Optimal. Leaf size=147 \[ \frac{2 (7 A+5 C) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 B \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.152271, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {4064, 3021, 2748, 2636, 2639, 2641} \[ \frac{2 (7 A+5 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 B \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4064
Rule 3021
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx &=\int \frac{C+B \cos (c+d x)+A \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2}{7} \int \frac{\frac{7 B}{2}+\frac{1}{2} (7 A+5 C) \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+B \int \frac{1}{\cos ^{\frac{7}{2}}(c+d x)} \, dx+\frac{1}{7} (7 A+5 C) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{1}{5} (3 B) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{21} (7 A+5 C) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (7 A+5 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 B \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{1}{5} (3 B) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 (7 A+5 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 B \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.560619, size = 129, normalized size = 0.88 \[ \frac{10 (7 A+5 C) \cos ^{\frac{5}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+35 A \sin (2 (c+d x))+42 B \sin (c+d x)+126 B \sin (c+d x) \cos ^2(c+d x)-126 B \cos ^{\frac{5}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+25 C \sin (2 (c+d x))+30 C \tan (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.17, size = 684, normalized size = 4.7 \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 ) + A}{\cos \left (d x + c\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{C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{\cos \left (d x + c\right )^{\frac{5}{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{C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{\cos \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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