Optimal. Leaf size=188 \[ \frac{2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac{7 a^4 (10 A+7 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (10 A+7 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+7 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}-\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d} \]
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Rubi [A] time = 0.302775, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {4083, 4001, 3791, 3770, 3767, 8, 3768} \[ \frac{2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac{7 a^4 (10 A+7 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (10 A+7 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+7 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}-\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d} \]
Antiderivative was successfully verified.
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Rule 4083
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))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac{\int \sec (c+d x) (a+a \sec (c+d x))^4 (a (6 A+5 C)-a C \sec (c+d x)) \, dx}{6 a}\\ &=-\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac{C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac{1}{10} (10 A+7 C) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx\\ &=-\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac{C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac{1}{10} (10 A+7 C) \int \left (a^4 \sec (c+d x)+4 a^4 \sec ^2(c+d x)+6 a^4 \sec ^3(c+d x)+4 a^4 \sec ^4(c+d x)+a^4 \sec ^5(c+d x)\right ) \, dx\\ &=-\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac{C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac{1}{10} \left (a^4 (10 A+7 C)\right ) \int \sec (c+d x) \, dx+\frac{1}{10} \left (a^4 (10 A+7 C)\right ) \int \sec ^5(c+d x) \, dx+\frac{1}{5} \left (2 a^4 (10 A+7 C)\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{5} \left (2 a^4 (10 A+7 C)\right ) \int \sec ^4(c+d x) \, dx+\frac{1}{5} \left (3 a^4 (10 A+7 C)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{a^4 (10 A+7 C) \tanh ^{-1}(\sin (c+d x))}{10 d}+\frac{3 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{10 d}+\frac{a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac{C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac{1}{40} \left (3 a^4 (10 A+7 C)\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{10} \left (3 a^4 (10 A+7 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (2 a^4 (10 A+7 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{5 d}-\frac{\left (2 a^4 (10 A+7 C)\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac{2 a^4 (10 A+7 C) \tanh ^{-1}(\sin (c+d x))}{5 d}+\frac{4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac{27 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac{a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac{C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac{2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d}+\frac{1}{80} \left (3 a^4 (10 A+7 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac{7 a^4 (10 A+7 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac{27 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac{a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac{C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac{2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [B] time = 3.45619, size = 387, normalized size = 2.06 \[ -\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) \sec ^6(c+d x) \left (A \cos ^2(c+d x)+C\right ) \left (3360 (10 A+7 C) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-640 (25 A+18 C) \sin (c)+1860 A \sin (2 c+d x)+17280 A \sin (c+2 d x)-6720 A \sin (3 c+2 d x)+2670 A \sin (2 c+3 d x)+2670 A \sin (4 c+3 d x)+8640 A \sin (3 c+4 d x)-960 A \sin (5 c+4 d x)+810 A \sin (4 c+5 d x)+810 A \sin (6 c+5 d x)+1600 A \sin (5 c+6 d x)+30 (62 A+125 C) \sin (d x)+3750 C \sin (2 c+d x)+15360 C \sin (c+2 d x)-1920 C \sin (3 c+2 d x)+3845 C \sin (2 c+3 d x)+3845 C \sin (4 c+3 d x)+6912 C \sin (3 c+4 d x)+735 C \sin (4 c+5 d x)+735 C \sin (6 c+5 d x)+1152 C \sin (5 c+6 d x))\right )}{61440 d (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 258, normalized size = 1.4 \begin{align*}{\frac{35\,A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{49\,{a}^{4}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{49\,{a}^{4}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{20\,A{a}^{4}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{24\,{a}^{4}C\tan \left ( dx+c \right ) }{5\,d}}+{\frac{12\,{a}^{4}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{5\,d}}+{\frac{27\,A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{41\,{a}^{4}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{4\,A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{4\,{a}^{4}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{{a}^{4}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 = 0.964257, size = 606, normalized size = 3.22 \begin{align*} \frac{640 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 128 \,{\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^{4} + 640 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 5 \, C a^{4}{\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^{4}{\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 \, C a^{4}{\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 )} - 720 \, A a^{4}{\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 )} - 120 \, C a^{4}{\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^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 1920 \, A a^{4} \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 = 0.533052, size = 471, normalized size = 2.51 \begin{align*} \frac{105 \,{\left (10 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (10 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (64 \,{\left (25 \, A + 18 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \,{\left (54 \, A + 49 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 64 \,{\left (5 \, A + 9 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \,{\left (6 \, A + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 192 \, C a^{4} \cos \left (d x + c\right ) + 40 \, C a^{4}\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^{4} \left (\int A \sec{\left (c + d x \right )}\, dx + \int 4 A \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{4}{\left (c + d x \right )}\, dx + \int A \sec ^{5}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 4 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 = 1.23798, size = 378, normalized size = 2.01 \begin{align*} \frac{105 \,{\left (10 \, A a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 105 \,{\left (10 \, A a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (1050 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 735 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 5950 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 4165 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 13860 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 9702 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 16860 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 11802 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 10690 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 7355 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2790 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3105 \, C a^{4} \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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