Optimal. Leaf size=209 \[ \frac {2 a^4 (10 A+8 B+7 C) \tan ^3(c+d x)}{15 d}+\frac {4 a^4 (10 A+8 B+7 C) \tan (c+d x)}{5 d}+\frac {7 a^4 (10 A+8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^4 (10 A+8 B+7 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac {27 a^4 (10 A+8 B+7 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac {(6 B-C) \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d} \]
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Rubi [A] time = 0.33, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.180, Rules used = {4082, 4001, 3791, 3770, 3767, 8, 3768} \[ \frac {2 a^4 (10 A+8 B+7 C) \tan ^3(c+d x)}{15 d}+\frac {4 a^4 (10 A+8 B+7 C) \tan (c+d x)}{5 d}+\frac {7 a^4 (10 A+8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^4 (10 A+8 B+7 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac {27 a^4 (10 A+8 B+7 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac {(6 B-C) \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3768
Rule 3770
Rule 3791
Rule 4001
Rule 4082
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+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 (6 B-C) \sec (c+d x)) \, dx}{6 a}\\ &=\frac {(6 B-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+8 B+7 C) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx\\ &=\frac {(6 B-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+8 B+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 {(6 B-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+8 B+7 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{10} \left (a^4 (10 A+8 B+7 C)\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (10 A+8 B+7 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (10 A+8 B+7 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{5} \left (3 a^4 (10 A+8 B+7 C)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac {a^4 (10 A+8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{10 d}+\frac {3 a^4 (10 A+8 B+7 C) \sec (c+d x) \tan (c+d x)}{10 d}+\frac {a^4 (10 A+8 B+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {(6 B-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+8 B+7 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{10} \left (3 a^4 (10 A+8 B+7 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (2 a^4 (10 A+8 B+7 C)\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{5 d}-\frac {\left (2 a^4 (10 A+8 B+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+8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{5 d}+\frac {4 a^4 (10 A+8 B+7 C) \tan (c+d x)}{5 d}+\frac {27 a^4 (10 A+8 B+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (10 A+8 B+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {(6 B-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+8 B+7 C) \tan ^3(c+d x)}{15 d}+\frac {1}{80} \left (3 a^4 (10 A+8 B+7 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac {7 a^4 (10 A+8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {4 a^4 (10 A+8 B+7 C) \tan (c+d x)}{5 d}+\frac {27 a^4 (10 A+8 B+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (10 A+8 B+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {(6 B-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+8 B+7 C) \tan ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 6.33, size = 359, normalized size = 1.72 \[ -\frac {a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \sec ^6(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (105 (10 A+8 B+7 C) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-\sec (c) \cos ^5(c+d x) (15 \sin (c) (54 A+56 B+49 C)+16 (100 A+83 B+72 C) \sin (d x))-\sec (c) \cos ^4(c+d x) (32 \sin (c) (10 A+17 B+18 C)+15 (54 A+56 B+49 C) \sin (d x))-2 \sec (c) \cos ^3(c+d x) (5 \sin (c) (6 A+24 B+41 C)+16 (10 A+17 B+18 C) \sin (d x))-2 \sec (c) \cos ^2(c+d x) (5 (6 A+24 B+41 C) \sin (d x)+24 (B+4 C) \sin (c))-8 \sec (c) \cos (c+d x) (6 (B+4 C) \sin (d x)+5 C \sin (c))-40 C \sec (c) \sin (d x)\right )}{1920 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 203, normalized size = 0.97 \[ \frac {105 \, {\left (10 \, A + 8 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (10 \, A + 8 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (100 \, A + 83 \, B + 72 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (54 \, A + 56 \, B + 49 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 32 \, {\left (10 \, A + 17 \, B + 18 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (6 \, A + 24 \, B + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 48 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 40 \, C a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 392, normalized size = 1.88 \[ \frac {105 \, {\left (10 \, A a^{4} + 8 \, B a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (10 \, A a^{4} + 8 \, B a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (1050 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 735 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 5950 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4760 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4165 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13860 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9702 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 16860 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 13488 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11802 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 10690 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7355 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2790 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3000 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.94, size = 385, normalized size = 1.84 \[ \frac {49 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d}+\frac {7 a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {27 A \,a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {41 a^{4} C \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{24 d}+\frac {a^{4} B \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{d}+\frac {A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {a^{4} C \tan \left (d x +c \right ) \left (\sec ^{5}\left (d x +c \right )\right )}{6 d}+\frac {20 A \,a^{4} \tan \left (d x +c \right )}{3 d}+\frac {35 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {49 a^{4} C \sec \left (d x +c \right ) \tan \left (d x +c \right )}{16 d}+\frac {7 a^{4} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {83 a^{4} B \tan \left (d x +c \right )}{15 d}+\frac {24 a^{4} C \tan \left (d x +c \right )}{5 d}+\frac {12 a^{4} C \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{5 d}+\frac {34 a^{4} B \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{15 d}+\frac {4 A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {4 a^{4} C \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d}+\frac {a^{4} B \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 638, normalized size = 3.05 \[ \frac {640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 32 \, {\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} + 960 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B 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 )} - 120 \, 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 )} - 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 )} - 480 \, 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 )} - 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 \, B a^{4} \tan \left (d x + c\right )}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.56, size = 338, normalized size = 1.62 \[ \frac {7\,a^4\,\mathrm {atanh}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (10\,A+8\,B+7\,C\right )}{4\,\left (\frac {35\,A\,a^4}{2}+14\,B\,a^4+\frac {49\,C\,a^4}{4}\right )}\right )\,\left (10\,A+8\,B+7\,C\right )}{8\,d}-\frac {\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4+\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (-\frac {595\,A\,a^4}{12}-\frac {119\,B\,a^4}{3}-\frac {833\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {231\,A\,a^4}{2}+\frac {462\,B\,a^4}{5}+\frac {1617\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (-\frac {281\,A\,a^4}{2}-\frac {562\,B\,a^4}{5}-\frac {1967\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1069\,A\,a^4}{12}+\frac {233\,B\,a^4}{3}+\frac {1471\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (-\frac {93\,A\,a^4}{4}-25\,B\,a^4-\frac {207\,C\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{4} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 4 A \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{4}{\left (c + d x \right )}\, dx + \int A \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 B \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{7}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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