Optimal. Leaf size=197 \[ \frac{a^3 (30 A+23 C) \tan ^3(c+d x)}{120 d}+\frac{a^3 (30 A+23 C) \tan (c+d x)}{10 d}+\frac{a^3 (30 A+23 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{3 a^3 (30 A+23 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac{(30 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{120 d}+\frac{C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^3}{6 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{10 a d} \]
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Rubi [A] time = 0.416659, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4089, 4010, 4001, 3791, 3770, 3767, 8, 3768} \[ \frac{a^3 (30 A+23 C) \tan ^3(c+d x)}{120 d}+\frac{a^3 (30 A+23 C) \tan (c+d x)}{10 d}+\frac{a^3 (30 A+23 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{3 a^3 (30 A+23 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac{(30 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{120 d}+\frac{C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^3}{6 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{10 a d} \]
Antiderivative was successfully verified.
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Rule 4089
Rule 4010
Rule 4001
Rule 3791
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac{\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (2 a (3 A+C)+3 a C \sec (c+d x)) \, dx}{6 a}\\ &=\frac{C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac{\int \sec (c+d x) (a+a \sec (c+d x))^3 \left (12 a^2 C+a^2 (30 A+7 C) \sec (c+d x)\right ) \, dx}{30 a^2}\\ &=\frac{(30 A+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac{C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac{1}{40} (30 A+23 C) \int \sec (c+d x) (a+a \sec (c+d x))^3 \, dx\\ &=\frac{(30 A+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac{C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac{1}{40} (30 A+23 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{(30 A+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac{C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac{1}{40} \left (a^3 (30 A+23 C)\right ) \int \sec (c+d x) \, dx+\frac{1}{40} \left (a^3 (30 A+23 C)\right ) \int \sec ^4(c+d x) \, dx+\frac{1}{40} \left (3 a^3 (30 A+23 C)\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{40} \left (3 a^3 (30 A+23 C)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{a^3 (30 A+23 C) \tanh ^{-1}(\sin (c+d x))}{40 d}+\frac{3 a^3 (30 A+23 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac{(30 A+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac{C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac{1}{80} \left (3 a^3 (30 A+23 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^3 (30 A+23 C)\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{40 d}-\frac{\left (3 a^3 (30 A+23 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{40 d}\\ &=\frac{a^3 (30 A+23 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (30 A+23 C) \tan (c+d x)}{10 d}+\frac{3 a^3 (30 A+23 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac{(30 A+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac{C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac{a^3 (30 A+23 C) \tan ^3(c+d x)}{120 d}\\ \end{align*}
Mathematica [A] time = 2.78739, size = 387, normalized size = 1.96 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^6(c+d x) \left (A \cos ^2(c+d x)+C\right ) \left (480 (30 A+23 C) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-160 (45 A+34 C) \sin (c)+1140 A \sin (2 c+d x)+8160 A \sin (c+2 d x)-2640 A \sin (3 c+2 d x)+1590 A \sin (2 c+3 d x)+1590 A \sin (4 c+3 d x)+4080 A \sin (3 c+4 d x)-240 A \sin (5 c+4 d x)+450 A \sin (4 c+5 d x)+450 A \sin (6 c+5 d x)+720 A \sin (5 c+6 d x)+30 (38 A+75 C) \sin (d x)+2250 C \sin (2 c+d x)+7680 C \sin (c+2 d x)-480 C \sin (3 c+2 d x)+1955 C \sin (2 c+3 d x)+1955 C \sin (4 c+3 d x)+3264 C \sin (3 c+4 d x)+345 C \sin (4 c+5 d x)+345 C \sin (6 c+5 d x)+544 C \sin (5 c+6 d x))\right )}{30720 d (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 257, normalized size = 1.3 \begin{align*} 3\,{\frac{A{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{34\,{a}^{3}C\tan \left ( dx+c \right ) }{15\,d}}+{\frac{17\,{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{15\,A{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{15\,A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{23\,{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{23\,{a}^{3}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{23\,{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{A{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 ) ^{4}}{5\,d}}+{\frac{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}}{6\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.960638, size = 516, normalized size = 2.62 \begin{align*} \frac{480 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 96 \,{\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} + 160 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 5 \, C a^{3}{\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^{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 )} - 90 \, 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 )} - 360 \, A 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 )} + 480 \, A a^{3} \tan \left (d x + c\right )}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.531258, size = 474, normalized size = 2.41 \begin{align*} \frac{15 \,{\left (30 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (30 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (45 \, A + 34 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 15 \,{\left (30 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 16 \,{\left (15 \, A + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 10 \,{\left (6 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 144 \, C a^{3} \cos \left (d x + c\right ) + 40 \, C a^{3}\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^{3} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{4}{\left (c + d x \right )}\, dx + \int A \sec ^{5}{\left (c + d x \right )}\, dx + \int C \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 3 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 = 1.24728, size = 378, normalized size = 1.92 \begin{align*} \frac{15 \,{\left (30 \, A a^{3} + 23 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (30 \, A a^{3} + 23 \, 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 (450 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 345 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 2550 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1955 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 5940 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 4554 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 7500 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 5814 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 5130 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3165 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1470 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1575 \, 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 )}^{6}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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