Optimal. Leaf size=132 \[ \frac {a^2 \sec ^8(c+d x)}{8 d}+\frac {2 a^2 \sec ^7(c+d x)}{7 d}-\frac {a^2 \sec ^6(c+d x)}{3 d}-\frac {6 a^2 \sec ^5(c+d x)}{5 d}+\frac {2 a^2 \sec ^3(c+d x)}{d}+\frac {a^2 \sec ^2(c+d x)}{d}-\frac {2 a^2 \sec (c+d x)}{d}+\frac {a^2 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac {a^2 \sec ^8(c+d x)}{8 d}+\frac {2 a^2 \sec ^7(c+d x)}{7 d}-\frac {a^2 \sec ^6(c+d x)}{3 d}-\frac {6 a^2 \sec ^5(c+d x)}{5 d}+\frac {2 a^2 \sec ^3(c+d x)}{d}+\frac {a^2 \sec ^2(c+d x)}{d}-\frac {2 a^2 \sec (c+d x)}{d}+\frac {a^2 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^2 \tan ^7(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^3 (a+a x)^5}{x^9} \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^8}{x^9}+\frac {2 a^8}{x^8}-\frac {2 a^8}{x^7}-\frac {6 a^8}{x^6}+\frac {6 a^8}{x^4}+\frac {2 a^8}{x^3}-\frac {2 a^8}{x^2}-\frac {a^8}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=\frac {a^2 \log (\cos (c+d x))}{d}-\frac {2 a^2 \sec (c+d x)}{d}+\frac {a^2 \sec ^2(c+d x)}{d}+\frac {2 a^2 \sec ^3(c+d x)}{d}-\frac {6 a^2 \sec ^5(c+d x)}{5 d}-\frac {a^2 \sec ^6(c+d x)}{3 d}+\frac {2 a^2 \sec ^7(c+d x)}{7 d}+\frac {a^2 \sec ^8(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 110, normalized size = 0.83 \[ \frac {a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (105 \sec ^8(c+d x)+240 \sec ^7(c+d x)-280 \sec ^6(c+d x)-1008 \sec ^5(c+d x)+1680 \sec ^3(c+d x)+840 \sec ^2(c+d x)-1680 \sec (c+d x)+840 \log (\cos (c+d x))\right )}{3360 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 117, normalized size = 0.89 \[ \frac {840 \, a^{2} \cos \left (d x + c\right )^{8} \log \left (-\cos \left (d x + c\right )\right ) - 1680 \, a^{2} \cos \left (d x + c\right )^{7} + 840 \, a^{2} \cos \left (d x + c\right )^{6} + 1680 \, a^{2} \cos \left (d x + c\right )^{5} - 1008 \, a^{2} \cos \left (d x + c\right )^{3} - 280 \, a^{2} \cos \left (d x + c\right )^{2} + 240 \, a^{2} \cos \left (d x + c\right ) + 105 \, a^{2}}{840 \, d \cos \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 11.00, size = 292, normalized size = 2.21 \[ -\frac {840 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 840 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {3819 \, a^{2} + \frac {32232 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {120372 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {261464 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {258370 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {175448 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {77364 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {19944 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {2283 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{8}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.78, size = 264, normalized size = 2.00 \[ \frac {a^{2} \left (\tan ^{6}\left (d x +c \right )\right )}{6 d}-\frac {a^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {2 a^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )^{7}}-\frac {2 a^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{5}}+\frac {2 a^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{3}}-\frac {2 a^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )}-\frac {32 a^{2} \cos \left (d x +c \right )}{35 d}-\frac {2 a^{2} \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{7 d}-\frac {12 a^{2} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{35 d}-\frac {16 a^{2} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35 d}+\frac {a^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 110, normalized size = 0.83 \[ \frac {840 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {1680 \, a^{2} \cos \left (d x + c\right )^{7} - 840 \, a^{2} \cos \left (d x + c\right )^{6} - 1680 \, a^{2} \cos \left (d x + c\right )^{5} + 1008 \, a^{2} \cos \left (d x + c\right )^{3} + 280 \, a^{2} \cos \left (d x + c\right )^{2} - 240 \, a^{2} \cos \left (d x + c\right ) - 105 \, a^{2}}{\cos \left (d x + c\right )^{8}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.82, size = 249, normalized size = 1.89 \[ \frac {2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-16\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\frac {170\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}-\frac {352\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {2386\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}-\frac {336\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {582\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}-\frac {64\,a^2}{35}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {2\,a^2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.18, size = 252, normalized size = 1.91 \[ \begin {cases} - \frac {a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{2} \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} + \frac {2 a^{2} \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{7 d} + \frac {a^{2} \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac {a^{2} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} - \frac {12 a^{2} \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} - \frac {a^{2} \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac {a^{2} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} + \frac {16 a^{2} \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} + \frac {a^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac {a^{2} \sec ^{2}{\left (c + d x \right )}}{8 d} - \frac {32 a^{2} \sec {\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right )^{2} \tan ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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