Optimal. Leaf size=149 \[ \frac {a^2 \sec ^6(c+d x)}{6 d}-\frac {3 a^2 \sec ^4(c+d x)}{4 d}+\frac {3 a^2 \sec ^2(c+d x)}{2 d}+\frac {a^2 \log (\cos (c+d x))}{d}+\frac {2 a b \sec ^7(c+d x)}{7 d}-\frac {6 a b \sec ^5(c+d x)}{5 d}+\frac {2 a b \sec ^3(c+d x)}{d}-\frac {2 a b \sec (c+d x)}{d}+\frac {b^2 \tan ^8(c+d x)}{8 d} \]
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Rubi [A] time = 0.11, antiderivative size = 169, normalized size of antiderivative = 1.13, 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 = {3885, 948} \[ \frac {\left (a^2-3 b^2\right ) \sec ^6(c+d x)}{6 d}-\frac {3 \left (a^2-b^2\right ) \sec ^4(c+d x)}{4 d}+\frac {\left (3 a^2-b^2\right ) \sec ^2(c+d x)}{2 d}+\frac {a^2 \log (\cos (c+d x))}{d}+\frac {2 a b \sec ^7(c+d x)}{7 d}-\frac {6 a b \sec ^5(c+d x)}{5 d}+\frac {2 a b \sec ^3(c+d x)}{d}-\frac {2 a b \sec (c+d x)}{d}+\frac {b^2 \sec ^8(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 948
Rule 3885
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^2 \tan ^7(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a+x)^2 \left (b^2-x^2\right )^3}{x} \, dx,x,b \sec (c+d x)\right )}{b^6 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (2 a b^6+\frac {a^2 b^6}{x}-b^4 \left (3 a^2-b^2\right ) x-6 a b^4 x^2+3 b^2 \left (a^2-b^2\right ) x^3+6 a b^2 x^4-\left (a^2-3 b^2\right ) x^5-2 a x^6-x^7\right ) \, dx,x,b \sec (c+d x)\right )}{b^6 d}\\ &=\frac {a^2 \log (\cos (c+d x))}{d}-\frac {2 a b \sec (c+d x)}{d}+\frac {\left (3 a^2-b^2\right ) \sec ^2(c+d x)}{2 d}+\frac {2 a b \sec ^3(c+d x)}{d}-\frac {3 \left (a^2-b^2\right ) \sec ^4(c+d x)}{4 d}-\frac {6 a b \sec ^5(c+d x)}{5 d}+\frac {\left (a^2-3 b^2\right ) \sec ^6(c+d x)}{6 d}+\frac {2 a b \sec ^7(c+d x)}{7 d}+\frac {b^2 \sec ^8(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 138, normalized size = 0.93 \[ \frac {140 \left (a^2-3 b^2\right ) \sec ^6(c+d x)-630 \left (a^2-b^2\right ) \sec ^4(c+d x)+420 \left (3 a^2-b^2\right ) \sec ^2(c+d x)+840 a^2 \log (\cos (c+d x))+240 a b \sec ^7(c+d x)-1008 a b \sec ^5(c+d x)+1680 a b \sec ^3(c+d x)-1680 a b \sec (c+d x)+105 b^2 \sec ^8(c+d x)}{840 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 146, normalized size = 0.98 \[ \frac {840 \, a^{2} \cos \left (d x + c\right )^{8} \log \left (-\cos \left (d x + c\right )\right ) - 1680 \, a b \cos \left (d x + c\right )^{7} + 1680 \, a b \cos \left (d x + c\right )^{5} + 420 \, {\left (3 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{6} - 1008 \, a b \cos \left (d x + c\right )^{3} - 630 \, {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 240 \, a b \cos \left (d x + c\right ) + 140 \, {\left (a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 105 \, b^{2}}{840 \, d \cos \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 8.89, size = 415, normalized size = 2.79 \[ -\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 {2283 \, a^{2} + 1536 \, a b + \frac {19944 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {12288 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {77364 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {43008 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {175448 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {86016 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {231490 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {53760 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {26880 \, b^{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 [A] time = 0.64, size = 256, normalized size = 1.72 \[ \frac {a^{2} \left (\tan ^{6}\left (d x +c \right )\right )}{6 d}-\frac {a^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {2 a b \left (\sin ^{8}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )^{7}}-\frac {2 a b \left (\sin ^{8}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{5}}+\frac {2 a b \left (\sin ^{8}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{3}}-\frac {2 a b \left (\sin ^{8}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )}-\frac {32 a b \cos \left (d x +c \right )}{35 d}-\frac {2 a b \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{7 d}-\frac {12 a b \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{35 d}-\frac {16 a b \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35 d}+\frac {b^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 139, normalized size = 0.93 \[ \frac {840 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {1680 \, a b \cos \left (d x + c\right )^{7} - 1680 \, a b \cos \left (d x + c\right )^{5} - 420 \, {\left (3 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{6} + 1008 \, a b \cos \left (d x + c\right )^{3} + 630 \, {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} - 240 \, a b \cos \left (d x + c\right ) - 140 \, {\left (a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 105 \, b^{2}}{\cos \left (d x + c\right )^{8}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.17, size = 280, normalized size = 1.88 \[ -\frac {\frac {64\,a\,b}{35}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (16\,a^2+\frac {256\,b\,a}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^2+\frac {512\,b\,a}{35}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {170\,a^2}{3}+\frac {512\,b\,a}{5}\right )-\frac {170\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}+16\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {256\,a^2}{3}+64\,a\,b-32\,b^2\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {2\,a^2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.24, size = 252, normalized size = 1.69 \[ \begin {cases} - \frac {a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{2} \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac {a^{2} \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac {a^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac {2 a b \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{7 d} - \frac {12 a b \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} + \frac {16 a b \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} - \frac {32 a b \sec {\left (c + d x \right )}}{35 d} + \frac {b^{2} \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} - \frac {b^{2} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} + \frac {b^{2} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} - \frac {b^{2} \sec ^{2}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \sec {\relax (c )}\right )^{2} \tan ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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