Optimal. Leaf size=147 \[ \frac{2 B \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{3 b^3 d}+\frac{6 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.10415, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3787, 3769, 3771, 2639, 2641} \[ \frac{6 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 b^3 d} \]
Antiderivative was successfully verified.
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Rule 3787
Rule 3769
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)}{(b \sec (c+d x))^{5/2}} \, dx &=A \int \frac{1}{(b \sec (c+d x))^{5/2}} \, dx+\frac{B \int \frac{1}{(b \sec (c+d x))^{3/2}} \, dx}{b}\\ &=\frac{2 A \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{(3 A) \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx}{5 b^2}+\frac{B \int \sqrt{b \sec (c+d x)} \, dx}{3 b^3}\\ &=\frac{2 A \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{(3 A) \int \sqrt{\cos (c+d x)} \, dx}{5 b^2 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\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{6 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 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 A \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.508454, size = 88, normalized size = 0.6 \[ \frac{2 \left (5 B \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} (3 A \cos (c+d x)+5 B)+9 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{15 d \cos ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.198, size = 482, normalized size = 3.3 \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{B \sec \left (d x + c\right ) + 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 (B \sec \left (d x + c\right ) + 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B \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{B \sec \left (d x + c\right ) + 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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