Integrand size = 45, antiderivative size = 363 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {\left (2 a^2 C d^2-a b d (2 c C+B d) (3+m)+b^2 (2+m) \left (2 c^2 C+2 B c d (3+m)+(A-C) d^2 (3+m)\right )\right ) (a+b \tan (e+f x))^{1+m}}{b^3 f (1+m) (2+m) (3+m)}+\frac {(A-i B-C) (c-i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) f (1+m)}+\frac {(i A-B-i C) (c+i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) f (1+m)}-\frac {d (2 a C d-b (2 c C+B d (3+m))) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^2 f (2+m) (3+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b f (3+m)} \]
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Time = 1.19 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3728, 3718, 3711, 3620, 3618, 70} \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {(a+b \tan (e+f x))^{m+1} \left (2 a^2 C d^2-a b d (m+3) (B d+2 c C)+b^2 (m+2) \left (d^2 (m+3) (A-C)+2 B c d (m+3)+2 c^2 C\right )\right )}{b^3 f (m+1) (m+2) (m+3)}+\frac {(c-i d)^2 (A-i B-C) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a)}+\frac {(c+i d)^2 (i A-B-i C) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}+\frac {d \tan (e+f x) (-2 a C d+b B d (m+3)+2 b c C) (a+b \tan (e+f x))^{m+1}}{b^2 f (m+2) (m+3)}+\frac {C (c+d \tan (e+f x))^2 (a+b \tan (e+f x))^{m+1}}{b f (m+3)} \]
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Rule 70
Rule 3618
Rule 3620
Rule 3711
Rule 3718
Rule 3728
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b f (3+m)}+\frac {\int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \left (A b c (3+m)-C (2 a d+b c (1+m))+b (B c+(A-C) d) (3+m) \tan (e+f x)+(2 b c C-2 a C d+b B d (3+m)) \tan ^2(e+f x)\right ) \, dx}{b (3+m)} \\ & = \frac {d (2 b c C-2 a C d+b B d (3+m)) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^2 f (2+m) (3+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b f (3+m)}-\frac {\int (a+b \tan (e+f x))^m \left (a d (2 b c C-2 a C d+b B d (3+m))-b c (2+m) (A b c (3+m)-C (2 a d+b c (1+m)))-b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) \tan (e+f x)-\left (b c (2+m) (2 b c C-2 a C d+b B d (3+m))+d \left (b^2 (B c+(A-C) d) (2+m) (3+m)-a (2 b c C-2 a C d+b B d (3+m))\right )\right ) \tan ^2(e+f x)\right ) \, dx}{b^2 (2+m) (3+m)} \\ & = \frac {\left (2 a^2 C d^2-a b d (2 c C+B d) (3+m)+b^2 (2+m) \left (2 c^2 C+2 B c d (3+m)+(A-C) d^2 (3+m)\right )\right ) (a+b \tan (e+f x))^{1+m}}{b^3 f (1+m) (2+m) (3+m)}+\frac {d (2 b c C-2 a C d+b B d (3+m)) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^2 f (2+m) (3+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b f (3+m)}-\frac {\int (a+b \tan (e+f x))^m \left (-b^2 \left (A c^2-c^2 C-2 B c d-A d^2+C d^2\right ) (2+m) (3+m)-b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) \tan (e+f x)\right ) \, dx}{b^2 (2+m) (3+m)} \\ & = \frac {\left (2 a^2 C d^2-a b d (2 c C+B d) (3+m)+b^2 (2+m) \left (2 c^2 C+2 B c d (3+m)+(A-C) d^2 (3+m)\right )\right ) (a+b \tan (e+f x))^{1+m}}{b^3 f (1+m) (2+m) (3+m)}+\frac {d (2 b c C-2 a C d+b B d (3+m)) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^2 f (2+m) (3+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b f (3+m)}+\frac {1}{2} \left ((A-i B-C) (c-i d)^2\right ) \int (1+i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx+\frac {1}{2} \left ((A+i B-C) (c+i d)^2\right ) \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx \\ & = \frac {\left (2 a^2 C d^2-a b d (2 c C+B d) (3+m)+b^2 (2+m) \left (2 c^2 C+2 B c d (3+m)+(A-C) d^2 (3+m)\right )\right ) (a+b \tan (e+f x))^{1+m}}{b^3 f (1+m) (2+m) (3+m)}+\frac {d (2 b c C-2 a C d+b B d (3+m)) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^2 f (2+m) (3+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b f (3+m)}+\frac {\left ((i A+B-i C) (c-i d)^2\right ) \text {Subst}\left (\int \frac {(a-i b x)^m}{-1+x} \, dx,x,i \tan (e+f x)\right )}{2 f}-\frac {\left (i (A+i B-C) (c+i d)^2\right ) \text {Subst}\left (\int \frac {(a+i b x)^m}{-1+x} \, dx,x,-i \tan (e+f x)\right )}{2 f} \\ & = \frac {\left (2 a^2 C d^2-a b d (2 c C+B d) (3+m)+b^2 (2+m) \left (2 c^2 C+2 B c d (3+m)+(A-C) d^2 (3+m)\right )\right ) (a+b \tan (e+f x))^{1+m}}{b^3 f (1+m) (2+m) (3+m)}-\frac {(i A+B-i C) (c-i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a-i b) f (1+m)}-\frac {(A+i B-C) (c+i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a-b) f (1+m)}+\frac {d (2 b c C-2 a C d+b B d (3+m)) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^2 f (2+m) (3+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b f (3+m)} \\ \end{align*}
Time = 6.37 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.39 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b f (3+m)}+\frac {\frac {d (2 b c C-2 a C d+b B d (3+m)) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b f (2+m)}-\frac {\frac {\left (-b c (2+m) (2 b c C-2 a C d+b B d (3+m))-d \left (b^2 (B c+(A-C) d) (2+m) (3+m)-a (2 b c C-2 a C d+b B d (3+m))\right )\right ) (a+b \tan (e+f x))^{1+m}}{b f (1+m)}+\frac {i \left (-b^2 \left (A c^2-c^2 C-2 B c d-A d^2+C d^2\right ) (2+m) (3+m)-i b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m)\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {-i a-i b \tan (e+f x)}{-i a+b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) f (1+m)}-\frac {i \left (-b^2 \left (A c^2-c^2 C-2 B c d-A d^2+C d^2\right ) (2+m) (3+m)+i b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m)\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {i a+i b \tan (e+f x)}{-i a-b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a-i b) f (1+m)}}{b (2+m)}}{b (3+m)} \]
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\[\int \left (a +b \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )^{2} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )d x\]
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\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int { {\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{m} \left (c + d \tan {\left (e + f x \right )}\right )^{2} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
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\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int { {\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int { {\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^2\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right ) \,d x \]
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