Optimal. Leaf size=238 \[ \frac {8 (83 A-216 B) \tan (c+d x)}{105 a^4 d}-\frac {(8 A-21 B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac {(52 A-129 B) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {4 (83 A-216 B) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(8 A-21 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}+\frac {(A-B) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {(A-2 B) \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.66, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4019, 3787, 3767, 8, 3768, 3770} \[ \frac {8 (83 A-216 B) \tan (c+d x)}{105 a^4 d}-\frac {(8 A-21 B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac {(52 A-129 B) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {4 (83 A-216 B) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(8 A-21 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}+\frac {(A-B) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {(A-2 B) \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3768
Rule 3770
Rule 3787
Rule 4019
Rubi steps
\begin {align*} \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx &=\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\sec ^5(c+d x) (5 a (A-B)-a (2 A-9 B) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^4(c+d x) \left (28 a^2 (A-2 B)-a^2 (24 A-73 B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^3(c+d x) \left (3 a^3 (52 A-129 B)-a^3 (176 A-477 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {\int \sec ^2(c+d x) \left (8 a^4 (83 A-216 B)-105 a^4 (8 A-21 B) \sec (c+d x)\right ) \, dx}{105 a^8}\\ &=\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(8 (83 A-216 B)) \int \sec ^2(c+d x) \, dx}{105 a^4}-\frac {(8 A-21 B) \int \sec ^3(c+d x) \, dx}{a^4}\\ &=-\frac {(8 A-21 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}+\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(8 A-21 B) \int \sec (c+d x) \, dx}{2 a^4}-\frac {(8 (83 A-216 B)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d}\\ &=-\frac {(8 A-21 B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac {8 (83 A-216 B) \tan (c+d x)}{105 a^4 d}-\frac {(8 A-21 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}+\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 6.52, size = 880, normalized size = 3.70 \[ -\frac {8 (21 B-8 A) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^3(c+d x) (A+B \sec (c+d x)) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac {8 (21 B-8 A) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^3(c+d x) (A+B \sec (c+d x)) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac {\sec \left (\frac {c}{2}\right ) \sec (c) \sec ^5(c+d x) (A+B \sec (c+d x)) \left (-38668 A \sin \left (\frac {d x}{2}\right )+73206 B \sin \left (\frac {d x}{2}\right )+64384 A \sin \left (\frac {3 d x}{2}\right )-166668 B \sin \left (\frac {3 d x}{2}\right )-70896 A \sin \left (c-\frac {d x}{2}\right )+183162 B \sin \left (c-\frac {d x}{2}\right )+50316 A \sin \left (c+\frac {d x}{2}\right )-100842 B \sin \left (c+\frac {d x}{2}\right )-59248 A \sin \left (2 c+\frac {d x}{2}\right )+155526 B \sin \left (2 c+\frac {d x}{2}\right )-22820 A \sin \left (c+\frac {3 d x}{2}\right )+37380 B \sin \left (c+\frac {3 d x}{2}\right )+48004 A \sin \left (2 c+\frac {3 d x}{2}\right )-101148 B \sin \left (2 c+\frac {3 d x}{2}\right )-39200 A \sin \left (3 c+\frac {3 d x}{2}\right )+102900 B \sin \left (3 c+\frac {3 d x}{2}\right )+46032 A \sin \left (c+\frac {5 d x}{2}\right )-119364 B \sin \left (c+\frac {5 d x}{2}\right )-8750 A \sin \left (2 c+\frac {5 d x}{2}\right )+8820 B \sin \left (2 c+\frac {5 d x}{2}\right )+35742 A \sin \left (3 c+\frac {5 d x}{2}\right )-78204 B \sin \left (3 c+\frac {5 d x}{2}\right )-19040 A \sin \left (4 c+\frac {5 d x}{2}\right )+49980 B \sin \left (4 c+\frac {5 d x}{2}\right )+24664 A \sin \left (2 c+\frac {7 d x}{2}\right )-64053 B \sin \left (2 c+\frac {7 d x}{2}\right )-1050 A \sin \left (3 c+\frac {7 d x}{2}\right )-3885 B \sin \left (3 c+\frac {7 d x}{2}\right )+19834 A \sin \left (4 c+\frac {7 d x}{2}\right )-44733 B \sin \left (4 c+\frac {7 d x}{2}\right )-5880 A \sin \left (5 c+\frac {7 d x}{2}\right )+15435 B \sin \left (5 c+\frac {7 d x}{2}\right )+8456 A \sin \left (3 c+\frac {9 d x}{2}\right )-21987 B \sin \left (3 c+\frac {9 d x}{2}\right )+630 A \sin \left (4 c+\frac {9 d x}{2}\right )-3675 B \sin \left (4 c+\frac {9 d x}{2}\right )+6986 A \sin \left (5 c+\frac {9 d x}{2}\right )-16107 B \sin \left (5 c+\frac {9 d x}{2}\right )-840 A \sin \left (6 c+\frac {9 d x}{2}\right )+2205 B \sin \left (6 c+\frac {9 d x}{2}\right )+1328 A \sin \left (4 c+\frac {11 d x}{2}\right )-3456 B \sin \left (4 c+\frac {11 d x}{2}\right )+210 A \sin \left (5 c+\frac {11 d x}{2}\right )-840 B \sin \left (5 c+\frac {11 d x}{2}\right )+1118 A \sin \left (6 c+\frac {11 d x}{2}\right )-2616 B \sin \left (6 c+\frac {11 d x}{2}\right )\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{6720 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 358, normalized size = 1.50 \[ -\frac {105 \, {\left ({\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (83 \, A - 216 \, B\right )} \cos \left (d x + c\right )^{5} + {\left (4472 \, A - 11619 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (1318 \, A - 3411 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (592 \, A - 1509 \, B\right )} \cos \left (d x + c\right )^{2} + 210 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right ) + 105 \, B\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 267, normalized size = 1.12 \[ -\frac {\frac {420 \, {\left (8 \, A - 21 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {420 \, {\left (8 \, A - 21 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {840 \, {\left (2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 147 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 189 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1365 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5145 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 11655 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.63, size = 374, normalized size = 1.57 \[ \frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}-\frac {B \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {7 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {9 B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{24 d \,a^{4}}-\frac {13 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}+\frac {49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {111 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) A}{d \,a^{4}}-\frac {21 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) B}{2 d \,a^{4}}+\frac {9 B}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {A}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {B}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {9 B}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {A}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) A}{d \,a^{4}}+\frac {21 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) B}{2 d \,a^{4}}-\frac {B}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 419, normalized size = 1.76 \[ -\frac {3 \, B {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - A {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.05, size = 272, normalized size = 1.14 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{4\,a^4}+\frac {4\,A-6\,B}{8\,a^4}+\frac {5\,A-15\,B}{24\,a^4}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A-B\right )}{4\,a^4}-\frac {5\,B}{2\,a^4}+\frac {3\,\left (4\,A-6\,B\right )}{4\,a^4}+\frac {3\,\left (5\,A-15\,B\right )}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,A-9\,B\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,A-7\,B\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A-B\right )}{40\,a^4}+\frac {4\,A-6\,B}{40\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (8\,A-21\,B\right )}{a^4\,d} \]
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 \[ \frac {\int \frac {A \sec ^{6}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{7}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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