Optimal. Leaf size=307 \[ \frac{2 \left (a^2 A b+2 a^3 B-6 a b^2 B+3 A b^3\right ) \sin (c+d x)}{3 b d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a (A b-a B) \sin (c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (2 a^2 B+a A b-3 b^2 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (a^2 A b+2 a^3 B-6 a b^2 B+3 A b^3\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.471744, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {2968, 3021, 2754, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (a^2 A b+2 a^3 B-6 a b^2 B+3 A b^3\right ) \sin (c+d x)}{3 b d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a (A b-a B) \sin (c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (2 a^2 B+a A b-3 b^2 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (a^2 A b+2 a^3 B-6 a b^2 B+3 A b^3\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3021
Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx &=\int \frac{A \cos (c+d x)+B \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx\\ &=\frac{2 a (A b-a B) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{2 \int \frac{\frac{3}{2} b (A b-a B)-\frac{1}{2} \left (a A b+2 a^2 B-3 b^2 B\right ) \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 b \left (a^2-b^2\right )}\\ &=\frac{2 a (A b-a B) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 B\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{4 \int \frac{-\frac{1}{4} b \left (4 a A b-a^2 B-3 b^2 B\right )-\frac{1}{4} \left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 B\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{3 b \left (a^2-b^2\right )^2}\\ &=\frac{2 a (A b-a B) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 B\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\left (a A b+2 a^2 B-3 b^2 B\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )}-\frac{\left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 B\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{3 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{2 a (A b-a B) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 B\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}-\frac{\left (\left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 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}{3 b^2 \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (\left (a A b+2 a^2 B-3 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}{3 b^2 \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}\\ &=-\frac{2 \left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (a A b+2 a^2 B-3 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 )}{3 b^2 \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}+\frac{2 a (A b-a B) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 B\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.85905, size = 224, normalized size = 0.73 \[ \frac{2 \left (\frac{b \sin (c+d x) \left (b \left (a^2 A b+2 a^3 B-6 a b^2 B+3 A b^3\right ) \cos (c+d x)+a \left (2 a^2 A b+a^3 B-5 a b^2 B+2 A b^3\right )\right )}{\left (a^2-b^2\right )^2}-\frac{\left (\frac{a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (\left (a^2 A b+2 a^3 B-6 a b^2 B+3 A b^3\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-(a-b) \left (2 a^2 B+a A b-3 b^2 B\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )}{(a-b)^2}\right )}{3 b^2 d (a+b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 14.194, size = 860, normalized size = 2.8 \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 \frac{{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{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 (B \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}}, 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 (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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