Optimal. Leaf size=217 \[ \frac{2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 b^3 (5 A+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 b^3 B \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 b^2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.267899, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195, Rules used = {16, 3021, 2748, 2636, 2640, 2639, 2642, 2641} \[ \frac{2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 b^3 (5 A+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 b^3 B \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 b^2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3021
Rule 2748
Rule 2636
Rule 2640
Rule 2639
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx &=b^7 \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{9/2}} \, dx\\ &=\frac{2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{1}{7} \left (2 b^4\right ) \int \frac{\frac{7 b^2 B}{2}+\frac{1}{2} b^2 (5 A+7 C) \cos (c+d x)}{(b \cos (c+d x))^{7/2}} \, dx\\ &=\frac{2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\left (b^6 B\right ) \int \frac{1}{(b \cos (c+d x))^{7/2}} \, dx+\frac{1}{7} \left (b^5 (5 A+7 C)\right ) \int \frac{1}{(b \cos (c+d x))^{5/2}} \, dx\\ &=\frac{2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{1}{5} \left (3 b^4 B\right ) \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx+\frac{1}{21} \left (b^3 (5 A+7 C)\right ) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx\\ &=\frac{2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{6 b^3 B \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{1}{5} \left (3 b^2 B\right ) \int \sqrt{b \cos (c+d x)} \, dx+\frac{\left (b^3 (5 A+7 C) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 \sqrt{b \cos (c+d x)}}\\ &=\frac{2 b^3 (5 A+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{6 b^3 B \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{\left (3 b^2 B \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \sqrt{\cos (c+d x)}}\\ &=-\frac{6 b^2 B \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^3 (5 A+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{6 b^3 B \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.911582, size = 134, normalized size = 0.62 \[ \frac{\sec ^6(c+d x) (b \cos (c+d x))^{5/2} \left (2 \sin (c+d x) (10 (5 A+7 C) \cos (2 (c+d x))+110 A+273 B \cos (c+d x)+63 B \cos (3 (c+d x))+70 C)+40 (5 A+7 C) \cos ^{\frac{7}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-504 B \cos ^{\frac{7}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{420 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 11.339, size = 727, normalized size = 3.4 \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{4} + B b^{2} \cos \left (d x + c\right )^{3} + A b^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt{b \cos \left (d x + c\right )} \sec \left (d x + c\right )^{7}, 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{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}} \sec \left (d x + c\right )^{7}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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