Optimal. Leaf size=318 \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right )}{5 a^4 d}+\frac{2 b^2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d (a+b)}+\frac{2 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left (a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right )}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 1.74405, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {3055, 3059, 2639, 3002, 2641, 2805} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right )}{5 a^4 d}+\frac{2 b^2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d (a+b)}+\frac{2 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left (a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right )}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3055
Rule 3059
Rule 2639
Rule 3002
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx &=\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \int \frac{-\frac{7}{2} (A b-a B)+\frac{1}{2} a (5 A+7 C) \cos (c+d x)+\frac{5}{2} A b \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{7 a}\\ &=\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 \int \frac{\frac{5}{4} \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right )+\frac{1}{4} a (4 A b+21 a B) \cos (c+d x)-\frac{21}{4} b (A b-a B) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{35 a^2}\\ &=\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 \int \frac{-\frac{21}{8} \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right )-\frac{1}{8} a \left (28 A b^2-28 a b B-5 a^2 (5 A+7 C)\right ) \cos (c+d x)+\frac{5}{8} b \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{105 a^3}\\ &=\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt{\cos (c+d x)}}+\frac{16 \int \frac{\frac{5}{16} \left (21 A b^4-7 a^3 b B-21 a b^3 B+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )+\frac{1}{16} a \left (140 A b^3-63 a^3 B-140 a b^2 B+4 a^2 b (22 A+35 C)\right ) \cos (c+d x)+\frac{21}{16} b \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{105 a^4}\\ &=\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{16 \int \frac{-\frac{5}{16} b \left (21 A b^4-7 a^3 b B-21 a b^3 B+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )-\frac{5}{16} a b^2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{105 a^4 b}+\frac{\left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 a^4}\\ &=\frac{2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^4 d}+\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt{\cos (c+d x)}}+\frac{\left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 a^3}+\frac{\left (b^2 \left (A b^2-a (b B-a C)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^4}\\ &=\frac{2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^4 d}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a^3 d}+\frac{2 b^2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 (a+b) d}+\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 4.79384, size = 420, normalized size = 1.32 \[ \frac{\frac{2 \left (7 a^2 b^2 (19 A+45 C)+10 a^4 (5 A+7 C)-133 a^3 b B-315 a b^3 B+315 A b^4\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}+\frac{4 a \left (4 a^2 b (22 A+35 C)-63 a^3 B-140 a b^2 B+140 A b^3\right ) \left ((a+b) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-a \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{b (a+b)}+\frac{2 \left (5 \left (a \sin (2 (c+d x)) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )+6 a^3 A \tan (c+d x)\right )+42 \sin (c+d x) \left (\cos ^2(c+d x) \left (-a^2 b (3 A+5 C)+3 a^3 B+5 a b^2 B-5 A b^3\right )+a^2 (a B-A b)\right )\right )}{\cos ^{\frac{5}{2}}(c+d x)}-\frac{42 \sin (c+d x) \left (-a^2 b (3 A+5 C)+3 a^3 B+5 a b^2 B-5 A b^3\right ) \left (\left (2 a^2-b^2\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{a b \sqrt{\sin ^2(c+d x)}}}{210 a^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.277, size = 1003, normalized size = 3.2 \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(-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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Fricas [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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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} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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