Optimal. Leaf size=375 \[ -\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (6 a^2 C+5 A b^2-b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left (a^2-b^2\right )}-\frac{2 a \left (8 a^2 C+5 A b^2-3 b^2 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^3 d \left (a^2-b^2\right )}-\frac{4 a \left (2 C \left (4 a^2+b^2\right )+5 A b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{5 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (2 a^2 b^2 (5 A-4 C)+16 a^4 C-b^4 (5 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{5 b^4 d \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.735158, antiderivative size = 375, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {3048, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (6 a^2 C+5 A b^2-b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left (a^2-b^2\right )}-\frac{2 a \left (8 a^2 C+5 A b^2-3 b^2 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^3 d \left (a^2-b^2\right )}-\frac{4 a \left (2 C \left (4 a^2+b^2\right )+5 A b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{5 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (2 a^2 b^2 (5 A-4 C)+16 a^4 C-b^4 (5 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{5 b^4 d \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 3048
Rule 3049
Rule 3023
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}-\frac{2 \int \frac{\cos (c+d x) \left (2 \left (A b^2+a^2 C\right )-\frac{1}{2} a b (A+C) \cos (c+d x)-\frac{1}{2} \left (5 A b^2+6 a^2 C-b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{a+b \cos (c+d x)}} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (5 A b^2+6 a^2 C-b^2 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d}-\frac{4 \int \frac{-\frac{1}{2} a \left (b^2 (5 A-C)+6 a^2 C\right )+\frac{1}{4} b \left (5 A b^2+2 a^2 C+3 b^2 C\right ) \cos (c+d x)+\frac{3}{4} a \left (5 A b^2+8 a^2 C-3 b^2 C\right ) \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{5 b^2 \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left (5 A b^2+8 a^2 C-3 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (5 A b^2+6 a^2 C-b^2 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d}-\frac{8 \int \frac{-\frac{3}{8} a b \left (5 A b^2+\left (4 a^2+b^2\right ) C\right )-\frac{3}{8} \left (2 a^2 b^2 (5 A-4 C)+16 a^4 C-b^4 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^3 \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left (5 A b^2+8 a^2 C-3 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (5 A b^2+6 a^2 C-b^2 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d}-\frac{\left (2 a \left (5 A b^2+2 \left (4 a^2+b^2\right ) C\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{5 b^4}+\frac{\left (2 a^2 b^2 (5 A-4 C)+16 a^4 C-b^4 (5 A+3 C)\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{5 b^4 \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left (5 A b^2+8 a^2 C-3 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (5 A b^2+6 a^2 C-b^2 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d}+\frac{\left (\left (2 a^2 b^2 (5 A-4 C)+16 a^4 C-b^4 (5 A+3 C)\right ) \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{5 b^4 \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (2 a \left (5 A b^2+2 \left (4 a^2+b^2\right ) C\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}{5 b^4 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (2 a^2 b^2 (5 A-4 C)+16 a^4 C-b^4 (5 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{5 b^4 \left (a^2-b^2\right ) d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a \left (5 A b^2+2 \left (4 a^2+b^2\right ) C\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 )}{5 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left (5 A b^2+8 a^2 C-3 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (5 A b^2+6 a^2 C-b^2 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 1.60632, size = 289, normalized size = 0.77 \[ \frac{\frac{10 a^2 b \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2-a^2}+\frac{2 a b^2 \left (C \left (4 a^2+b^2\right )+5 A b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{(a-b) (a+b)}+\frac{2 \left (2 a^2 b^2 (5 A-4 C)+16 a^4 C-b^4 (5 A+3 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )}{(a-b) (a+b)}+b^2 C \sin (2 (c+d x)) (a+b \cos (c+d x))-6 a b C \sin (c+d x) (a+b \cos (c+d x))}{5 b^4 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.456, size = 1289, normalized size = 3.4 \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 \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\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 )^{4} + A \cos \left (d x + c\right )^{2}\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{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} + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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