Optimal. Leaf size=389 \[ \frac{2 \left (-a^2 b^2 (23 A+19 C)+2 a^4 C-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{15 b d \left (a^2-b^2\right )^3 \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \sin (c+d x)}{15 b d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^{3/2}}-\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}-\frac{4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (-a^2 b^2 (23 A+19 C)+2 a^4 C-3 b^4 (3 A+5 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 d \left (a^2-b^2\right )^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.605387, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {3022, 2754, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (-a^2 b^2 (23 A+19 C)+2 a^4 C-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{15 b d \left (a^2-b^2\right )^3 \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \sin (c+d x)}{15 b d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^{3/2}}-\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}-\frac{4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (-a^2 b^2 (23 A+19 C)+2 a^4 C-3 b^4 (3 A+5 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 d \left (a^2-b^2\right )^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 3022
Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx &=-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{5 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac{2 \int \frac{-\frac{5}{2} a b (A+C)+\frac{1}{2} \left (3 A b^2-2 a^2 C+5 b^2 C\right ) \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{5 b \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{5 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac{4 a \left (4 A b^2-a^2 C+5 b^2 C\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}+\frac{4 \int \frac{\frac{3}{4} b \left (a^2 (5 A+3 C)+b^2 (3 A+5 C)\right )-\frac{1}{2} a \left (4 A b^2-\left (a^2-5 b^2\right ) C\right ) \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{15 b \left (a^2-b^2\right )^2}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{5 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac{4 a \left (4 A b^2-a^2 C+5 b^2 C\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^3 d \sqrt{a+b \cos (c+d x)}}-\frac{8 \int \frac{-\frac{1}{8} a b \left (a^2 (15 A+7 C)+b^2 (17 A+25 C)\right )+\frac{1}{8} \left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b \left (a^2-b^2\right )^3}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{5 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac{4 a \left (4 A b^2-a^2 C+5 b^2 C\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^3 d \sqrt{a+b \cos (c+d x)}}-\frac{\left (2 a \left (4 A b^2-a^2 C+5 b^2 C\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^2 \left (a^2-b^2\right )^2}-\frac{\left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{15 b^2 \left (a^2-b^2\right )^3}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{5 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac{4 a \left (4 A b^2-a^2 C+5 b^2 C\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^3 d \sqrt{a+b \cos (c+d x)}}-\frac{\left (\left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 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}{15 b^2 \left (a^2-b^2\right )^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (2 a \left (4 A b^2-a^2 C+5 b^2 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}{15 b^2 \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}\\ &=-\frac{2 \left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 \left (a^2-b^2\right )^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a \left (4 A b^2-\left (a^2-5 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 )}{15 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{5 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac{4 a \left (4 A b^2-a^2 C+5 b^2 C\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^3 d \sqrt{a+b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.56816, size = 314, normalized size = 0.81 \[ \frac{2 \left (\frac{b \sin (c+d x) \left (-4 a b \left (-a^2 b^2 (27 A+25 C)+3 a^4 C-5 b^4 (A+2 C)\right ) \cos (c+d x)+\left (a^2 b^4 (23 A+19 C)-2 a^4 b^2 C+3 b^6 (3 A+5 C)\right ) \cos (2 (c+d x))+68 a^4 A b^2+13 a^2 A b^4+48 a^4 b^2 C+35 a^2 b^4 C-2 a^6 C+15 A b^6+15 b^6 C\right )}{2 \left (b^2-a^2\right )^3}+\frac{\left (\frac{a+b \cos (c+d x)}{a+b}\right )^{5/2} \left (2 a (a-b) \left (C \left (a^2-5 b^2\right )-4 A b^2\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+\left (a^2 b^2 (23 A+19 C)-2 a^4 C+3 b^4 (3 A+5 C)\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )}{(a-b)^3}\right )}{15 b^2 d (a+b \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.747, size = 1305, 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{C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{7}{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 )^{2} + A\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{b^{4} \cos \left (d x + c\right )^{4} + 4 \, a b^{3} \cos \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \cos \left (d x + c\right )^{2} + 4 \, a^{3} b \cos \left (d x + c\right ) + a^{4}}, 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{C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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