Optimal. Leaf size=163 \[ \frac{a^3 (15 B+13 C) \tan ^3(c+d x)}{60 d}+\frac{a^3 (15 B+13 C) \tan (c+d x)}{5 d}+\frac{a^3 (15 B+13 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 (15 B+13 C) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{(5 B-C) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 a d} \]
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Rubi [A] time = 0.312872, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {4072, 4010, 4001, 3791, 3770, 3767, 8, 3768} \[ \frac{a^3 (15 B+13 C) \tan ^3(c+d x)}{60 d}+\frac{a^3 (15 B+13 C) \tan (c+d x)}{5 d}+\frac{a^3 (15 B+13 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 (15 B+13 C) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{(5 B-C) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 a d} \]
Antiderivative was successfully verified.
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Rule 4072
Rule 4010
Rule 4001
Rule 3791
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sec (c+d x))^3 \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))^3 (B+C \sec (c+d x)) \, dx\\ &=\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac{\int \sec (c+d x) (a+a \sec (c+d x))^3 (4 a C+a (5 B-C) \sec (c+d x)) \, dx}{5 a}\\ &=\frac{(5 B-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac{1}{20} (15 B+13 C) \int \sec (c+d x) (a+a \sec (c+d x))^3 \, dx\\ &=\frac{(5 B-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac{1}{20} (15 B+13 C) \int \left (a^3 \sec (c+d x)+3 a^3 \sec ^2(c+d x)+3 a^3 \sec ^3(c+d x)+a^3 \sec ^4(c+d x)\right ) \, dx\\ &=\frac{(5 B-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac{1}{20} \left (a^3 (15 B+13 C)\right ) \int \sec (c+d x) \, dx+\frac{1}{20} \left (a^3 (15 B+13 C)\right ) \int \sec ^4(c+d x) \, dx+\frac{1}{20} \left (3 a^3 (15 B+13 C)\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{20} \left (3 a^3 (15 B+13 C)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{a^3 (15 B+13 C) \tanh ^{-1}(\sin (c+d x))}{20 d}+\frac{3 a^3 (15 B+13 C) \sec (c+d x) \tan (c+d x)}{40 d}+\frac{(5 B-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac{1}{40} \left (3 a^3 (15 B+13 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^3 (15 B+13 C)\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{20 d}-\frac{\left (3 a^3 (15 B+13 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{20 d}\\ &=\frac{a^3 (15 B+13 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (15 B+13 C) \tan (c+d x)}{5 d}+\frac{3 a^3 (15 B+13 C) \sec (c+d x) \tan (c+d x)}{40 d}+\frac{(5 B-C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac{a^3 (15 B+13 C) \tan ^3(c+d x)}{60 d}\\ \end{align*}
Mathematica [B] time = 0.80796, size = 391, normalized size = 2.4 \[ -\frac{a^3 \sec ^5(c+d x) \left (150 (15 B+13 C) \cos (c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+75 (15 B+13 C) \cos (3 (c+d x)) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-1200 B \sin (c+d x)-1140 B \sin (2 (c+d x))-1560 B \sin (3 (c+d x))-450 B \sin (4 (c+d x))-360 B \sin (5 (c+d x))+225 B \cos (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-225 B \cos (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-1600 C \sin (c+d x)-1500 C \sin (2 (c+d x))-1520 C \sin (3 (c+d x))-390 C \sin (4 (c+d x))-304 C \sin (5 (c+d x))+195 C \cos (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-195 C \cos (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{1920 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 234, normalized size = 1.4 \begin{align*} 3\,{\frac{B{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{13\,{a}^{3}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{13\,{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{15\,B{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{15\,B{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{38\,{a}^{3}C\tan \left ( dx+c \right ) }{15\,d}}+{\frac{19\,{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{3\,{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.958117, size = 455, normalized size = 2.79 \begin{align*} \frac{240 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 16 \,{\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^{3} + 240 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 15 \, B a^{3}{\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 )} - 45 \, C a^{3}{\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 )} - 180 \, B a^{3}{\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 )} - 60 \, C a^{3}{\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 )} + 240 \, B a^{3} \tan \left (d x + c\right )}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.522507, size = 431, normalized size = 2.64 \begin{align*} \frac{15 \,{\left (15 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (15 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (45 \, B + 38 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \,{\left (15 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \,{\left (15 \, B + 19 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 30 \,{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 24 \, C a^{3}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \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^{3} \left (\int B \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{4}{\left (c + d x \right )}\, dx + \int B \sec ^{5}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{5}{\left (c + d x \right )}\, dx + \int C \sec ^{6}{\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 = 1.19213, size = 332, normalized size = 2.04 \begin{align*} \frac{15 \,{\left (15 \, B a^{3} + 13 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (15 \, B a^{3} + 13 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (225 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 195 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1050 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 910 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1920 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1664 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1830 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1330 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 735 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 765 \, C a^{3} \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 )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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