Optimal. Leaf size=187 \[ \frac{2 a (9 B+8 C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{4 (9 B+8 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 a d}-\frac{8 (9 B+8 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{4 a (9 B+8 C) \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a C \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.418851, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4072, 4016, 3803, 3800, 4001, 3792} \[ \frac{2 a (9 B+8 C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{4 (9 B+8 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 a d}-\frac{8 (9 B+8 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{4 a (9 B+8 C) \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a C \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 4072
Rule 4016
Rule 3803
Rule 3800
Rule 4001
Rule 3792
Rubi steps
\begin{align*} \int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} (B+C \sec (c+d x)) \, dx\\ &=\frac{2 a C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{9} (9 B+8 C) \int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a (9 B+8 C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{21} (2 (9 B+8 C)) \int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a (9 B+8 C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{4 (9 B+8 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}+\frac{(4 (9 B+8 C)) \int \sec (c+d x) \left (\frac{3 a}{2}-a \sec (c+d x)\right ) \sqrt{a+a \sec (c+d x)} \, dx}{105 a}\\ &=\frac{2 a (9 B+8 C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}-\frac{8 (9 B+8 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac{4 (9 B+8 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}+\frac{1}{45} (2 (9 B+8 C)) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{4 a (9 B+8 C) \tan (c+d x)}{45 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (9 B+8 C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}-\frac{8 (9 B+8 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac{4 (9 B+8 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}\\ \end{align*}
Mathematica [A] time = 0.536767, size = 98, normalized size = 0.52 \[ \frac{2 a \tan (c+d x) \left (5 (9 B+8 C) \sec ^3(c+d x)+6 (9 B+8 C) \sec ^2(c+d x)+8 (9 B+8 C) \sec (c+d x)+16 (9 B+8 C)+35 C \sec ^4(c+d x)\right )}{315 d \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.398, size = 138, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( 144\,B \left ( \cos \left ( dx+c \right ) \right ) ^{4}+128\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+72\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+64\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+54\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+48\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+45\,B\cos \left ( dx+c \right ) +40\,C\cos \left ( dx+c \right ) +35\,C \right ) }{315\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}} \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 = 0.505987, size = 308, normalized size = 1.65 \begin{align*} \frac{2 \,{\left (16 \,{\left (9 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{4} + 8 \,{\left (9 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{3} + 6 \,{\left (9 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{2} + 5 \,{\left (9 \, B + 8 \, C\right )} \cos \left (d x + c\right ) + 35 \, C\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 )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )} \left (B + C \sec{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 4.56227, size = 362, normalized size = 1.94 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{2} B a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 315 \, \sqrt{2} C a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (630 \, \sqrt{2} B a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 420 \, \sqrt{2} C a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (756 \, \sqrt{2} B a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 882 \, \sqrt{2} C a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (522 \, \sqrt{2} B a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 324 \, \sqrt{2} C a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (81 \, \sqrt{2} B a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 107 \, \sqrt{2} C a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{315 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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