Optimal. Leaf size=147 \[ \frac{a^2 (12 A+8 B+7 C) \tan (c+d x)}{6 d}+\frac{a^2 (12 A+8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (12 A+8 B+7 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(4 B-C) \tan (c+d x) (a \sec (c+d x)+a)^2}{12 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 a d} \]
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Rubi [A] time = 0.234871, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.18, Rules used = {4082, 4001, 3788, 3767, 8, 4046, 3770} \[ \frac{a^2 (12 A+8 B+7 C) \tan (c+d x)}{6 d}+\frac{a^2 (12 A+8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (12 A+8 B+7 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(4 B-C) \tan (c+d x) (a \sec (c+d x)+a)^2}{12 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 a d} \]
Antiderivative was successfully verified.
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Rule 4082
Rule 4001
Rule 3788
Rule 3767
Rule 8
Rule 4046
Rule 3770
Rubi steps
\begin{align*} \int \sec (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 (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac{\int \sec (c+d x) (a+a \sec (c+d x))^2 (a (4 A+3 C)+a (4 B-C) \sec (c+d x)) \, dx}{4 a}\\ &=\frac{(4 B-C) (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac{C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac{1}{12} (12 A+8 B+7 C) \int \sec (c+d x) (a+a \sec (c+d x))^2 \, dx\\ &=\frac{(4 B-C) (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac{C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac{1}{12} (12 A+8 B+7 C) \int \sec (c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx+\frac{1}{6} \left (a^2 (12 A+8 B+7 C)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{a^2 (12 A+8 B+7 C) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(4 B-C) (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac{C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac{1}{8} \left (a^2 (12 A+8 B+7 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^2 (12 A+8 B+7 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d}\\ &=\frac{a^2 (12 A+8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (12 A+8 B+7 C) \tan (c+d x)}{6 d}+\frac{a^2 (12 A+8 B+7 C) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(4 B-C) (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac{C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}\\ \end{align*}
Mathematica [B] time = 2.41386, size = 386, normalized size = 2.63 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (24 (12 A+8 B+7 C) \cos ^4(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-24 \sin (c) (6 A+5 B+4 C)+3 (4 A+8 B+15 C) \sin (d x)+12 A \sin (2 c+d x)+144 A \sin (c+2 d x)-48 A \sin (3 c+2 d x)+12 A \sin (2 c+3 d x)+12 A \sin (4 c+3 d x)+48 A \sin (3 c+4 d x)+24 B \sin (2 c+d x)+136 B \sin (c+2 d x)-24 B \sin (3 c+2 d x)+24 B \sin (2 c+3 d x)+24 B \sin (4 c+3 d x)+40 B \sin (3 c+4 d x)+45 C \sin (2 c+d x)+128 C \sin (c+2 d x)+21 C \sin (2 c+3 d x)+21 C \sin (4 c+3 d x)+32 C \sin (3 c+4 d x))\right )}{384 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 246, normalized size = 1.7 \begin{align*}{\frac{3\,{a}^{2}A\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{5\,B{a}^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{7\,{a}^{2}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{7\,{a}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+2\,{\frac{{a}^{2}A\tan \left ( dx+c \right ) }{d}}+{\frac{B{a}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+{\frac{B{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{4\,{a}^{2}C\tan \left ( dx+c \right ) }{3\,d}}+{\frac{2\,{a}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{a}^{2}A\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{B{a}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{a}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.959035, size = 417, normalized size = 2.84 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} + 32 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} - 3 \, 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 )} - 12 \, 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 )} - 24 \, B 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 )} - 12 \, C 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 )} + 48 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 96 \, A a^{2} \tan \left (d x + c\right ) + 48 \, B a^{2} \tan \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.518893, size = 397, normalized size = 2.7 \begin{align*} \frac{3 \,{\left (12 \, A + 8 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (12 \, A + 8 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (6 \, A + 5 \, B + 4 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (4 \, A + 8 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \,{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right ) + 6 \, C a^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \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{\left (c + d x \right )}\, dx + \int 2 A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 B \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 C \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26903, size = 392, normalized size = 2.67 \begin{align*} \frac{3 \,{\left (12 \, A a^{2} + 8 \, B a^{2} + 7 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (12 \, A a^{2} + 8 \, B a^{2} + 7 \, 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 (36 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 24 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 21 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 132 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 88 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 77 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 156 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 136 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 83 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 60 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 72 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 75 \, 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 )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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