Optimal. Leaf size=158 \[ \frac{2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d (a+b)}+\frac{2 (A b-a B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac{2 (A b-a B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 A F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.837433, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d (a+b)}+\frac{2 (A b-a B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac{2 (A b-a B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 A F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)} \]
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{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx &=\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \int \frac{-\frac{3}{2} (A b-a B)+\frac{1}{2} a (A+3 C) \cos (c+d x)+\frac{1}{2} A b \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{3 a}\\ &=\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{4 \int \frac{\frac{1}{4} \left (3 A b^2-3 a b B+a^2 (A+3 C)\right )+\frac{1}{4} a (4 A b-3 a B) \cos (c+d x)+\frac{3}{4} b (A b-a B) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 a^2}\\ &=\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{4 \int \frac{-\frac{1}{4} b \left (3 A b^2-3 a b B+a^2 (A+3 C)\right )-\frac{1}{4} a A b^2 \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 a^2 b}+\frac{(A b-a B) \int \sqrt{\cos (c+d x)} \, dx}{a^2}\\ &=\frac{2 (A b-a B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{A \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a}+\frac{\left (A b^2-a (b B-a C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^2}\\ &=\frac{2 (A b-a B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 A F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{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^2 (a+b) d}+\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.46276, size = 266, normalized size = 1.68 \[ \frac{\frac{2 a \left (2 a^2 (A+3 C)-9 a b B+9 A b^2\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}-\frac{6 (A b-a B) \sin (c+d x) \left (\left (b^2-2 a^2\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{b \sqrt{\sin ^2(c+d x)}}+\frac{a \left (8 a A b-6 a^2 B\right ) \left (2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-\frac{2 a \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}\right )}{b}+\frac{4 a^2 A \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{12 a (a B-A b) \sin (c+d x)}{\sqrt{\cos (c+d x)}}}{6 a^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.854, size = 474, normalized size = 3. \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} + 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{5}{2}}}\,{d x} \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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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