Optimal. Leaf size=147 \[ \frac{2 C \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{3 b^4 d}+\frac{2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac{6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 C \sin (c+d x)}{3 b^3 d \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.134684, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {4047, 3769, 3771, 2639, 12, 16, 2641} \[ \frac{2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac{6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 C \sin (c+d x)}{3 b^3 d \sqrt{b \sec (c+d x)}}+\frac{2 C \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 b^4 d} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 3769
Rule 3771
Rule 2639
Rule 12
Rule 16
Rule 2641
Rubi steps
\begin{align*} \int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx &=\frac{B \int \frac{1}{(b \sec (c+d x))^{5/2}} \, dx}{b}+\int \frac{C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx\\ &=\frac{2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac{(3 B) \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx}{5 b^3}+C \int \frac{\sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx\\ &=\frac{2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac{C \int \frac{1}{(b \sec (c+d x))^{3/2}} \, dx}{b^2}+\frac{(3 B) \int \sqrt{\cos (c+d x)} \, dx}{5 b^3 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}\\ &=\frac{6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac{2 C \sin (c+d x)}{3 b^3 d \sqrt{b \sec (c+d x)}}+\frac{C \int \sqrt{b \sec (c+d x)} \, dx}{3 b^4}\\ &=\frac{6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac{2 C \sin (c+d x)}{3 b^3 d \sqrt{b \sec (c+d x)}}+\frac{\left (C \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b^4}\\ &=\frac{6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 C \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^4 d}+\frac{2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac{2 C \sin (c+d x)}{3 b^3 d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.531016, size = 91, normalized size = 0.62 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)} \left (5 C \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} (3 B \cos (c+d x)+5 C)+9 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{15 b^4 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.25, 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{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{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 \sec \left (d x + c\right ) + B\right )} \sqrt{b \sec \left (d x + c\right )}}{b^{4} \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 \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )}{\left (b \sec \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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