Optimal. Leaf size=140 \[ \frac{2 \left (3 a^2 B+2 a b C+b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 \left (5 a (a C+2 b B)+3 b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 b (7 a C+5 b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d} \]
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Rubi [A] time = 0.356083, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3029, 2990, 3023, 2748, 2641, 2639} \[ \frac{2 \left (3 a^2 B+2 a b C+b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 \left (5 a (a C+2 b B)+3 b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 b (7 a C+5 b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2990
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\int \frac{(a+b \cos (c+d x))^2 (B+C \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b C \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \sin (c+d x)}{5 d}+\frac{2}{5} \int \frac{\frac{1}{2} a (5 a B+b C)+\frac{1}{2} \left (3 b^2 C+5 a (2 b B+a C)\right ) \cos (c+d x)+\frac{1}{2} b (5 b B+7 a C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b (5 b B+7 a C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 b C \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \sin (c+d x)}{5 d}+\frac{4}{15} \int \frac{\frac{5}{4} \left (3 a^2 B+b^2 B+2 a b C\right )+\frac{3}{4} \left (3 b^2 C+5 a (2 b B+a C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b (5 b B+7 a C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 b C \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \sin (c+d x)}{5 d}+\frac{1}{3} \left (3 a^2 B+b^2 B+2 a b C\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{5} \left (3 b^2 C+5 a (2 b B+a C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (3 b^2 C+5 a (2 b B+a C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (3 a^2 B+b^2 B+2 a b C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 b (5 b B+7 a C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 b C \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.569829, size = 106, normalized size = 0.76 \[ \frac{2 \left (5 \left (3 a^2 B+2 a b C+b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+3 \left (5 a^2 C+10 a b B+3 b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+b \sin (c+d x) \sqrt{\cos (c+d x)} (10 a C+5 b B+3 b C \cos (c+d x))\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.742, size = 487, normalized size = 3.5 \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} + B \cos \left (d x + c\right )\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac{3}{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{C b^{2} \cos \left (d x + c\right )^{3} + B a^{2} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}, 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} + B \cos \left (d x + c\right )\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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