Optimal. Leaf size=175 \[ \frac{16 a^2 (15 B+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{64 a^3 (15 B+13 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (9 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 B+13 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d} \]
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Rubi [A] time = 0.409784, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4072, 4010, 4001, 3793, 3792} \[ \frac{16 a^2 (15 B+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{64 a^3 (15 B+13 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (9 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 B+13 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d} \]
Antiderivative was successfully verified.
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Rule 4072
Rule 4010
Rule 4001
Rule 3793
Rule 3792
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} (B+C \sec (c+d x)) \, dx\\ &=\frac{2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac{2 \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac{7 a C}{2}+\frac{1}{2} a (9 B-2 C) \sec (c+d x)\right ) \, dx}{9 a}\\ &=\frac{2 (9 B-2 C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac{2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac{1}{21} (15 B+13 C) \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx\\ &=\frac{2 a (15 B+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac{2 (9 B-2 C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac{2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac{1}{105} (8 a (15 B+13 C)) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac{16 a^2 (15 B+13 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac{2 a (15 B+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac{2 (9 B-2 C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac{2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac{1}{315} \left (32 a^2 (15 B+13 C)\right ) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{64 a^3 (15 B+13 C) \tan (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{16 a^2 (15 B+13 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac{2 a (15 B+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac{2 (9 B-2 C) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac{2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}\\ \end{align*}
Mathematica [A] time = 0.647338, size = 96, normalized size = 0.55 \[ \frac{2 a^3 \tan (c+d x) \left (5 (9 B+26 C) \sec ^3(c+d x)+3 (60 B+73 C) \sec ^2(c+d x)+(345 B+292 C) \sec (c+d x)+690 B+35 C \sec ^4(c+d x)+584 C\right )}{315 d \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.295, size = 141, normalized size = 0.8 \begin{align*} -{\frac{2\,{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 690\,B \left ( \cos \left ( dx+c \right ) \right ) ^{4}+584\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+345\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+292\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+180\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+219\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+45\,B\cos \left ( dx+c \right ) +130\,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 = 0.511441, size = 346, normalized size = 1.98 \begin{align*} \frac{2 \,{\left (2 \,{\left (345 \, B + 292 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} +{\left (345 \, B + 292 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (60 \, B + 73 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 5 \,{\left (9 \, B + 26 \, C\right )} a^{2} \cos \left (d x + c\right ) + 35 \, 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 )}{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(-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 [A] time = 5.02846, size = 362, normalized size = 2.07 \begin{align*} \frac{8 \,{\left (315 \, \sqrt{2} B a^{7} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 315 \, \sqrt{2} C a^{7} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (840 \, \sqrt{2} B a^{7} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 630 \, \sqrt{2} C a^{7} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (945 \, \sqrt{2} B a^{7} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 819 \, \sqrt{2} C a^{7} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) - 4 \,{\left (135 \, \sqrt{2} B a^{7} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 117 \, \sqrt{2} C a^{7} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) - 2 \,{\left (15 \, \sqrt{2} B a^{7} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, \sqrt{2} C a^{7} \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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