Optimal. Leaf size=253 \[ \frac{a^3 (432 A+392 B+299 C) \sin (c+d x)}{192 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{32 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{a^{5/2} (304 A+200 B+163 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{64 d}+\frac{a (8 B+5 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.891092, antiderivative size = 253, 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 = {4265, 4088, 4018, 4016, 3801, 215} \[ \frac{a^3 (432 A+392 B+299 C) \sin (c+d x)}{192 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{32 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{a^{5/2} (304 A+200 B+163 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{64 d}+\frac{a (8 B+5 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4265
Rule 4088
Rule 4018
Rule 4016
Rule 3801
Rule 215
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \left (\frac{1}{2} a (8 A+C)+\frac{1}{2} a (8 B+5 C) \sec (c+d x)\right ) \, dx}{4 a}\\ &=\frac{a (8 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \left (\frac{1}{4} a^2 (48 A+8 B+11 C)+\frac{3}{4} a^2 (16 A+24 B+17 C) \sec (c+d x)\right ) \, dx}{12 a}\\ &=\frac{a^2 (16 A+24 B+17 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{a (8 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \left (\frac{1}{8} a^3 (240 A+104 B+95 C)+\frac{1}{8} a^3 (432 A+392 B+299 C) \sec (c+d x)\right ) \, dx}{24 a}\\ &=\frac{a^3 (432 A+392 B+299 C) \sin (c+d x)}{192 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (16 A+24 B+17 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{a (8 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{1}{128} \left (a^2 (304 A+200 B+163 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^3 (432 A+392 B+299 C) \sin (c+d x)}{192 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (16 A+24 B+17 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{a (8 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{\left (a^2 (304 A+200 B+163 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 )}{64 d}\\ &=\frac{a^{5/2} (304 A+200 B+163 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)}}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x)}{192 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (16 A+24 B+17 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{a (8 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 4.07389, size = 178, normalized size = 0.7 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) ((1584 A+2056 B+2203 C) \cos (c+d x)+4 (48 A+136 B+163 C) \cos (2 (c+d x))+528 A \cos (3 (c+d x))+192 A+600 B \cos (3 (c+d x))+544 B+489 C \cos (3 (c+d x))+844 C)+6 \sqrt{2} (304 A+200 B+163 C) \cos ^4(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{768 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.348, size = 629, normalized size = 2.5 \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 = 1.64701, size = 1415, normalized size = 5.59 \begin{align*} \left [\frac{4 \,{\left (3 \,{\left (176 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \,{\left (48 \, A + 136 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \,{\left (8 \, B + 23 \, C\right )} a^{2} \cos \left (d x + c\right ) + 48 \, C a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 3 \,{\left ({\left (304 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} +{\left (304 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{4}\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 4 \, \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}{\left (\cos \left (d x + c\right ) - 2\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right )^{2} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right )}{768 \,{\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}}, \frac{2 \,{\left (3 \,{\left (176 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \,{\left (48 \, A + 136 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \,{\left (8 \, B + 23 \, C\right )} a^{2} \cos \left (d x + c\right ) + 48 \, C a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 3 \,{\left ({\left (304 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} +{\left (304 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{4}\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right )}{384 \,{\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}}\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{{\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{5}{2}}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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