Optimal. Leaf size=116 \[ \frac{2 B \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{3 b^3 d}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.10671, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {4047, 3769, 3771, 2641, 12, 16, 2639} \[ \frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 b^3 d}+\frac{2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 3769
Rule 3771
Rule 2641
Rule 12
Rule 16
Rule 2639
Rubi steps
\begin{align*} \int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx &=\frac{B \int \frac{1}{(b \sec (c+d x))^{3/2}} \, dx}{b}+\int \frac{C \sec ^2(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx\\ &=\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{B \int \sqrt{b \sec (c+d x)} \, dx}{3 b^3}+C \int \frac{\sec ^2(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx\\ &=\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{C \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx}{b^2}+\frac{\left (B \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b^3}\\ &=\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 b^3 d}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{C \int \sqrt{\cos (c+d x)} \, dx}{b^2 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}\\ &=\frac{2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 b^3 d}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.170775, size = 81, normalized size = 0.7 \[ \frac{2 B \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\frac{B \sin (2 (c+d x))}{\sqrt{\cos (c+d x)}}+6 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.204, size = 470, normalized size = 4.1 \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 \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )}{\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 \sec \left (d x + c\right ) + B\right )} \sqrt{b \sec \left (d x + c\right )}}{b^{3} \sec \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (B + C \sec{\left (c + d x \right )}\right ) \sec{\left (c + d x \right )}}{\left (b \sec{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx \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 \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )}{\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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