Optimal. Leaf size=228 \[ \frac {a^4 (14 A+11 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d} \]
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Rubi [A]
time = 0.32, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4174, 4095,
4086, 3876, 3855, 3852, 8, 3853} \begin {gather*} \frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {a^4 (14 A+11 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a^4 (14 A+11 C) \tan (c+d x) \sec ^3(c+d x)}{70 d}+\frac {27 a^4 (14 A+11 C) \tan (c+d x) \sec (c+d x)}{140 d}+\frac {(21 A+4 C) \tan (c+d x) (a \sec (c+d x)+a)^4}{105 d}+\frac {C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^4}{7 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^5}{21 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3876
Rule 4086
Rule 4095
Rule 4174
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 (a (7 A+2 C)+4 a C \sec (c+d x)) \, dx}{7 a}\\ &=\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^4 \left (20 a^2 C+2 a^2 (21 A+4 C) \sec (c+d x)\right ) \, dx}{42 a^2}\\ &=\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{35} (2 (14 A+11 C)) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx\\ &=\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{35} (2 (14 A+11 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 {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{35} \left (2 a^4 (14 A+11 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{35} \left (2 a^4 (14 A+11 C)\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{35} \left (8 a^4 (14 A+11 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{35} \left (8 a^4 (14 A+11 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{35} \left (12 a^4 (14 A+11 C)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac {2 a^4 (14 A+11 C) \tanh ^{-1}(\sin (c+d x))}{35 d}+\frac {6 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{35 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{70} \left (3 a^4 (14 A+11 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{35} \left (6 a^4 (14 A+11 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (8 a^4 (14 A+11 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{35 d}-\frac {\left (8 a^4 (14 A+11 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d}\\ &=\frac {8 a^4 (14 A+11 C) \tanh ^{-1}(\sin (c+d x))}{35 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d}+\frac {1}{140} \left (3 a^4 (14 A+11 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac {a^4 (14 A+11 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d}\\ \end {align*}
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Mathematica [A]
time = 5.10, size = 419, normalized size = 1.84 \begin {gather*} -\frac {a^4 (1+\cos (c+d x))^4 \left (C+A \cos ^2(c+d x)\right ) \sec ^8\left (\frac {1}{2} (c+d x)\right ) \sec ^7(c+d x) \left (6720 (14 A+11 C) \cos ^7(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 )-\sec (c) (560 (91 A+83 C) \sin (d x)-140 (217 A+122 C) \sin (2 c+d x)+10710 A \sin (c+2 d x)+16415 C \sin (c+2 d x)+10710 A \sin (3 c+2 d x)+16415 C \sin (3 c+2 d x)+41244 A \sin (2 c+3 d x)+37296 C \sin (2 c+3 d x)-7560 A \sin (4 c+3 d x)-840 C \sin (4 c+3 d x)+7560 A \sin (3 c+4 d x)+7700 C \sin (3 c+4 d x)+7560 A \sin (5 c+4 d x)+7700 C \sin (5 c+4 d x)+15848 A \sin (4 c+5 d x)+12712 C \sin (4 c+5 d x)-420 A \sin (6 c+5 d x)+1470 A \sin (5 c+6 d x)+1155 C \sin (5 c+6 d x)+1470 A \sin (7 c+6 d x)+1155 C \sin (7 c+6 d x)+2324 A \sin (6 c+7 d x)+1816 C \sin (6 c+7 d x))\right )}{215040 d (A+2 C+A \cos (2 (c+d x)))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.00, size = 374, normalized size = 1.64 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 462 vs.
\(2 (212) = 424\).
time = 0.28, size = 462, normalized size = 2.03 \begin {gather*} \frac {56 \, {\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 a^{4} + 1680 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 24 \, {\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 a^{4} + 336 \, {\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} + 280 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 35 \, 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 )} - 210 \, 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 )} - 210 \, 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 )} - 840 \, 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 )} + 840 \, A a^{4} \tan \left (d x + c\right )}{840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.99, size = 201, normalized size = 0.88 \begin {gather*} \frac {105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, {\left (581 \, A + 454 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 4 \, {\left (238 \, A + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (6 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 12 \, {\left (7 \, A + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 280 \, C a^{4} \cos \left (d x + c\right ) + 60 \, C a^{4}\right )} \sin \left (d x + c\right )}{840 \, 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*} a^{4} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{6}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{8}{\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 314, normalized size = 1.38 \begin {gather*} \frac {105 \, {\left (14 \, A a^{4} + 11 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (14 \, A a^{4} + 11 \, 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 (1470 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1155 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 9800 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 7700 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 27734 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 21791 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 43008 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 33792 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 39914 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31521 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21560 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 14700 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5250 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5565 \, 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 )}^{7}}}{420 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.41, size = 301, normalized size = 1.32 \begin {gather*} \frac {a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (14\,A+11\,C\right )}{2\,d}-\frac {\left (7\,A\,a^4+\frac {11\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (-\frac {140\,A\,a^4}{3}-\frac {110\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^4}{15}+\frac {3113\,C\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {1024\,A\,a^4}{5}-\frac {5632\,C\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {2851\,A\,a^4}{15}+\frac {1501\,C\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {308\,A\,a^4}{3}-70\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {53\,C\,a^4}{2}\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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