Optimal. Leaf size=392 \[ -\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left (a^2-b^2\right )}-\frac{4 a \left (5 a^2 b^2 C-3 a^4 C+2 A b^4\right ) \sin (c+d x)}{3 b^3 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (2 a^2 b^2 (A-8 C)+16 a^4 C-b^4 (3 A+C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^4 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left (a^2 b^2 (A-14 C)+8 a^4 C-b^4 (3 A-4 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^4 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.829636, antiderivative size = 392, 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, 3031, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left (a^2-b^2\right )}-\frac{4 a \left (5 a^2 b^2 C-3 a^4 C+2 A b^4\right ) \sin (c+d x)}{3 b^3 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (2 a^2 b^2 (A-8 C)+16 a^4 C-b^4 (3 A+C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^4 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left (a^2 b^2 (A-14 C)+8 a^4 C-b^4 (3 A-4 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^4 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 3048
Rule 3031
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))^{5/2}} \, dx &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^2(c+d x) \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{\cos (c+d x) \left (2 \left (A b^2+a^2 C\right )-\frac{3}{2} a b (A+C) \cos (c+d x)-\frac{3}{2} \left (A b^2+2 a^2 C-b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 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)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{4 a \left (2 A b^4-3 a^4 C+5 a^2 b^2 C\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 \int \frac{-\frac{1}{2} b \left (2 A b^4-3 a^4 C+5 a^2 b^2 C\right )-\frac{1}{2} a \left (2 A b^4-\left (6 a^4-11 a^2 b^2+3 b^4\right ) C\right ) \cos (c+d x)-\frac{3}{4} b \left (a^2-b^2\right ) \left (A b^2+2 a^2 C-b^2 C\right ) \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{4 a \left (2 A b^4-3 a^4 C+5 a^2 b^2 C\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (A b^2+2 a^2 C-b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d}-\frac{8 \int \frac{\frac{3}{8} b^2 \left (4 a^4 C-b^4 (3 A+C)-a^2 b^2 (A+7 C)\right )+\frac{3}{4} a b \left (a^2 b^2 (A-14 C)-b^4 (3 A-4 C)+8 a^4 C\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{9 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{4 a \left (2 A b^4-3 a^4 C+5 a^2 b^2 C\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (A b^2+2 a^2 C-b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d}-\frac{\left (2 a \left (a^2 b^2 (A-14 C)-b^4 (3 A-4 C)+8 a^4 C\right )\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{3 b^4 \left (a^2-b^2\right )^2}+\frac{\left (2 a^2 b^2 (A-8 C)+16 a^4 C-b^4 (3 A+C)\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{3 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)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{4 a \left (2 A b^4-3 a^4 C+5 a^2 b^2 C\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (A b^2+2 a^2 C-b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d}-\frac{\left (2 a \left (a^2 b^2 (A-14 C)-b^4 (3 A-4 C)+8 a^4 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}{3 b^4 \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (\left (2 a^2 b^2 (A-8 C)+16 a^4 C-b^4 (3 A+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}{3 b^4 \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}\\ &=-\frac{4 a \left (a^2 b^2 (A-14 C)-b^4 (3 A-4 C)+8 a^4 C\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^4 \left (a^2-b^2\right )^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (2 a^2 b^2 (A-8 C)+16 a^4 C-b^4 (3 A+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 )}{3 b^4 \left (a^2-b^2\right ) 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)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{4 a \left (2 A b^4-3 a^4 C+5 a^2 b^2 C\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (A b^2+2 a^2 C-b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 2.71695, size = 306, normalized size = 0.78 \[ \frac{2 \left (\frac{b \sin (c+d x) \left (4 a b \left (a^2 b^2 (A-8 C)+5 a^4 C+b^4 (C-3 A)\right ) \cos (c+d x)+2 a^4 A b^2-10 a^2 A b^4+C \left (b^3-a^2 b\right )^2 \cos (2 (c+d x))-25 a^4 b^2 C+16 a^6 C+b^6 C\right )}{2 \left (a^2-b^2\right )^2}+\frac{\left (\frac{a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b \left (a^2 b^3 (A+7 C)-4 a^4 b C+b^5 (3 A+C)\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-2 a \left (a^2 b^2 (A-14 C)+8 a^4 C+b^4 (4 C-3 A)\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}\right )}{3 b^4 d (a+b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.934, size = 1323, 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{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 (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^{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 (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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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