\int (b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b^3}\\ &=\frac{2 C (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^4 d}+\frac{2 \int (b \cos (c+d x))^{3/2} \left (\frac{1}{2} b (7 A+5 C)+\frac{7}{2} b B \cos (c+d x)\right ) \, dx}{7 b^4}\\ &=\frac{2 C (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^4 d}+\frac{B \int (b \cos (c+d x))^{5/2} \, dx}{b^4}+\frac{(7 A+5 C) \int (b \cos (c+d x))^{3/2} \, dx}{7 b^3}\\ &=\frac{2 (7 A+5 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 b^2 d}+\frac{2 B (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b^3 d}+\frac{2 C (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^4 d}+\frac{(3 B) \int \sqrt{b \cos (c+d x)} \, dx}{5 b^2}+\frac{(7 A+5 C) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx}{21 b}\\ &=\frac{2 (7 A+5 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 b^2 d}+\frac{2 B (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b^3 d}+\frac{2 C (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^4 d}+\frac{\left ((7 A+5 C) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 b \sqrt{b \cos (c+d x)}}+\frac{\left (3 B \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 b^2 \sqrt{\cos (c+d x)}}\\ &=\frac{6 B \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b d \sqrt{b \cos (c+d x)}}+\frac{2 (7 A+5 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 b^2 d}+\frac{2 B (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b^3 d}+\frac{2 C (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^4 d}\\ \end{align*}
Mathematica [A] time = 0.623786, size = 108, normalized size = 0.57 \[ \frac{\cos ^{\frac{3}{2}}(c+d x) \left (\sin (c+d x) \sqrt{\cos (c+d x)} (70 A+42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+10 (7 A+5 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+126 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{105 d (b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.393, size = 353, normalized size = 1.9 \begin{align*} -{\frac{2}{105\,bd}\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\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}\cos \left ( 1/2\,dx+c/2 \right ) + \left ( -168\,B-360\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( 140\,A+168\,B+280\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -70\,A-42\,B-80\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +35\,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 ) -63\,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}+25\,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} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac{3}{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 (\frac{{\left (C \cos \left (d x + c\right )^{3} + B \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \sqrt{b \cos \left (d x + c\right )}}{b^{2}}, 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} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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