Optimal. Leaf size=378 \[ -\frac{2 \left (-8 a^2 B+18 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left (18 a^2 A b-8 a^3 B-39 a b^2 B-75 A b^3\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}+\frac{2 \left (a^2-b^2\right ) \left (18 a^2 A b-8 a^3 B-39 a b^2 B-75 A b^3\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (18 a^3 A b-33 a^2 b^2 B-8 a^4 B-246 a A b^3-147 b^4 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]
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Rubi [A] time = 0.733061, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {2990, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (-8 a^2 B+18 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left (18 a^2 A b-8 a^3 B-39 a b^2 B-75 A b^3\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}+\frac{2 \left (a^2-b^2\right ) \left (18 a^2 A b-8 a^3 B-39 a b^2 B-75 A b^3\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (18 a^3 A b-33 a^2 b^2 B-8 a^4 B-246 a A b^3-147 b^4 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]
Antiderivative was successfully verified.
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Rule 2990
Rule 3023
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx &=\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac{2 \int (a+b \cos (c+d x))^{3/2} \left (a B+\frac{7}{2} b B \cos (c+d x)+\frac{1}{2} (9 A b-4 a B) \cos ^2(c+d x)\right ) \, dx}{9 b}\\ &=\frac{2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac{4 \int (a+b \cos (c+d x))^{3/2} \left (\frac{3}{4} b (15 A b-2 a B)-\frac{1}{4} \left (18 a A b-8 a^2 B-49 b^2 B\right ) \cos (c+d x)\right ) \, dx}{63 b^2}\\ &=-\frac{2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac{2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac{8 \int \sqrt{a+b \cos (c+d x)} \left (\frac{3}{8} b \left (57 a A b-2 a^2 B+49 b^2 B\right )-\frac{3}{8} \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \cos (c+d x)\right ) \, dx}{315 b^2}\\ &=-\frac{2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac{2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac{2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac{16 \int \frac{\frac{3}{16} b \left (153 a^2 A b+75 A b^3+2 a^3 B+186 a b^2 B\right )-\frac{3}{16} \left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{945 b^2}\\ &=-\frac{2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac{2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac{2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac{\left (\left (a^2-b^2\right ) \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{315 b^3}-\frac{\left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{315 b^3}\\ &=-\frac{2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac{2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac{2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}-\frac{\left (\left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{315 b^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (\left (a^2-b^2\right ) \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}}} \, dx}{315 b^3 \sqrt{a+b \cos (c+d x)}}\\ &=-\frac{2 \left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (a^2-b^2\right ) \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac{2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac{2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}\\ \end{align*}
Mathematica [A] time = 1.44465, size = 291, normalized size = 0.77 \[ \frac{b (a+b \cos (c+d x)) \left (\left (72 a^2 A b-32 a^3 B+804 a b^2 B+690 A b^3\right ) \sin (c+d x)+b \left (2 \left (6 a^2 B+144 a A b+133 b^2 B\right ) \sin (2 (c+d x))+5 b (2 (10 a B+9 A b) \sin (3 (c+d x))+7 b B \sin (4 (c+d x)))\right )\right )+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b^2 \left (153 a^2 A b+2 a^3 B+186 a b^2 B+75 A b^3\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+\left (-18 a^3 A b+33 a^2 b^2 B+8 a^4 B+246 a A b^3+147 b^4 B\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{1260 b^3 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.509, size = 1635, normalized size = 4.3 \begin{align*} \text{result too large to display} \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 )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \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 b \cos \left (d x + c\right )^{4} + A a \cos \left (d x + c\right )^{2} +{\left (B a + A b\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt{b \cos \left (d x + c\right ) + a}, 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 (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \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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