Optimal. Leaf size=231 \[ \frac{64 a^3 (13 A+15 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (13 A+15 B+21 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d \sqrt{\sec (c+d x)}}+\frac{2 a (13 A+15 B+21 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (5 A+9 B) \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.555245, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {4086, 4013, 3809, 3804} \[ \frac{64 a^3 (13 A+15 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (13 A+15 B+21 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d \sqrt{\sec (c+d x)}}+\frac{2 a (13 A+15 B+21 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (5 A+9 B) \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 4013
Rule 3809
Rule 3804
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 )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \sec (c+d x))^{5/2} \left (\frac{1}{2} a (5 A+9 B)+\frac{1}{2} a (2 A+9 C) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (5 A+9 B) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{21} (13 A+15 B+21 C) \int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a (13 A+15 B+21 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (5 A+9 B) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{105} (8 a (13 A+15 B+21 C)) \int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{16 a^2 (13 A+15 B+21 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}+\frac{2 a (13 A+15 B+21 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (5 A+9 B) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{315} \left (32 a^2 (13 A+15 B+21 C)\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{64 a^3 (13 A+15 B+21 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{16 a^2 (13 A+15 B+21 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}+\frac{2 a (13 A+15 B+21 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (5 A+9 B) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 1.63969, size = 124, normalized size = 0.54 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} ((3116 A+3030 B+2352 C) \cos (c+d x)+4 (254 A+180 B+63 C) \cos (2 (c+d x))+260 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+5653 A+90 B \cos (3 (c+d x))+6240 B+7476 C)}{1260 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.371, size = 166, normalized size = 0.7 \begin{align*} -{\frac{2\,{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 35\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+130\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+45\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+219\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+180\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+63\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+292\,A\cos \left ( dx+c \right ) +345\,B\cos \left ( dx+c \right ) +294\,C\cos \left ( dx+c \right ) +584\,A+690\,B+903\,C \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{315\,d\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.41028, size = 1085, normalized size = 4.7 \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 = 0.497328, size = 400, normalized size = 1.73 \begin{align*} \frac{2 \,{\left (35 \, A a^{2} \cos \left (d x + c\right )^{5} + 5 \,{\left (26 \, A + 9 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 3 \,{\left (73 \, A + 60 \, B + 21 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} +{\left (292 \, A + 345 \, B + 294 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (584 \, A + 690 \, B + 903 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \,{\left (d \cos \left (d x + c\right ) + d\right )} \sqrt{\cos \left (d x + c\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}}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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