Integrand size = 23, antiderivative size = 96 \[ \int \sec ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {a^2 \tan (c+d x)}{d}+\frac {2 a (a+b) \tan ^3(c+d x)}{3 d}+\frac {\left (a^2+4 a b+b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {2 b (a+b) \tan ^7(c+d x)}{7 d}+\frac {b^2 \tan ^9(c+d x)}{9 d} \]
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Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3756, 380} \[ \int \sec ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {\left (a^2+4 a b+b^2\right ) \tan ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d}+\frac {2 b (a+b) \tan ^7(c+d x)}{7 d}+\frac {2 a (a+b) \tan ^3(c+d x)}{3 d}+\frac {b^2 \tan ^9(c+d x)}{9 d} \]
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Rule 380
Rule 3756
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 {\text {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*}
Time = 1.53 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.10 \[ \int \sec ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {\left (8 \left (21 a^2-6 a b+b^2\right )+4 \left (21 a^2-6 a b+b^2\right ) \sec ^2(c+d x)+3 \left (21 a^2-6 a b+b^2\right ) \sec ^4(c+d x)+10 (9 a-5 b) b \sec ^6(c+d x)+35 b^2 \sec ^8(c+d x)\right ) \tan (c+d x)}{315 d} \]
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Time = 11.66 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.64
method | result | size |
derivativedivides | \(\frac {b^{2} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )+2 a b \left (\frac {\sin \left (d x +c \right )^{3}}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \sin \left (d x +c \right )^{3}}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{3}}\right )-a^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(157\) |
default | \(\frac {b^{2} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )+2 a b \left (\frac {\sin \left (d x +c \right )^{3}}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \sin \left (d x +c \right )^{3}}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{3}}\right )-a^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(157\) |
risch | \(\frac {16 i \left (210 a^{2} {\mathrm e}^{12 i \left (d x +c \right )}-420 a b \,{\mathrm e}^{12 i \left (d x +c \right )}+210 b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+945 a^{2} {\mathrm e}^{10 i \left (d x +c \right )}-630 a b \,{\mathrm e}^{10 i \left (d x +c \right )}-315 b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+1701 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-126 a b \,{\mathrm e}^{8 i \left (d x +c \right )}+441 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+1554 \,{\mathrm e}^{6 i \left (d x +c \right )} a^{2}-84 a b \,{\mathrm e}^{6 i \left (d x +c \right )}-126 \,{\mathrm e}^{6 i \left (d x +c \right )} b^{2}+756 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-216 a b \,{\mathrm e}^{4 i \left (d x +c \right )}+36 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+189 \,{\mathrm e}^{2 i \left (d x +c \right )} a^{2}-54 a b \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{2 i \left (d x +c \right )} b^{2}+21 a^{2}-6 a b +b^{2}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{9}}\) | \(279\) |
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Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.19 \[ \int \sec ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {{\left (8 \, {\left (21 \, a^{2} - 6 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{8} + 4 \, {\left (21 \, a^{2} - 6 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (21 \, a^{2} - 6 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (9 \, a b - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 35 \, b^{2}\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{9}} \]
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\[ \int \sec ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\int \left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2} \sec ^{6}{\left (c + d x \right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \sec ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {35 \, b^{2} \tan \left (d x + c\right )^{9} + 90 \, {\left (a b + b^{2}\right )} \tan \left (d x + c\right )^{7} + 63 \, {\left (a^{2} + 4 \, a b + b^{2}\right )} \tan \left (d x + c\right )^{5} + 210 \, {\left (a^{2} + a b\right )} \tan \left (d x + c\right )^{3} + 315 \, a^{2} \tan \left (d x + c\right )}{315 \, d} \]
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Time = 0.71 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.23 \[ \int \sec ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {35 \, b^{2} \tan \left (d x + c\right )^{9} + 90 \, a b \tan \left (d x + c\right )^{7} + 90 \, b^{2} \tan \left (d x + c\right )^{7} + 63 \, a^{2} \tan \left (d x + c\right )^{5} + 252 \, a b \tan \left (d x + c\right )^{5} + 63 \, b^{2} \tan \left (d x + c\right )^{5} + 210 \, a^{2} \tan \left (d x + c\right )^{3} + 210 \, a b \tan \left (d x + c\right )^{3} + 315 \, a^{2} \tan \left (d x + c\right )}{315 \, d} \]
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Time = 11.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.83 \[ \int \sec ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )+\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^9}{9}+{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (\frac {a^2}{5}+\frac {4\,a\,b}{5}+\frac {b^2}{5}\right )+\frac {2\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a+b\right )}{3}+\frac {2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^7\,\left (a+b\right )}{7}}{d} \]
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