Optimal. Leaf size=188 \[ \frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}-\frac{2 (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^3 d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.188985, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {3021, 2748, 2636, 2642, 2641, 2640, 2639} \[ \frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}-\frac{2 (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^3 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3021
Rule 2748
Rule 2636
Rule 2642
Rule 2641
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/2}} \, dx &=\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 \int \frac{\frac{5 b^2 B}{2}+\frac{1}{2} b^2 (3 A+5 C) \cos (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx}{5 b^3}\\ &=\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{B \int \frac{1}{(b \cos (c+d x))^{5/2}} \, dx}{b}+\frac{(3 A+5 C) \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx}{5 b^2}\\ &=\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d (b \cos (c+d x))^{3/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}+\frac{B \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx}{3 b^3}-\frac{(3 A+5 C) \int \sqrt{b \cos (c+d x)} \, dx}{5 b^4}\\ &=\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d (b \cos (c+d x))^{3/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}+\frac{\left (B \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b^3 \sqrt{b \cos (c+d x)}}-\frac{\left ((3 A+5 C) \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 b^4 \sqrt{\cos (c+d x)}}\\ &=-\frac{2 (3 A+5 C) \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^3 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d (b \cos (c+d x))^{3/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.118158, size = 119, normalized size = 0.63 \[ \frac{2 \left (-3 (3 A+5 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+9 A \sin (c+d x)+3 A \tan (c+d x) \sec (c+d x)+5 B \tan (c+d x)+5 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+15 C \sin (c+d x)\right )}{15 b^3 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 10.546, size = 807, normalized size = 4.3 \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 )\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{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt{b \cos \left (d x + c\right )}}{b^{4} \cos \left (d x + c\right )^{4}}, 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{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{\left (b \cos \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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