Optimal. Leaf size=244 \[ \frac{a^3 (34 A+38 B+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+26 B+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{a^3 (23 A+26 B+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{120 d}+\frac{(A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{10 a d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d} \]
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Rubi [A] time = 0.703489, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.22, Rules used = {3043, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{a^3 (34 A+38 B+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+26 B+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{a^3 (23 A+26 B+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{120 d}+\frac{(A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{10 a d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 3043
Rule 2975
Rule 2968
Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int (a+a \cos (c+d x))^3 (3 a (A+2 B)+2 a (A+3 C) \cos (c+d x)) \sec ^6(c+d x) \, dx}{6 a}\\ &=\frac{(A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int (a+a \cos (c+d x))^2 \left (a^2 (31 A+42 B+30 C)+2 a^2 (8 A+6 B+15 C) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx}{30 a}\\ &=\frac{(31 A+42 B+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int (a+a \cos (c+d x)) \left (3 a^3 (73 A+86 B+90 C)+6 a^3 (21 A+22 B+30 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{120 a}\\ &=\frac{(31 A+42 B+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int \left (3 a^4 (73 A+86 B+90 C)+\left (6 a^4 (21 A+22 B+30 C)+3 a^4 (73 A+86 B+90 C)\right ) \cos (c+d x)+6 a^4 (21 A+22 B+30 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx}{120 a}\\ &=\frac{a^3 (73 A+86 B+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(31 A+42 B+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int \left (45 a^4 (23 A+26 B+30 C)+24 a^4 (34 A+38 B+45 C) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{360 a}\\ &=\frac{a^3 (73 A+86 B+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(31 A+42 B+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{8} \left (a^3 (23 A+26 B+30 C)\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{15} \left (a^3 (34 A+38 B+45 C)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{a^3 (23 A+26 B+30 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 (73 A+86 B+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(31 A+42 B+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{16} \left (a^3 (23 A+26 B+30 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^3 (34 A+38 B+45 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d}\\ &=\frac{a^3 (23 A+26 B+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (34 A+38 B+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+26 B+30 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 (73 A+86 B+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(31 A+42 B+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{(A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 1.88904, size = 265, normalized size = 1.09 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^6(c+d x) \left (240 (23 A+26 B+30 C) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-2 \sin (c+d x) (16 (344 A+328 B+315 C) \cos (c+d x)+20 (115 A+114 B+102 C) \cos (2 (c+d x))+1904 A \cos (3 (c+d x))+345 A \cos (4 (c+d x))+272 A \cos (5 (c+d x))+2275 A+2128 B \cos (3 (c+d x))+390 B \cos (4 (c+d x))+304 B \cos (5 (c+d x))+1890 B+2280 C \cos (3 (c+d x))+450 C \cos (4 (c+d x))+360 C \cos (5 (c+d x))+1590 C)\right )}{30720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.098, size = 385, normalized size = 1.6 \begin{align*}{\frac{A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{6\,d}}+{\frac{23\,A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{23\,A{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{23\,A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{38\,{a}^{3}B\tan \left ( dx+c \right ) }{15\,d}}+{\frac{{a}^{3}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{19\,{a}^{3}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{15\,{a}^{3}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{15\,{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{34\,A{a}^{3}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{3\,A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{17\,A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{3\,{a}^{3}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{13\,{a}^{3}B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{13\,{a}^{3}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+3\,{\frac{{a}^{3}C\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08061, size = 755, normalized size = 3.09 \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 = 2.09937, size = 540, normalized size = 2.21 \begin{align*} \frac{15 \,{\left (23 \, A + 26 \, B + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (23 \, A + 26 \, B + 30 \, 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 (34 \, A + 38 \, B + 45 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 15 \,{\left (23 \, A + 26 \, B + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 16 \,{\left (17 \, A + 19 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 10 \,{\left (23 \, A + 18 \, B + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 48 \,{\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + 40 \, A 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27767, size = 529, normalized size = 2.17 \begin{align*} \frac{15 \,{\left (23 \, A a^{3} + 26 \, B a^{3} + 30 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (23 \, A a^{3} + 26 \, B a^{3} + 30 \, 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 (345 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 390 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 450 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 1955 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 2210 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 2550 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 4554 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 5148 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 5940 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 5814 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 5988 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 7500 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3165 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4190 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5130 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1575 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1530 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1470 \, 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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