Optimal. Leaf size=303 \[ \frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a (10 B+3 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.868542, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {4265, 4088, 4018, 4016, 3803, 3801, 215} \[ \frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a (10 B+3 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4265
Rule 4088
Rule 4018
Rule 4016
Rule 3803
Rule 3801
Rule 215
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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 (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a (10 B+3 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{96} \left (a (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a (10 B+3 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{128} \left (a (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a (10 B+3 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{256} \left (a (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a (10 B+3 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{\left (a (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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 ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{128 d}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a (10 B+3 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 5.92914, size = 210, normalized size = 0.69 \[ \frac{a \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) (12 (880 A+1070 B+1273 C) \cos (c+d x)+4 (3280 A+3450 B+3059 C) \cos (2 (c+d x))+3520 A \cos (3 (c+d x))+2640 A \cos (4 (c+d x))+10480 A+3000 B \cos (3 (c+d x))+2250 B \cos (4 (c+d x))+11550 B+2660 C \cos (3 (c+d x))+1995 C \cos (4 (c+d x))+13313 C)+60 \sqrt{2} (176 A+150 B+133 C) \cos ^5(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{15360 d \cos ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.382, size = 720, normalized size = 2.4 \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 = 1.65031, size = 1523, normalized size = 5.03 \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 \frac{{\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}}}{\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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