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 = 0.655504, antiderivative size = 238, normalized size of antiderivative = 1., 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 4019
Rule 3787
Rule 3767
Rule 8
Rule 3768
Rule 3770
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*}
Mathematica [B] time = 6.47336, size = 880, normalized size = 3.7 \[ -\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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Maple [A] time = 0.069, size = 374, normalized size = 1.6 \begin{align*}{\frac{A}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{B}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{7\,A}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{9\,B}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{23\,A}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{13\,B}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{49\,A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{111\,B}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{9\,B}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{A}{d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-4\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) A}{d{a}^{4}}}+{\frac{21\,B}{2\,d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{B}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+4\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) A}{d{a}^{4}}}-{\frac{21\,B}{2\,d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{9\,B}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{A}{d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{B}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998842, size = 566, normalized size = 2.38 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.516238, size = 940, normalized size = 3.95 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39089, size = 360, normalized size = 1.51 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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