Optimal. Leaf size=113 \[ -\frac{2 (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}} \]
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Rubi [A] time = 0.131559, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {16, 3012, 2636, 2640, 2639} \[ -\frac{2 (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3012
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx &=b^2 \int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/2}} \, dx\\ &=\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{1}{5} (3 A+5 C) \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx\\ &=\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}-\frac{(3 A+5 C) \int \sqrt{b \cos (c+d x)} \, dx}{5 b^2}\\ &=\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b d \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^2 \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^2 d \sqrt{\cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.272408, size = 81, normalized size = 0.72 \[ \frac{2 \left ((3 A+5 C) \sin (c+d x)-(3 A+5 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+A \tan (c+d x) \sec (c+d x)\right )}{5 b d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 8.447, size = 601, normalized size = 5.3 \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{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac{3}{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 \cos \left (d x + c\right )} \sec \left (d x + c\right )^{2}}{b^{2} \cos \left (d x + c\right )^{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 )} \sec \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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