Optimal. Leaf size=210 \[ \frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac{2 b (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{10 b^2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^2 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac{10 b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d} \]
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Rubi [A] time = 0.232338, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.205, Rules used = {16, 3023, 2748, 2635, 2640, 2639, 2642, 2641} \[ \frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac{2 b (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{10 b^2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^2 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac{10 b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3023
Rule 2748
Rule 2635
Rule 2640
Rule 2639
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \cos (c+d x) (b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{\int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b}\\ &=\frac{2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac{2 \int (b \cos (c+d x))^{5/2} \left (\frac{1}{2} b (9 A+7 C)+\frac{9}{2} b B \cos (c+d x)\right ) \, dx}{9 b^2}\\ &=\frac{2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac{B \int (b \cos (c+d x))^{7/2} \, dx}{b^2}+\frac{(9 A+7 C) \int (b \cos (c+d x))^{5/2} \, dx}{9 b}\\ &=\frac{2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac{2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac{1}{7} (5 B) \int (b \cos (c+d x))^{3/2} \, dx+\frac{1}{15} (b (9 A+7 C)) \int \sqrt{b \cos (c+d x)} \, dx\\ &=\frac{10 b B \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac{2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac{1}{21} \left (5 b^2 B\right ) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx+\frac{\left (b (9 A+7 C) \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{15 \sqrt{\cos (c+d x)}}\\ &=\frac{2 b (9 A+7 C) \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d \sqrt{\cos (c+d x)}}+\frac{10 b B \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac{2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac{\left (5 b^2 B \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 (9 A+7 C) \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d \sqrt{\cos (c+d x)}}+\frac{10 b^2 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 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac{2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}\\ \end{align*}
Mathematica [A] time = 0.875592, size = 128, normalized size = 0.61 \[ \frac{(b \cos (c+d x))^{5/2} \left (\sin (c+d x) \sqrt{\cos (c+d x)} (7 (36 A+43 C) \cos (c+d x)+5 (18 B \cos (2 (c+d x))+78 B+7 C \cos (3 (c+d x))))+84 (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+300 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{630 b d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.683, size = 384, normalized size = 1.8 \begin{align*} -{\frac{2\,{b}^{2}}{315\,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 ( -1120\,C\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+ \left ( 720\,B+2240\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -504\,A-1080\,B-2072\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( 504\,A+840\,B+952\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -126\,A-240\,B-168\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -189\,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 ) +75\,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 ) -147\,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{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}} \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 b \cos \left (d x + c\right )^{4} + B b \cos \left (d x + c\right )^{3} + A b \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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