Optimal. Leaf size=274 \[ \frac{a^3 (133 A+119 B+108 C) \tan ^3(c+d x)}{105 d}+\frac{a^3 (133 A+119 B+108 C) \tan (c+d x)}{35 d}+\frac{a^3 (26 A+23 B+21 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{280 d}+\frac{(3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac{a^3 (26 A+23 B+21 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{42 a d}+\frac{C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d} \]
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Rubi [A] time = 0.598019, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{a^3 (133 A+119 B+108 C) \tan ^3(c+d x)}{105 d}+\frac{a^3 (133 A+119 B+108 C) \tan (c+d x)}{35 d}+\frac{a^3 (26 A+23 B+21 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{280 d}+\frac{(3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac{a^3 (26 A+23 B+21 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{42 a d}+\frac{C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d} \]
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))^3 \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))^3 \tan (c+d x)}{7 d}+\frac{\int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (a (7 A+3 C)+a (7 B+3 C) \sec (c+d x)) \, dx}{7 a}\\ &=\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac{(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac{\int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \left (3 a^2 (14 A+7 B+9 C)+14 a^2 (3 A+4 B+3 C) \sec (c+d x)\right ) \, dx}{42 a}\\ &=\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac{(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \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^3 (112 A+91 B+87 C)+3 a^3 (154 A+147 B+129 C) \sec (c+d x)\right ) \, dx}{210 a}\\ &=\frac{a^3 (154 A+147 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac{(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{\int \sec ^3(c+d x) \left (105 a^4 (26 A+23 B+21 C)+24 a^4 (133 A+119 B+108 C) \sec (c+d x)\right ) \, dx}{840 a}\\ &=\frac{a^3 (154 A+147 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac{(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{1}{8} \left (a^3 (26 A+23 B+21 C)\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{35} \left (a^3 (133 A+119 B+108 C)\right ) \int \sec ^4(c+d x) \, dx\\ &=\frac{a^3 (26 A+23 B+21 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 (154 A+147 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac{(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{1}{16} \left (a^3 (26 A+23 B+21 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^3 (133 A+119 B+108 C)\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d}\\ &=\frac{a^3 (26 A+23 B+21 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (133 A+119 B+108 C) \tan (c+d x)}{35 d}+\frac{a^3 (26 A+23 B+21 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 (154 A+147 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac{C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac{(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac{a^3 (133 A+119 B+108 C) \tan ^3(c+d x)}{105 d}\\ \end{align*}
Mathematica [A] time = 6.16201, size = 402, normalized size = 1.47 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^7(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (105 (26 A+23 B+21 C) \cos ^7(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) \cos ^6(c+d x) (105 \sin (c) (26 A+23 B+21 C)+32 (133 A+119 B+108 C) \sin (d x))-\sec (c) \cos ^5(c+d x) (16 \sin (c) (133 A+119 B+108 C)+105 (26 A+23 B+21 C) \sin (d x))-2 \sec (c) \cos ^4(c+d x) (35 \sin (c) (18 A+23 B+21 C)+8 (133 A+119 B+108 C) \sin (d x))-2 \sec (c) \cos ^3(c+d x) (24 \sin (c) (7 A+21 B+27 C)+35 (18 A+23 B+21 C) \sin (d x))-8 \sec (c) \cos ^2(c+d x) (6 (7 A+21 B+27 C) \sin (d x)+35 (B+3 C) \sin (c))-40 \sec (c) \cos (c+d x) (7 (B+3 C) \sin (d x)+6 C \sin (c))-240 C \sec (c) \sin (d x)\right )}{6720 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 455, normalized size = 1.7 \begin{align*}{\frac{23\,B{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{23\,B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{23\,B{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{3\,A{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 ) ^{5}}{2\,d}}+{\frac{B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{6\,d}}+{\frac{34\,B{a}^{3}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{17\,B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{38\,A{a}^{3}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{19\,A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{72\,{a}^{3}C\tan \left ( dx+c \right ) }{35\,d}}+{\frac{27\,{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{35\,d}}+{\frac{36\,{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{35\,d}}+{\frac{3\,B{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{7\,d}}+{\frac{13\,A{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{7\,{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{21\,{a}^{3}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{13\,A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{21\,{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.989039, size = 876, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.556788, size = 617, normalized size = 2.25 \begin{align*} \frac{105 \,{\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (32 \,{\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} + 105 \,{\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 16 \,{\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 70 \,{\left (18 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 48 \,{\left (7 \, A + 21 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 280 \,{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 240 \, C a^{3}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \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 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{5}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{6}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{8}{\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 = 1.3511, size = 598, normalized size = 2.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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