Optimal. Leaf size=283 \[ \frac{a^2 (80 A+90 B+67 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (176 A+150 B+133 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a (10 B+3 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d} \]
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Rubi [A] time = 0.748112, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4088, 4018, 4016, 3803, 3801, 215} \[ \frac{a^2 (80 A+90 B+67 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (176 A+150 B+133 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a (10 B+3 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 4088
Rule 4018
Rule 4016
Rule 3803
Rule 3801
Rule 215
Rubi steps
\begin{align*} \int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{5}{2} a (2 A+C)+\frac{1}{2} a (10 B+3 C) \sec (c+d x)\right ) \, dx}{5 a}\\ &=\frac{a (10 B+3 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{\int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{5}{4} a^2 (16 A+10 B+11 C)+\frac{1}{4} a^2 (80 A+90 B+67 C) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac{a^2 (80 A+90 B+67 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt{a+a \sec (c+d x)}}+\frac{a (10 B+3 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{1}{96} (a (176 A+150 B+133 C)) \int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (176 A+150 B+133 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (80 A+90 B+67 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt{a+a \sec (c+d x)}}+\frac{a (10 B+3 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{1}{128} (a (176 A+150 B+133 C)) \int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (176 A+150 B+133 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{128 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+150 B+133 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (80 A+90 B+67 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt{a+a \sec (c+d x)}}+\frac{a (10 B+3 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{1}{256} (a (176 A+150 B+133 C)) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (176 A+150 B+133 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{128 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+150 B+133 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (80 A+90 B+67 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt{a+a \sec (c+d x)}}+\frac{a (10 B+3 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}-\frac{(a (176 A+150 B+133 C)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{128 d}\\ &=\frac{a^{3/2} (176 A+150 B+133 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{128 d}+\frac{a^2 (176 A+150 B+133 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{128 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+150 B+133 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (80 A+90 B+67 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt{a+a \sec (c+d x)}}+\frac{a (10 B+3 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 3.89925, size = 211, normalized size = 0.75 \[ \frac{a \sec \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left (4 \sin \left (\frac{1}{2} (c+d x)\right ) (12 (880 A+1070 B+1273 C) \cos (c+d x)+4 (3280 A+3450 B+3059 C) \cos (2 (c+d x))+3520 A \cos (3 (c+d x))+2640 A \cos (4 (c+d x))+10480 A+3000 B \cos (3 (c+d x))+2250 B \cos (4 (c+d x))+11550 B+2660 C \cos (3 (c+d x))+1995 C \cos (4 (c+d x))+13313 C)+240 \sqrt{2} (176 A+150 B+133 C) \cos ^5(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{61440 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.417, size = 732, normalized size = 2.6 \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(-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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Fricas [A] time = 2.41543, size = 1543, normalized size = 5.45 \begin{align*} \text{result too large to display} \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{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sec \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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