Optimal. Leaf size=344 \[ -\frac{2 b \left (a^2 (5 A+7 C)+7 A b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a^4 d}-\frac{2 \left (3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^5 d}-\frac{2 b^3 \left (a^2 C+A b^2\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^5 d (a+b)}-\frac{2 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right ) \sin (c+d x)}{15 a^5 d \sqrt{\cos (c+d x)}}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)} \]
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Rubi [A] time = 1.9357, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3056, 3055, 3059, 2639, 3002, 2641, 2805} \[ -\frac{2 b \left (a^2 (5 A+7 C)+7 A b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a^4 d}-\frac{2 \left (3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^5 d}-\frac{2 b^3 \left (a^2 C+A b^2\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^5 d (a+b)}-\frac{2 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right ) \sin (c+d x)}{15 a^5 d \sqrt{\cos (c+d x)}}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3056
Rule 3055
Rule 3059
Rule 2639
Rule 3002
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx &=\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 \int \frac{-\frac{9 A b}{2}+\frac{1}{2} a (7 A+9 C) \cos (c+d x)+\frac{7}{2} A b \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{9 a}\\ &=\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 \int \frac{\frac{7}{4} \left (9 A b^2+a^2 (7 A+9 C)\right )+a A b \cos (c+d x)-\frac{45}{4} A b^2 \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{63 a^2}\\ &=\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 \int \frac{-\frac{45}{8} b \left (7 A b^2+a^2 (5 A+7 C)\right )-\frac{3}{8} a \left (12 A b^2-7 a^2 (7 A+9 C)\right ) \cos (c+d x)+\frac{21}{8} b \left (9 A b^2+a^2 (7 A+9 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{315 a^3}\\ &=\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 \int \frac{\frac{63}{16} \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right )+\frac{9}{4} a b \left (7 A b^2+a^2 (6 A+7 C)\right ) \cos (c+d x)-\frac{45}{16} b^2 \left (7 A b^2+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}{945 a^4}\\ &=\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sin (c+d x)}{15 a^5 d \sqrt{\cos (c+d x)}}+\frac{32 \int \frac{-\frac{45}{32} b \left (21 A b^4+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )-\frac{9}{32} a \left (140 A b^4+7 a^4 (7 A+9 C)+4 a^2 b^2 (22 A+35 C)\right ) \cos (c+d x)-\frac{63}{32} b \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{945 a^5}\\ &=\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sin (c+d x)}{15 a^5 d \sqrt{\cos (c+d x)}}-\frac{32 \int \frac{\frac{45}{32} b^2 \left (21 A b^4+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )+\frac{45}{32} a b^3 \left (7 A b^2+a^2 (5 A+7 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{945 a^5 b}-\frac{\left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \int \sqrt{\cos (c+d x)} \, dx}{15 a^5}\\ &=-\frac{2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^5 d}+\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sin (c+d x)}{15 a^5 d \sqrt{\cos (c+d x)}}-\frac{\left (b^3 \left (A b^2+a^2 C\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^5}-\frac{\left (b \left (7 A b^2+a^2 (5 A+7 C)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 a^4}\\ &=-\frac{2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^5 d}-\frac{2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a^4 d}-\frac{2 b^3 \left (A b^2+a^2 C\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^5 (a+b) d}+\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sin (c+d x)}{15 a^5 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 4.11208, size = 431, normalized size = 1.25 \[ \frac{\frac{2 \left (7 \left (9 a^2 A b^2+a^4 (7 A+9 C)\right ) \sin (2 (c+d x))+6 \sin (c+d x) \left (-5 a b \left (a^2 (5 A+7 C)+7 A b^2\right ) \cos ^2(c+d x)+7 \left (3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right ) \cos ^3(c+d x)-15 a^3 A b\right )+70 a^4 A \tan (c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)}-3 \left (\frac{2 \left (7 a^2 b^3 (19 A+45 C)+a^4 b (99 A+133 C)+315 A b^5\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}+\frac{4 \left (4 a^3 b^2 (22 A+35 C)+7 a^5 (7 A+9 C)+140 a A b^4\right ) \left ((a+b) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-a \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{b (a+b)}+\frac{14 \left (3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right ) \sin (c+d x) \left (\left (2 a^2-b^2\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{a b \sqrt{\sin ^2(c+d x)}}\right )}{630 a^5 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.906, size = 1320, normalized size = 3.8 \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(-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} + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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