Optimal. Leaf size=218 \[ \frac{a^3 (24 A-49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+31 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{24 d}+\frac{5 a^{5/2} (8 A+5 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{8 d}+\frac{5 a C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{12 d}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}{3 d} \]
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Rubi [A] time = 0.658138, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {4089, 4018, 4015, 3801, 215} \[ \frac{a^3 (24 A-49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+31 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{24 d}+\frac{5 a^{5/2} (8 A+5 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{8 d}+\frac{5 a C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{12 d}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 4089
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 )}{\sqrt{\sec (c+d x)}} \, dx &=\frac{C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac{\int \frac{(a+a \sec (c+d x))^{5/2} \left (\frac{1}{2} a (6 A-C)+\frac{5}{2} a C \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{3 a}\\ &=\frac{5 a C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac{\int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{3}{4} a^2 (8 A-3 C)+\frac{1}{4} a^2 (24 A+31 C) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{6 a}\\ &=\frac{a^2 (24 A+31 C) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{5 a C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac{\int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{1}{8} a^3 (24 A-49 C)+\frac{15}{8} a^3 (8 A+5 C) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{6 a}\\ &=\frac{a^3 (24 A-49 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+31 C) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{5 a C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac{1}{16} \left (5 a^2 (8 A+5 C)\right ) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^3 (24 A-49 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+31 C) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{5 a C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}-\frac{\left (5 a^2 (8 A+5 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 )}{8 d}\\ &=\frac{5 a^{5/2} (8 A+5 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{8 d}+\frac{a^3 (24 A-49 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+31 C) \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{5 a C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac{C \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 6.70738, size = 411, normalized size = 1.89 \[ \frac{5 (8 A+5 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{\sec ^2(c+d x)-1} (a (\sec (c+d x)+1))^{5/2} \left (\log \left (\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right )-\log (\sec (c+d x)+1)\right ) \left (A+C \sec ^2(c+d x)\right )}{4 d \left (1-\cos ^2(c+d x)\right ) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{(a (\sec (c+d x)+1))^{5/2} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \left (\frac{\sec \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right ) \left (49 C \sin \left (\frac{d x}{2}\right )-24 A \sin \left (\frac{d x}{2}\right )\right )}{12 d}-\frac{\tan \left (\frac{c}{2}\right ) \sec (c) (24 A \cos (c)-75 C \cos (c)-26 C)}{12 d}+\frac{4 A \sin (c) \cos (d x)}{d}+\frac{4 A \cos (c) \sin (d x)}{d}+\frac{2 C \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) \sec (c+d x) (4 C \sin (c)+13 C \sin (d x))}{6 d}\right )}{\sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.375, size = 399, normalized size = 1.8 \begin{align*} -{\frac{{a}^{2}}{96\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 120\,A\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 ) \sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}-120\,A\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 ) \sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}+75\,C\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 ) \sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}-75\,C\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 ) \sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}+192\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}-96\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+300\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}-96\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}-164\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}-104\,C\cos \left ( dx+c \right ) -32\,C \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}} \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 = 0.804891, size = 1257, normalized size = 5.77 \begin{align*} \left [\frac{15 \,{\left ({\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} +{\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{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 (48 \, A a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (8 \, A + 25 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 34 \, C a^{2} \cos \left (d x + c\right ) + 8 \, 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 )}}}{96 \,{\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}}, \frac{15 \,{\left ({\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} +{\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{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 (48 \, A a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (8 \, A + 25 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 34 \, C a^{2} \cos \left (d x + c\right ) + 8 \, 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 )}}}{48 \,{\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\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}}}{\sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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