Optimal. Leaf size=159 \[ \frac{4 a^4 (5 A+4 B) \tan ^3(c+d x)}{15 d}+\frac{8 a^4 (5 A+4 B) \tan (c+d x)}{5 d}+\frac{7 a^4 (5 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (5 A+4 B) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{27 a^4 (5 A+4 B) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{B \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d} \]
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Rubi [A] time = 0.17923, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {4001, 3791, 3770, 3767, 8, 3768} \[ \frac{4 a^4 (5 A+4 B) \tan ^3(c+d x)}{15 d}+\frac{8 a^4 (5 A+4 B) \tan (c+d x)}{5 d}+\frac{7 a^4 (5 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (5 A+4 B) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{27 a^4 (5 A+4 B) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{B \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d} \]
Antiderivative was successfully verified.
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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 (A+B \sec (c+d x)) \, dx &=\frac{B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{1}{5} (5 A+4 B) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx\\ &=\frac{B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{1}{5} (5 A+4 B) \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{B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{1}{5} \left (a^4 (5 A+4 B)\right ) \int \sec (c+d x) \, dx+\frac{1}{5} \left (a^4 (5 A+4 B)\right ) \int \sec ^5(c+d x) \, dx+\frac{1}{5} \left (4 a^4 (5 A+4 B)\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{5} \left (4 a^4 (5 A+4 B)\right ) \int \sec ^4(c+d x) \, dx+\frac{1}{5} \left (6 a^4 (5 A+4 B)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{a^4 (5 A+4 B) \tanh ^{-1}(\sin (c+d x))}{5 d}+\frac{3 a^4 (5 A+4 B) \sec (c+d x) \tan (c+d x)}{5 d}+\frac{a^4 (5 A+4 B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{1}{20} \left (3 a^4 (5 A+4 B)\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{5} \left (3 a^4 (5 A+4 B)\right ) \int \sec (c+d x) \, dx-\frac{\left (4 a^4 (5 A+4 B)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{5 d}-\frac{\left (4 a^4 (5 A+4 B)\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac{4 a^4 (5 A+4 B) \tanh ^{-1}(\sin (c+d x))}{5 d}+\frac{8 a^4 (5 A+4 B) \tan (c+d x)}{5 d}+\frac{27 a^4 (5 A+4 B) \sec (c+d x) \tan (c+d x)}{40 d}+\frac{a^4 (5 A+4 B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{4 a^4 (5 A+4 B) \tan ^3(c+d x)}{15 d}+\frac{1}{40} \left (3 a^4 (5 A+4 B)\right ) \int \sec (c+d x) \, dx\\ &=\frac{7 a^4 (5 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{8 a^4 (5 A+4 B) \tan (c+d x)}{5 d}+\frac{27 a^4 (5 A+4 B) \sec (c+d x) \tan (c+d x)}{40 d}+\frac{a^4 (5 A+4 B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{B (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{4 a^4 (5 A+4 B) \tan ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 1.61498, size = 306, normalized size = 1.92 \[ -\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) \sec ^5(c+d x) \left (1680 (5 A+4 B) \cos ^5(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-960 (3 A+2 B) \sin (2 c+d x)+80 (64 A+59 B) \sin (d x)+930 A \sin (c+2 d x)+930 A \sin (3 c+2 d x)+3520 A \sin (2 c+3 d x)-480 A \sin (4 c+3 d x)+405 A \sin (3 c+4 d x)+405 A \sin (5 c+4 d x)+800 A \sin (4 c+5 d x)+1320 B \sin (c+2 d x)+1320 B \sin (3 c+2 d x)+3200 B \sin (2 c+3 d x)-120 B \sin (4 c+3 d x)+420 B \sin (3 c+4 d x)+420 B \sin (5 c+4 d x)+664 B \sin (4 c+5 d x))\right )}{30720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 234, normalized size = 1.5 \begin{align*}{\frac{35\,A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{83\,B{a}^{4}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{20\,A{a}^{4}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{7\,B{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{7\,B{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{27\,A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{34\,B{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{4\,A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{B{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{B{a}^{4}\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 = 1.03204, size = 498, normalized size = 3.13 \begin{align*} \frac{320 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 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 )} B a^{4} + 480 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} - 15 \, 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 )} - 60 \, B 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 )} - 360 \, 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 )} - 240 \, B 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 )} + 240 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 960 \, A a^{4} \tan \left (d x + c\right ) + 240 \, B a^{4} \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 = 0.499388, size = 431, normalized size = 2.71 \begin{align*} \frac{105 \,{\left (5 \, A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (5 \, A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (100 \, A + 83 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 15 \,{\left (27 \, A + 28 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \,{\left (10 \, A + 17 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 30 \,{\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right ) + 24 \, B a^{4}\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^{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 B \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 B \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{5}{\left (c + d x \right )}\, dx + \int B \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 = 1.24877, size = 332, normalized size = 2.09 \begin{align*} \frac{105 \,{\left (5 \, A a^{4} + 4 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 105 \,{\left (5 \, A a^{4} + 4 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (525 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 420 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 2450 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1960 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 4480 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3584 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3950 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3160 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1395 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1500 \, B 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 )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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