Optimal. Leaf size=230 \[ \frac{2 a^3 (32 A+49 C) \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.781066, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {4265, 4087, 4017, 4015, 3801, 215} \[ \frac{2 a^3 (32 A+49 C) \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 4265
Rule 4087
Rule 4017
Rule 4015
Rule 3801
Rule 215
Rubi steps
\begin{align*} \int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 A \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{5/2} \left (\frac{5 a A}{2}+\frac{7}{2} a C \sec (c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{7 a}\\ &=\frac{2 a A \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{5}{4} a^2 (8 A+7 C)+\frac{35}{4} a^2 C \sec (c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac{2 a^2 (8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{5}{8} a^3 (32 A+49 C)+\frac{105}{8} a^3 C \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{105 a}\\ &=\frac{2 a^3 (32 A+49 C) \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}+\left (a^2 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{2 a^3 (32 A+49 C) \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}-\frac{\left (2 a^2 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 )}{d}\\ &=\frac{2 a^{5/2} 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)}}{d}+\frac{2 a^3 (32 A+49 C) \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 1.50132, size = 125, normalized size = 0.54 \[ \frac{a^2 \sqrt{\cos (c+d x)} \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (2 \sin \left (\frac{1}{2} (c+d x)\right ) ((101 A+28 C) \cos (c+d x)+24 A \cos (2 (c+d x))+3 A \cos (3 (c+d x))+208 A+224 C)+84 \sqrt{2} C \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{84 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.272, size = 236, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{42\,d\sin \left ( dx+c \right ) }\sqrt{\cos \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 12\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}-21\,C\sqrt{2}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( \cos \left ( dx+c \right ) +1-\sin \left ( dx+c \right ) \right ) \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sin \left ( dx+c \right ) +21\,C\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( \cos \left ( dx+c \right ) +1+\sin \left ( dx+c \right ) \right ) \right ) \sin \left ( dx+c \right ) +36\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+44\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+28\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+92\,A\cos \left ( dx+c \right ) +196\,C\cos \left ( dx+c \right ) -184\,A-224\,C \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.22792, size = 1146, normalized size = 4.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.587811, size = 1115, normalized size = 4.85 \begin{align*} \left [\frac{4 \,{\left (3 \, A a^{2} \cos \left (d x + c\right )^{3} + 12 \, A a^{2} \cos \left (d x + c\right )^{2} +{\left (23 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \,{\left (23 \, A + 28 \, C\right )} 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 ) + 21 \,{\left (C a^{2} \cos \left (d x + c\right ) + C a^{2}\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 )}{42 \,{\left (d \cos \left (d x + c\right ) + d\right )}}, \frac{2 \,{\left (3 \, A a^{2} \cos \left (d x + c\right )^{3} + 12 \, A a^{2} \cos \left (d x + c\right )^{2} +{\left (23 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \,{\left (23 \, A + 28 \, C\right )} 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 ) + 21 \,{\left (C a^{2} \cos \left (d x + c\right ) + C a^{2}\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 )}{21 \,{\left (d \cos \left (d x + c\right ) + d\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{\left (C \sec \left (d x + c\right )^{2} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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