Optimal. Leaf size=252 \[ \frac {a^4 (56 A+49 B+44 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d} \]
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Rubi [A]
time = 0.36, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4173, 4095,
4086, 3876, 3855, 3852, 8, 3853} \begin {gather*} \frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {a^4 (56 A+49 B+44 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^4 (56 A+49 B+44 C) \tan (c+d x) \sec ^3(c+d x)}{280 d}+\frac {27 a^4 (56 A+49 B+44 C) \tan (c+d x) \sec (c+d x)}{560 d}+\frac {(42 A-7 B+8 C) \tan (c+d x) (a \sec (c+d x)+a)^4}{210 d}+\frac {(7 B+4 C) \tan (c+d x) (a \sec (c+d x)+a)^5}{42 a d}+\frac {C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^4}{7 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 4173
Rubi steps
\begin {align*} \int \sec ^2(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 \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)+a (7 B+4 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 {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^4 \left (5 a^2 (7 B+4 C)+a^2 (42 A-7 B+8 C) \sec (c+d x)\right ) \, dx}{42 a^2}\\ &=\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{70} (56 A+49 B+44 C) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx\\ &=\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{70} (56 A+49 B+44 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 {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{70} \left (a^4 (56 A+49 B+44 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{70} \left (a^4 (56 A+49 B+44 C)\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{35} \left (2 a^4 (56 A+49 B+44 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{35} \left (2 a^4 (56 A+49 B+44 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{35} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac {a^4 (56 A+49 B+44 C) \tanh ^{-1}(\sin (c+d x))}{70 d}+\frac {3 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{70 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{280} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{70} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (2 a^4 (56 A+49 B+44 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{35 d}-\frac {\left (2 a^4 (56 A+49 B+44 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d}\\ &=\frac {2 a^4 (56 A+49 B+44 C) \tanh ^{-1}(\sin (c+d x))}{35 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d}+\frac {1}{560} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac {a^4 (56 A+49 B+44 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1087\) vs. \(2(252)=504\).
time = 6.48, size = 1087, normalized size = 4.31 \begin {gather*} \frac {(-56 A-49 B-44 C) \cos ^6(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{128 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(56 A+49 B+44 C) \cos ^6(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{128 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {C \sec (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \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 ) \sin (d x)}{56 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {\sec (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) (6 C \sin (c)+7 B \sin (d x)+28 C \sin (d x))}{336 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {\cos (c+d x) \sec (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) (35 B \sin (c)+140 C \sin (c)+42 A \sin (d x)+168 B \sin (d x)+288 C \sin (d x))}{1680 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {\cos ^2(c+d x) \sec (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) (168 A \sin (c)+672 B \sin (c)+1152 C \sin (c)+840 A \sin (d x)+1435 B \sin (d x)+1540 C \sin (d x))}{6720 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {\cos ^3(c+d x) \sec (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) (840 A \sin (c)+1435 B \sin (c)+1540 C \sin (c)+1904 A \sin (d x)+2016 B \sin (d x)+1816 C \sin (d x))}{6720 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {\cos ^4(c+d x) \sec (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) (3808 A \sin (c)+4032 B \sin (c)+3632 C \sin (c)+5880 A \sin (d x)+5145 B \sin (d x)+4620 C \sin (d x))}{13440 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {\cos ^5(c+d x) \sec (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) (5880 A \sin (c)+5145 B \sin (c)+4620 C \sin (c)+9296 A \sin (d x)+8064 B \sin (d x)+7264 C \sin (d x))}{13440 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(576\) vs.
\(2(236)=472\).
time = 1.36, size = 577, normalized size = 2.29 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. 731 vs.
\(2 (236) = 472\).
time = 0.30, size = 731, normalized size = 2.90 \begin {gather*} \frac {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 a^{4} + 6720 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 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^{4} + 4480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{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 a^{4} + 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^{4} + 1120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 35 \, B 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 )} - 140 \, 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 )} - 840 \, 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 )} - 1260 \, 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 )} - 840 \, 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 )} - 3360 \, 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 \, 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^{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 = 2.23, size = 226, normalized size = 0.90 \begin {gather*} \frac {105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (581 \, A + 504 \, B + 454 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 16 \, {\left (238 \, A + 252 \, B + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (24 \, A + 41 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 48 \, {\left (7 \, A + 28 \, B + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 280 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 240 \, C a^{4}\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*} 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 B \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 B \sec ^{5}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{7}{\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 = 0.59, size = 443, normalized size = 1.76 \begin {gather*} \frac {105 \, {\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, 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 (5880 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5145 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 4620 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 39200 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 34300 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 30800 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 110936 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 97069 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 87164 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 172032 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 150528 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 135168 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 159656 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 134099 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 126084 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 86240 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 73220 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 58800 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21000 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21735 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 22260 \, 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}}}{1680 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.65, size = 381, normalized size = 1.51 \begin {gather*} \frac {a^4\,\mathrm {atanh}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (56\,A+49\,B+44\,C\right )}{4\,\left (14\,A\,a^4+\frac {49\,B\,a^4}{4}+11\,C\,a^4\right )}\right )\,\left (56\,A+49\,B+44\,C\right )}{8\,d}-\frac {\left (7\,A\,a^4+\frac {49\,B\,a^4}{8}+\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 {245\,B\,a^4}{6}-\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 {13867\,B\,a^4}{120}+\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 {896\,B\,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 {19157\,B\,a^4}{120}+\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}-\frac {523\,B\,a^4}{6}-70\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {207\,B\,a^4}{8}+\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.
[In]
[Out]
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