Optimal. Leaf size=491 \[ \frac {\left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac {\left (28 a^5 b B-847 a^3 b^3 B-896 a b^5 B-8 a^6 C-32 b^6 (7 A+6 C)-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)\right ) \tan (c+d x)}{420 b^2 d}-\frac {\left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{1680 b d}-\frac {\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac {\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac {\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac {(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d} \]
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Rubi [A]
time = 0.86, antiderivative size = 491, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4177, 4167,
4087, 4082, 3872, 3855, 3852, 8} \begin {gather*} \frac {\tan (c+d x) \left (2 a^2 C-7 a b B+42 A b^2+36 b^2 C\right ) (a+b \sec (c+d x))^4}{210 b^2 d}-\frac {\tan (c+d x) \left (-8 a^3 C+28 a^2 b B-4 a b^2 (42 A+31 C)-175 b^3 B\right ) (a+b \sec (c+d x))^3}{840 b^2 d}+\frac {\left (8 a^4 B+8 a^3 b (4 A+3 C)+36 a^2 b^2 B+4 a b^3 (6 A+5 C)+5 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac {\tan (c+d x) \left (-8 a^4 C+28 a^3 b B-12 a^2 b^2 (14 A+9 C)-371 a b^3 B-32 b^4 (7 A+6 C)\right ) (a+b \sec (c+d x))^2}{840 b^2 d}-\frac {\tan (c+d x) \sec (c+d x) \left (-16 a^5 C+56 a^4 b B-48 a^3 b^2 (7 A+4 C)-1246 a^2 b^3 B-4 a b^4 (406 A+333 C)-525 b^5 B\right )}{1680 b d}-\frac {\tan (c+d x) \left (-8 a^6 C+28 a^5 b B-4 a^4 b^2 (42 A+23 C)-847 a^3 b^3 B-32 a^2 b^4 (49 A+39 C)-896 a b^5 B-32 b^6 (7 A+6 C)\right )}{420 b^2 d}+\frac {(7 b B-2 a C) \tan (c+d x) (a+b \sec (c+d x))^5}{42 b^2 d}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^5}{7 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rule 4167
Rule 4177
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^4 \left (a C+b (7 A+6 C) \sec (c+d x)+(7 b B-2 a C) \sec ^2(c+d x)\right ) \, dx}{7 b}\\ &=\frac {(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^4 \left (b (35 b B-4 a C)+\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) \sec (c+d x)\right ) \, dx}{42 b^2}\\ &=\frac {\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac {(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (3 b \left (56 A b^2+49 a b B-4 a^2 C+48 b^2 C\right )-\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) \sec (c+d x)\right ) \, dx}{210 b^2}\\ &=-\frac {\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac {\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac {(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (3 b \left (168 a^2 b B+175 b^3 B-8 a^3 C+4 a b^2 (98 A+79 C)\right )-3 \left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) \sec (c+d x)\right ) \, dx}{840 b^2}\\ &=-\frac {\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac {\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac {\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac {(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x)) \left (3 b \left (448 a^3 b B+1267 a b^3 B-8 a^4 C+64 b^4 (7 A+6 C)+12 a^2 b^2 (126 A+97 C)\right )-3 \left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x)\right ) \, dx}{2520 b^2}\\ &=-\frac {\left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{1680 b d}-\frac {\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac {\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac {\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac {(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) \left (315 b^2 \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right )-12 \left (28 a^5 b B-847 a^3 b^3 B-896 a b^5 B-8 a^6 C-32 b^6 (7 A+6 C)-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)\right ) \sec (c+d x)\right ) \, dx}{5040 b^2}\\ &=-\frac {\left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{1680 b d}-\frac {\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac {\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac {\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac {(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {1}{16} \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (28 a^5 b B-847 a^3 b^3 B-896 a b^5 B-8 a^6 C-32 b^6 (7 A+6 C)-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)\right ) \int \sec ^2(c+d x) \, dx}{420 b^2}\\ &=\frac {\left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac {\left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{1680 b d}-\frac {\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac {\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac {\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac {(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\left (28 a^5 b B-847 a^3 b^3 B-896 a b^5 B-8 a^6 C-32 b^6 (7 A+6 C)-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{420 b^2 d}\\ &=\frac {\left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac {\left (28 a^5 b B-847 a^3 b^3 B-896 a b^5 B-8 a^6 C-32 b^6 (7 A+6 C)-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)\right ) \tan (c+d x)}{420 b^2 d}-\frac {\left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{1680 b d}-\frac {\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac {\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac {\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac {(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}\\ \end {align*}
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Mathematica [A]
time = 4.62, size = 486, normalized size = 0.99 \begin {gather*} -\frac {\left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right ) \sec ^6(c+d x) \left (105 \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \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 (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-70 b \left (36 a^2 b B+5 b^3 B+24 a^3 C+4 a b^2 (6 A+5 C)\right ) \cos ^2(c+d x) \sin (c+d x)-16 \left (140 a^3 b B+112 a b^3 B+35 a^4 C+42 a^2 b^2 (5 A+4 C)+4 b^4 (7 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)-105 \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \cos ^4(c+d x) \sin (c+d x)-16 \left (280 a^3 b B+224 a b^3 B+35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right ) \cos ^5(c+d x) \sin (c+d x)-8 b^2 \left (35 b (b B+4 a C) \sin (c+d x)+3 \left (7 A b^2+28 a b B+42 a^2 C+6 b^2 C\right ) \sin (2 (c+d x))+30 b^2 C \tan (c+d x)\right )\right )}{840 d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x)))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 592, normalized size = 1.21 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 746, normalized size = 1.52 \begin {gather*} \frac {1120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} + 4480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} b + 6720 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b^{2} + 1344 \, {\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^{2} b^{2} + 896 \, {\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 b^{3} + 224 \, {\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 )} A b^{4} + 96 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} C b^{4} - 140 \, C a b^{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 )} - 35 \, B b^{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 )} - 840 \, C a^{3} b {\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 )} - 1260 \, B a^{2} b^{2} {\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 )} - 840 \, A a b^{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 )} - 840 \, 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 )} - 3360 \, A a^{3} b {\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 )} + 3360 \, A a^{4} \tan \left (d x + c\right )}{3360 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.14, size = 450, normalized size = 0.92 \begin {gather*} \frac {105 \, {\left (8 \, B a^{4} + 8 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (8 \, B a^{4} + 8 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (35 \, {\left (3 \, A + 2 \, C\right )} a^{4} + 280 \, B a^{3} b + 84 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 224 \, B a b^{3} + 8 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} + 105 \, {\left (8 \, B a^{4} + 8 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 240 \, C b^{4} + 16 \, {\left (35 \, C a^{4} + 140 \, B a^{3} b + 42 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 112 \, B a b^{3} + 4 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (24 \, C a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 48 \, {\left (42 \, C a^{2} b^{2} + 28 \, B a b^{3} + {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 280 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \end {gather*}
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 \begin {gather*} \int \left (a + b \sec {\left (c + d x \right )}\right )^{4} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1888 vs.
\(2 (475) = 950\).
time = 0.60, size = 1888, normalized size = 3.85 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.75, size = 1044, normalized size = 2.13 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {B\,a^4}{2}+\frac {5\,B\,b^4}{16}+\frac {9\,B\,a^2\,b^2}{4}+\frac {3\,A\,a\,b^3}{2}+2\,A\,a^3\,b+\frac {5\,C\,a\,b^3}{4}+\frac {3\,C\,a^3\,b}{2}\right )}{2\,B\,a^4+\frac {5\,B\,b^4}{4}+9\,B\,a^2\,b^2+6\,A\,a\,b^3+8\,A\,a^3\,b+5\,C\,a\,b^3+6\,C\,a^3\,b}\right )\,\left (B\,a^4+\frac {5\,B\,b^4}{8}+\frac {9\,B\,a^2\,b^2}{2}+3\,A\,a\,b^3+4\,A\,a^3\,b+\frac {5\,C\,a\,b^3}{2}+3\,C\,a^3\,b\right )}{d}-\frac {\left (2\,A\,a^4+2\,A\,b^4-B\,a^4-\frac {11\,B\,b^4}{8}+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2-\frac {15\,B\,a^2\,b^2}{2}+12\,C\,a^2\,b^2-5\,A\,a\,b^3-4\,A\,a^3\,b+8\,B\,a\,b^3+8\,B\,a^3\,b-\frac {11\,C\,a\,b^3}{2}-5\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (4\,B\,a^4-\frac {20\,A\,b^4}{3}-12\,A\,a^4+\frac {7\,B\,b^4}{6}-\frac {28\,C\,a^4}{3}-4\,C\,b^4-56\,A\,a^2\,b^2+18\,B\,a^2\,b^2-40\,C\,a^2\,b^2+12\,A\,a\,b^3+16\,A\,a^3\,b-\frac {80\,B\,a\,b^3}{3}-\frac {112\,B\,a^3\,b}{3}+\frac {14\,C\,a\,b^3}{3}+12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}-5\,B\,a^4-\frac {85\,B\,b^4}{24}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2-\frac {27\,B\,a^2\,b^2}{2}+\frac {452\,C\,a^2\,b^2}{5}-9\,A\,a\,b^3-20\,A\,a^3\,b+\frac {904\,B\,a\,b^3}{15}+\frac {232\,B\,a^3\,b}{3}-\frac {85\,C\,a\,b^3}{6}-9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-40\,A\,a^4-\frac {104\,A\,b^4}{5}-24\,C\,a^4-\frac {424\,C\,b^4}{35}-144\,A\,a^2\,b^2-\frac {624\,C\,a^2\,b^2}{5}-\frac {416\,B\,a\,b^3}{5}-96\,B\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}+5\,B\,a^4+\frac {85\,B\,b^4}{24}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2+\frac {27\,B\,a^2\,b^2}{2}+\frac {452\,C\,a^2\,b^2}{5}+9\,A\,a\,b^3+20\,A\,a^3\,b+\frac {904\,B\,a\,b^3}{15}+\frac {232\,B\,a^3\,b}{3}+\frac {85\,C\,a\,b^3}{6}+9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-12\,A\,a^4-\frac {20\,A\,b^4}{3}-4\,B\,a^4-\frac {7\,B\,b^4}{6}-\frac {28\,C\,a^4}{3}-4\,C\,b^4-56\,A\,a^2\,b^2-18\,B\,a^2\,b^2-40\,C\,a^2\,b^2-12\,A\,a\,b^3-16\,A\,a^3\,b-\frac {80\,B\,a\,b^3}{3}-\frac {112\,B\,a^3\,b}{3}-\frac {14\,C\,a\,b^3}{3}-12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^4+2\,A\,b^4+B\,a^4+\frac {11\,B\,b^4}{8}+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+\frac {15\,B\,a^2\,b^2}{2}+12\,C\,a^2\,b^2+5\,A\,a\,b^3+4\,A\,a^3\,b+8\,B\,a\,b^3+8\,B\,a^3\,b+\frac {11\,C\,a\,b^3}{2}+5\,C\,a^3\,b\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 )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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