Optimal. Leaf size=372 \[ \frac{2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \left (45 a^2 A b-10 a^3 B+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{2 \left (a^2-b^2\right ) \left (45 a^2 A b-10 a^3 B+114 a b^2 B+75 A b^3\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (45 a^3 A b+279 a^2 b^2 B-10 a^4 B+435 a A b^3+147 b^4 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d} \]
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Rubi [A] time = 0.796367, antiderivative size = 372, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \left (45 a^2 A b-10 a^3 B+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{2 \left (a^2-b^2\right ) \left (45 a^2 A b-10 a^3 B+114 a b^2 B+75 A b^3\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (45 a^3 A b+279 a^2 b^2 B-10 a^4 B+435 a A b^3+147 b^4 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3023
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx &=\int (a+b \cos (c+d x))^{5/2} \left (A \cos (c+d x)+B \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac{2 \int (a+b \cos (c+d x))^{5/2} \left (\frac{7 b B}{2}+\frac{1}{2} (9 A b-2 a B) \cos (c+d x)\right ) \, dx}{9 b}\\ &=\frac{2 (9 A b-2 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac{2 B (a+b \cos (c+d x))^{7/2} \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+13 a B)+\frac{1}{4} \left (45 a A b-10 a^2 B+49 b^2 B\right ) \cos (c+d x)\right ) \, dx}{63 b}\\ &=\frac{2 \left (45 a A b-10 a^2 B+49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}+\frac{2 (9 A b-2 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac{2 B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac{8 \int \sqrt{a+b \cos (c+d x)} \left (\frac{3}{8} b \left (120 a A b+55 a^2 B+49 b^2 B\right )+\frac{3}{8} \left (45 a^2 A b+75 A b^3-10 a^3 B+114 a b^2 B\right ) \cos (c+d x)\right ) \, dx}{315 b}\\ &=\frac{2 \left (45 a^2 A b+75 A b^3-10 a^3 B+114 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b d}+\frac{2 \left (45 a A b-10 a^2 B+49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}+\frac{2 (9 A b-2 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac{2 B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac{16 \int \frac{\frac{3}{16} b \left (405 a^2 A b+75 A b^3+155 a^3 B+261 a b^2 B\right )+\frac{3}{16} \left (45 a^3 A b+435 a A b^3-10 a^4 B+279 a^2 b^2 B+147 b^4 B\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{945 b}\\ &=\frac{2 \left (45 a^2 A b+75 A b^3-10 a^3 B+114 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b d}+\frac{2 \left (45 a A b-10 a^2 B+49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}+\frac{2 (9 A b-2 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac{2 B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}-\frac{\left (\left (a^2-b^2\right ) \left (45 a^2 A b+75 A b^3-10 a^3 B+114 a b^2 B\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{315 b^2}+\frac{\left (45 a^3 A b+435 a A b^3-10 a^4 B+279 a^2 b^2 B+147 b^4 B\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{315 b^2}\\ &=\frac{2 \left (45 a^2 A b+75 A b^3-10 a^3 B+114 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b d}+\frac{2 \left (45 a A b-10 a^2 B+49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}+\frac{2 (9 A b-2 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac{2 B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac{\left (\left (45 a^3 A b+435 a A b^3-10 a^4 B+279 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^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (\left (a^2-b^2\right ) \left (45 a^2 A b+75 A b^3-10 a^3 B+114 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^2 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (45 a^3 A b+435 a A b^3-10 a^4 B+279 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^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left (a^2-b^2\right ) \left (45 a^2 A b+75 A b^3-10 a^3 B+114 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^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (45 a^2 A b+75 A b^3-10 a^3 B+114 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b d}+\frac{2 \left (45 a A b-10 a^2 B+49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}+\frac{2 (9 A b-2 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac{2 B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}\\ \end{align*}
Mathematica [A] time = 1.48195, size = 291, normalized size = 0.78 \[ \frac{b (a+b \cos (c+d x)) \left (2 \left (540 a^2 A b+20 a^3 B+747 a b^2 B+345 A b^3\right ) \sin (c+d x)+b \left (\left (300 a^2 B+540 a A b+266 b^2 B\right ) \sin (2 (c+d x))+5 b (2 (19 a B+9 A b) \sin (3 (c+d x))+7 b B \sin (4 (c+d x)))\right )\right )+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b^2 \left (405 a^2 A b+155 a^3 B+261 a b^2 B+75 A b^3\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+\left (45 a^3 A b+279 a^2 b^2 B-10 a^4 B+435 a A b^3+147 b^4 B\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{1260 b^2 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.468, size = 1635, normalized size = 4.4 \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{5}{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 (B b^{2} \cos \left (d x + c\right )^{4} + A a^{2} \cos \left (d x + c\right ) +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{3} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{2}\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{5}{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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