Optimal. Leaf size=132 \[ -\frac{2 a (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 A b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.195111, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3032, 3021, 2748, 2636, 2639, 2641} \[ -\frac{2 a (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 A b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3032
Rule 3021
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx &=\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2}{5} \int \frac{\frac{5 A b}{2}+\frac{1}{2} a (3 A+5 C) \cos (c+d x)+\frac{5}{2} b C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4}{15} \int \frac{\frac{3}{4} a (3 A+5 C)+\frac{5}{4} b (A+3 C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{1}{3} (b (A+3 C)) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{5} (a (3 A+5 C)) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 b (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{1}{5} (a (3 A+5 C)) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 a (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 b (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.733206, size = 122, normalized size = 0.92 \[ \frac{-6 a (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+9 a A \sin (2 (c+d x))+6 a A \tan (c+d x)+15 a C \sin (2 (c+d x))+10 b (A+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+10 A b \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.153, size = 732, normalized size = 5.6 \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} + A\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 )^{3} + C a \cos \left (d x + c\right )^{2} + A b \cos \left (d x + c\right ) + A a}{\cos \left (d x + c\right )^{\frac{7}{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} + A\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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