Optimal. Leaf size=96 \[ \frac {\left (a^2+4 a b+b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d}+\frac {2 b (a+b) \tan ^7(c+d x)}{7 d}+\frac {2 a (a+b) \tan ^3(c+d x)}{3 d}+\frac {b^2 \tan ^9(c+d x)}{9 d} \]
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Rubi [A] time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3675, 373} \[ \frac {\left (a^2+4 a b+b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d}+\frac {2 b (a+b) \tan ^7(c+d x)}{7 d}+\frac {2 a (a+b) \tan ^3(c+d x)}{3 d}+\frac {b^2 \tan ^9(c+d x)}{9 d} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3675
Rubi steps
\begin {align*} \int \sec ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (1+x^2\right )^2 \left (a+b x^2\right )^2 \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2+2 a (a+b) x^2+\left (a^2+4 a b+b^2\right ) x^4+2 b (a+b) x^6+b^2 x^8\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {a^2 \tan (c+d x)}{d}+\frac {2 a (a+b) \tan ^3(c+d x)}{3 d}+\frac {\left (a^2+4 a b+b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {2 b (a+b) \tan ^7(c+d x)}{7 d}+\frac {b^2 \tan ^9(c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 106, normalized size = 1.10 \[ \frac {\tan (c+d x) \left (3 \left (21 a^2-6 a b+b^2\right ) \sec ^4(c+d x)+4 \left (21 a^2-6 a b+b^2\right ) \sec ^2(c+d x)+8 \left (21 a^2-6 a b+b^2\right )+10 b (9 a-5 b) \sec ^6(c+d x)+35 b^2 \sec ^8(c+d x)\right )}{315 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 114, normalized size = 1.19 \[ \frac {{\left (8 \, {\left (21 \, a^{2} - 6 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{8} + 4 \, {\left (21 \, a^{2} - 6 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (21 \, a^{2} - 6 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (9 \, a b - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 35 \, b^{2}\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.14, size = 118, normalized size = 1.23 \[ \frac {35 \, b^{2} \tan \left (d x + c\right )^{9} + 90 \, a b \tan \left (d x + c\right )^{7} + 90 \, b^{2} \tan \left (d x + c\right )^{7} + 63 \, a^{2} \tan \left (d x + c\right )^{5} + 252 \, a b \tan \left (d x + c\right )^{5} + 63 \, b^{2} \tan \left (d x + c\right )^{5} + 210 \, a^{2} \tan \left (d x + c\right )^{3} + 210 \, a b \tan \left (d x + c\right )^{3} + 315 \, a^{2} \tan \left (d x + c\right )}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.76, size = 157, normalized size = 1.64 \[ \frac {-a^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+2 a b \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+b^{2} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 85, normalized size = 0.89 \[ \frac {35 \, b^{2} \tan \left (d x + c\right )^{9} + 90 \, {\left (a b + b^{2}\right )} \tan \left (d x + c\right )^{7} + 63 \, {\left (a^{2} + 4 \, a b + b^{2}\right )} \tan \left (d x + c\right )^{5} + 210 \, {\left (a^{2} + a b\right )} \tan \left (d x + c\right )^{3} + 315 \, a^{2} \tan \left (d x + c\right )}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.22, size = 80, normalized size = 0.83 \[ \frac {a^2\,\mathrm {tan}\left (c+d\,x\right )+\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^9}{9}+{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (\frac {a^2}{5}+\frac {4\,a\,b}{5}+\frac {b^2}{5}\right )+\frac {2\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a+b\right )}{3}+\frac {2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^7\,\left (a+b\right )}{7}}{d} \]
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 \[ \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2} \sec ^{6}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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