Optimal. Leaf size=222 \[ \frac{a^2 (10 A+9 B+8 C) \tan ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B+8 C) \tan (c+d x)}{5 d}+\frac{a^2 (14 A+12 B+11 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^2 (10 A+12 B+9 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{a^2 (14 A+12 B+11 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(3 B+C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{15 d}+\frac{C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^2}{6 d} \]
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Rubi [A] time = 0.429762, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4088, 4018, 3997, 3787, 3768, 3770, 3767} \[ \frac{a^2 (10 A+9 B+8 C) \tan ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B+8 C) \tan (c+d x)}{5 d}+\frac{a^2 (14 A+12 B+11 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^2 (10 A+12 B+9 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{a^2 (14 A+12 B+11 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(3 B+C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{15 d}+\frac{C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^2}{6 d} \]
Antiderivative was successfully verified.
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Rule 4088
Rule 4018
Rule 3997
Rule 3787
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{\int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (3 a (2 A+C)+2 a (3 B+C) \sec (c+d x)) \, dx}{6 a}\\ &=\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{(3 B+C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{\int \sec ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^2 (10 A+6 B+7 C)+3 a^2 (10 A+12 B+9 C) \sec (c+d x)\right ) \, dx}{30 a}\\ &=\frac{a^2 (10 A+12 B+9 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{(3 B+C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{\int \sec ^3(c+d x) \left (15 a^3 (14 A+12 B+11 C)+24 a^3 (10 A+9 B+8 C) \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac{a^2 (10 A+12 B+9 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{(3 B+C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{1}{5} \left (a^2 (10 A+9 B+8 C)\right ) \int \sec ^4(c+d x) \, dx+\frac{1}{8} \left (a^2 (14 A+12 B+11 C)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{a^2 (14 A+12 B+11 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^2 (10 A+12 B+9 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{(3 B+C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{1}{16} \left (a^2 (14 A+12 B+11 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^2 (10 A+9 B+8 C)\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac{a^2 (14 A+12 B+11 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^2 (10 A+9 B+8 C) \tan (c+d x)}{5 d}+\frac{a^2 (14 A+12 B+11 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^2 (10 A+12 B+9 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac{(3 B+C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{a^2 (10 A+9 B+8 C) \tan ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 3.29793, size = 359, normalized size = 1.62 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^6(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (15 (14 A+12 B+11 C) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) \cos ^5(c+d x) (15 \sin (c) (14 A+12 B+11 C)+32 (10 A+9 B+8 C) \sin (d x))-\sec (c) \cos ^4(c+d x) (16 \sin (c) (10 A+9 B+8 C)+15 (14 A+12 B+11 C) \sin (d x))-2 \sec (c) \cos ^3(c+d x) (5 \sin (c) (6 A+12 B+11 C)+8 (10 A+9 B+8 C) \sin (d x))-2 \sec (c) \cos ^2(c+d x) (5 (6 A+12 B+11 C) \sin (d x)+24 (B+2 C) \sin (c))-8 \sec (c) \cos (c+d x) (6 (B+2 C) \sin (d x)+5 C \sin (c))-40 C \sec (c) \sin (d x)\right )}{480 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.068, size = 386, normalized size = 1.7 \begin{align*}{\frac{7\,{a}^{2}A\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{7\,{a}^{2}A\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{6\,B{a}^{2}\tan \left ( dx+c \right ) }{5\,d}}+{\frac{3\,B{a}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{5\,d}}+{\frac{11\,{a}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{11\,{a}^{2}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{11\,{a}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{4\,{a}^{2}A\tan \left ( dx+c \right ) }{3\,d}}+{\frac{2\,{a}^{2}A\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{B{a}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{2\,d}}+{\frac{3\,B{a}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,B{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{16\,{a}^{2}C\tan \left ( dx+c \right ) }{15\,d}}+{\frac{2\,{a}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{8\,{a}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{{a}^{2}A\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{B{a}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{{a}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{6\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.977191, size = 644, normalized size = 2.9 \begin{align*} \frac{320 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} + 32 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{2} + 160 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} + 64 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{2} - 5 \, C a^{2}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, A a^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, B a^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, C a^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, A a^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.555197, size = 535, normalized size = 2.41 \begin{align*} \frac{15 \,{\left (14 \, A + 12 \, B + 11 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (14 \, A + 12 \, B + 11 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (32 \,{\left (10 \, A + 9 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 15 \,{\left (14 \, A + 12 \, B + 11 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 16 \,{\left (10 \, A + 9 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \,{\left (6 \, A + 12 \, B + 11 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 48 \,{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right ) + 40 \, C a^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \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*} a^{2} \left (\int A \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 A \sec ^{4}{\left (c + d x \right )}\, dx + \int A \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int 2 B \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx + \int 2 C \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{7}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36714, size = 529, normalized size = 2.38 \begin{align*} \frac{15 \,{\left (14 \, A a^{2} + 12 \, B a^{2} + 11 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (14 \, A a^{2} + 12 \, B a^{2} + 11 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (210 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 180 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 165 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 1190 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1020 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 935 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 2580 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 2568 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1986 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 3180 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2808 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3006 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2330 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1860 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1305 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 750 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 780 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 795 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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