Optimal. Leaf size=210 \[ \frac{a^3 (64 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (16 A-15 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{5 a^{5/2} C \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.656814, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {4087, 4017, 4018, 4015, 3801, 215} \[ \frac{a^3 (64 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (16 A-15 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{5 a^{5/2} C \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4087
Rule 4017
Rule 4018
Rule 4015
Rule 3801
Rule 215
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \sec (c+d x))^{5/2} \left (\frac{5 a A}{2}-\frac{1}{2} a (2 A-5 C) \sec (c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{5 a}\\ &=\frac{2 a A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 \int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{3}{4} a^2 (8 A+5 C)-\frac{1}{4} a^2 (16 A-15 C) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{15 a}\\ &=-\frac{a^2 (16 A-15 C) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{1}{8} a^3 (64 A+15 C)+\frac{75}{8} a^3 C \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{15 a}\\ &=\frac{a^3 (64 A+15 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d \sqrt{a+a \sec (c+d x)}}-\frac{a^2 (16 A-15 C) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{2} \left (5 a^2 C\right ) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^3 (64 A+15 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d \sqrt{a+a \sec (c+d x)}}-\frac{a^2 (16 A-15 C) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{\left (5 a^2 C\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{5 a^{5/2} C \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}+\frac{a^3 (64 A+15 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d \sqrt{a+a \sec (c+d x)}}-\frac{a^2 (16 A-15 C) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [B] time = 6.58031, size = 428, normalized size = 2.04 \[ \frac{10 C \sin (c+d x) \cos ^3(c+d x) \sqrt{\sec ^2(c+d x)-1} (a (\sec (c+d x)+1))^{5/2} \left (\log \left (\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right )-\log (\sec (c+d x)+1)\right ) \left (A+C \sec ^2(c+d x)\right )}{d \left (1-\cos ^2(c+d x)\right ) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{(a (\sec (c+d x)+1))^{5/2} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \left (\frac{(131 A+60 C) \sin (c) \cos (d x)}{15 d}+\frac{(131 A+60 C) \cos (c) \sin (d x)}{15 d}-\frac{2 \sec \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right ) \left (64 A \sin \left (\frac{d x}{2}\right )+15 C \sin \left (\frac{d x}{2}\right )\right )}{15 d}-\frac{2 (64 A+15 C) \tan \left (\frac{c}{2}\right )}{15 d}+\frac{22 A \sin (2 c) \cos (2 d x)}{15 d}+\frac{A \sin (3 c) \cos (3 d x)}{5 d}+\frac{22 A \cos (2 c) \sin (2 d x)}{15 d}+\frac{A \cos (3 c) \sin (3 d x)}{5 d}\right )}{\sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.413, size = 255, normalized size = 1.2 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{60\,d\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 75\,C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \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 ) \sqrt{2}-75\,C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \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 ) \sqrt{2}+24\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+88\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+232\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+120\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}-344\,A\cos \left ( dx+c \right ) -60\,C\cos \left ( dx+c \right ) -60\,C \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{5}{2}}}} \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 = 0.637299, size = 1114, normalized size = 5.3 \begin{align*} \left [\frac{75 \,{\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} - 7 \, a \cos \left (d x + c\right )^{2} - \frac{4 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + \frac{4 \,{\left (6 \, A a^{2} \cos \left (d x + c\right )^{3} + 28 \, A a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (43 \, A + 15 \, C\right )} a^{2} \cos \left (d x + c\right ) + 15 \, C a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{60 \,{\left (d \cos \left (d x + c\right ) + d\right )}}, \frac{75 \,{\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 ) + \frac{2 \,{\left (6 \, A a^{2} \cos \left (d x + c\right )^{3} + 28 \, A a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (43 \, A + 15 \, C\right )} a^{2} \cos \left (d x + c\right ) + 15 \, C a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{30 \,{\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 \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{5}{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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