Optimal. Leaf size=103 \[ \frac{2 (a B+3 b C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 (a C+b B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 (a C+b B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.227216, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {3029, 2968, 3021, 2748, 2636, 2639, 2641} \[ \frac{2 (a B+3 b C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 (a C+b B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 (a C+b B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2968
Rule 3021
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx &=\int \frac{(a+b \cos (c+d x)) (B+C \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\int \frac{a B+(b B+a C) \cos (c+d x)+b C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2}{3} \int \frac{\frac{3}{2} (b B+a C)+\frac{1}{2} (a B+3 b C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+(b B+a C) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{3} (a B+3 b C) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (a B+3 b C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (b B+a C) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+(-b B-a C) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 (b B+a C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 (a B+3 b C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (b B+a C) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.46622, size = 107, normalized size = 1.04 \[ \frac{2 \left ((a B+3 b C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-3 (a C+b B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+a B \tan (c+d x)+3 a C \sin (c+d x)+3 b B \sin (c+d x)\right )}{3 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.444, size = 428, normalized size = 4.2 \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] 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 )}}{\cos \left (d x + c\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{C b \cos \left (d x + c\right )^{2} + B a +{\left (C a + B b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac{5}{2}}}, 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 )}}{\cos \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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