Optimal. Leaf size=115 \[ \frac{2 (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 (9 A+7 C) \sin (c+d x)}{45 b d (b \sec (c+d x))^{3/2}}+\frac{2 b^2 C \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}} \]
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Rubi [A] time = 0.127748, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3238, 4045, 3769, 3771, 2639} \[ \frac{2 (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 (9 A+7 C) \sin (c+d x)}{45 b d (b \sec (c+d x))^{3/2}}+\frac{2 b^2 C \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 4045
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx &=b^2 \int \frac{C+A \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx\\ &=\frac{2 b^2 C \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}+\frac{1}{9} (9 A+7 C) \int \frac{1}{(b \sec (c+d x))^{5/2}} \, dx\\ &=\frac{2 (9 A+7 C) \sin (c+d x)}{45 b d (b \sec (c+d x))^{3/2}}+\frac{2 b^2 C \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}+\frac{(9 A+7 C) \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx}{15 b^2}\\ &=\frac{2 (9 A+7 C) \sin (c+d x)}{45 b d (b \sec (c+d x))^{3/2}}+\frac{2 b^2 C \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}+\frac{(9 A+7 C) \int \sqrt{\cos (c+d x)} \, dx}{15 b^2 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}\\ &=\frac{2 (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 (9 A+7 C) \sin (c+d x)}{45 b d (b \sec (c+d x))^{3/2}}+\frac{2 b^2 C \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.622843, size = 81, normalized size = 0.7 \[ \frac{4 \sin (2 (c+d x)) (18 A+5 C \cos (2 (c+d x))+19 C)+\frac{48 (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{\sqrt{\cos (c+d x)}}}{360 b^2 d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.547, size = 636, normalized size = 5.5 \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{C \cos \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac{5}{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} + A\right )} \sqrt{b \sec \left (d x + c\right )}}{b^{3} \sec \left (d x + c\right )^{3}}, 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} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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