Optimal. Leaf size=163 \[ \frac {5 \left (8 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac {\left (8 a^2-b^2\right ) \tan (c+d x) \sec ^5(c+d x)}{48 d}+\frac {5 \left (8 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{192 d}+\frac {5 \left (8 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{128 d}+\frac {9 a b \sec ^7(c+d x)}{56 d}+\frac {b \sec ^7(c+d x) (a+b \tan (c+d x))}{8 d} \]
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Rubi [A] time = 0.13, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3508, 3486, 3768, 3770} \[ \frac {5 \left (8 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac {\left (8 a^2-b^2\right ) \tan (c+d x) \sec ^5(c+d x)}{48 d}+\frac {5 \left (8 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{192 d}+\frac {5 \left (8 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{128 d}+\frac {9 a b \sec ^7(c+d x)}{56 d}+\frac {b \sec ^7(c+d x) (a+b \tan (c+d x))}{8 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3508
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \sec ^7(c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac {b \sec ^7(c+d x) (a+b \tan (c+d x))}{8 d}+\frac {1}{8} \int \sec ^7(c+d x) \left (8 a^2-b^2+9 a b \tan (c+d x)\right ) \, dx\\ &=\frac {9 a b \sec ^7(c+d x)}{56 d}+\frac {b \sec ^7(c+d x) (a+b \tan (c+d x))}{8 d}+\frac {1}{8} \left (8 a^2-b^2\right ) \int \sec ^7(c+d x) \, dx\\ &=\frac {9 a b \sec ^7(c+d x)}{56 d}+\frac {\left (8 a^2-b^2\right ) \sec ^5(c+d x) \tan (c+d x)}{48 d}+\frac {b \sec ^7(c+d x) (a+b \tan (c+d x))}{8 d}+\frac {1}{48} \left (5 \left (8 a^2-b^2\right )\right ) \int \sec ^5(c+d x) \, dx\\ &=\frac {9 a b \sec ^7(c+d x)}{56 d}+\frac {5 \left (8 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{192 d}+\frac {\left (8 a^2-b^2\right ) \sec ^5(c+d x) \tan (c+d x)}{48 d}+\frac {b \sec ^7(c+d x) (a+b \tan (c+d x))}{8 d}+\frac {1}{64} \left (5 \left (8 a^2-b^2\right )\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac {9 a b \sec ^7(c+d x)}{56 d}+\frac {5 \left (8 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{128 d}+\frac {5 \left (8 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{192 d}+\frac {\left (8 a^2-b^2\right ) \sec ^5(c+d x) \tan (c+d x)}{48 d}+\frac {b \sec ^7(c+d x) (a+b \tan (c+d x))}{8 d}+\frac {1}{128} \left (5 \left (8 a^2-b^2\right )\right ) \int \sec (c+d x) \, dx\\ &=\frac {5 \left (8 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac {9 a b \sec ^7(c+d x)}{56 d}+\frac {5 \left (8 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{128 d}+\frac {5 \left (8 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{192 d}+\frac {\left (8 a^2-b^2\right ) \sec ^5(c+d x) \tan (c+d x)}{48 d}+\frac {b \sec ^7(c+d x) (a+b \tan (c+d x))}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 131, normalized size = 0.80 \[ \frac {105 \left (8 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))+56 \left (8 a^2-b^2\right ) \tan (c+d x) \sec ^5(c+d x)+70 \left (8 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)+105 \left (8 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)+48 b \sec ^7(c+d x) (16 a+7 b \tan (c+d x))}{2688 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 163, normalized size = 1.00 \[ \frac {105 \, {\left (8 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (8 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 1536 \, a b \cos \left (d x + c\right ) + 14 \, {\left (15 \, {\left (8 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{6} + 10 \, {\left (8 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (8 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 48 \, b^{2}\right )} \sin \left (d x + c\right )}{5376 \, d \cos \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 437, normalized size = 2.68 \[ \frac {105 \, {\left (8 \, a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (8 \, a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (1848 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 105 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 5376 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 3416 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2779 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5376 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 6328 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 6265 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 26880 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 4760 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 12355 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 26880 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 4760 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12355 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 16128 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6328 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6265 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16128 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3416 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2779 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 768 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1848 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 768 \, a b\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{8}}}{2688 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 235, normalized size = 1.44 \[ \frac {a^{2} \tan \left (d x +c \right ) \left (\sec ^{5}\left (d x +c \right )\right )}{6 d}+\frac {5 a^{2} \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{24 d}+\frac {5 a^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{16 d}+\frac {5 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d}+\frac {2 a b}{7 d \cos \left (d x +c \right )^{7}}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{8}}+\frac {5 b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{48 d \cos \left (d x +c \right )^{6}}+\frac {5 b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{64 d \cos \left (d x +c \right )^{4}}+\frac {5 b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{128 d \cos \left (d x +c \right )^{2}}+\frac {5 b^{2} \sin \left (d x +c \right )}{128 d}-\frac {5 b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 220, normalized size = 1.35 \[ \frac {7 \, b^{2} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 56 \, a^{2} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {1536 \, a b}{\cos \left (d x + c\right )^{7}}}{5376 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.10, size = 432, normalized size = 2.65 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {5\,a^2}{8}-\frac {5\,b^2}{64}\right )}{d}+\frac {\left (\frac {11\,a^2}{8}+\frac {5\,b^2}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\left (\frac {397\,b^2}{192}-\frac {61\,a^2}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\left (\frac {113\,a^2}{24}+\frac {895\,b^2}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-20\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (\frac {1765\,b^2}{192}-\frac {85\,a^2}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+20\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\left (\frac {1765\,b^2}{192}-\frac {85\,a^2}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {113\,a^2}{24}+\frac {895\,b^2}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {397\,b^2}{192}-\frac {61\,a^2}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}+\left (\frac {11\,a^2}{8}+\frac {5\,b^2}{64}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {4\,a\,b}{7}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{2} \sec ^{7}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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