Integrand size = 31, antiderivative size = 403 \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {b \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^{1+m}(c+d x)}{d (1+m) (3+m) (4+m)}+\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+m)}+\frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) (3+m) (4+m)}+\frac {\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (2+m)}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)} \]
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Time = 1.87 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3688, 3728, 3718, 3711, 3619, 3557, 371} \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {b^2 \left (a^2 B \left (m^2+9 m+26\right )+2 a A b (m+4)^2-b^2 B \left (m^2+7 m+12\right )\right ) \tan ^{m+2}(c+d x)}{d (m+2) (m+3) (m+4)}-\frac {b \left (-2 a^3 B \left (m^2+8 m+19\right )-a^2 A b \left (5 m^2+37 m+68\right )+4 a b^2 B \left (m^2+7 m+12\right )+A b^3 \left (m^2+7 m+12\right )\right ) \tan ^{m+1}(c+d x)}{d (m+1) (m+3) (m+4)}+\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(c+d x)\right )}{d (m+1)}+\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \tan ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\tan ^2(c+d x)\right )}{d (m+2)}+\frac {b (a B (m+7)+A b (m+4)) \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^2}{d (m+3) (m+4)}+\frac {b B \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^3}{d (m+4)} \]
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Rule 371
Rule 3557
Rule 3619
Rule 3688
Rule 3711
Rule 3718
Rule 3728
Rubi steps \begin{align*} \text {integral}& = \frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}+\frac {\int \tan ^m(c+d x) (a+b \tan (c+d x))^2 \left (-a (b B (1+m)-a A (4+m))+\left (2 a A b+a^2 B-b^2 B\right ) (4+m) \tan (c+d x)+b (A b (4+m)+a B (7+m)) \tan ^2(c+d x)\right ) \, dx}{4+m} \\ & = \frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}+\frac {\int \tan ^m(c+d x) (a+b \tan (c+d x)) \left (-a (a (3+m) (b B (1+m)-a A (4+m))+b (1+m) (A b (4+m)+a B (7+m)))+\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) (3+m) (4+m) \tan (c+d x)+b \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^2(c+d x)\right ) \, dx}{12+7 m+m^2} \\ & = \frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}-\frac {\int \tan ^m(c+d x) \left (a^2 (2+m) (a (3+m) (b B (1+m)-a A (4+m))+b (1+m) (A b (4+m)+a B (7+m)))-\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) (2+m) (3+m) (4+m) \tan (c+d x)+b (2+m) \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^2(c+d x)\right ) \, dx}{24+26 m+9 m^2+m^3} \\ & = -\frac {b \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^{1+m}(c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}-\frac {\int \tan ^m(c+d x) \left (-\left (\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) (2+m) (3+m) (4+m)\right )-\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) (2+m) (3+m) (4+m) \tan (c+d x)\right ) \, dx}{24+26 m+9 m^2+m^3} \\ & = -\frac {b \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^{1+m}(c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}+\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \int \tan ^m(c+d x) \, dx+\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \int \tan ^{1+m}(c+d x) \, dx \\ & = -\frac {b \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^{1+m}(c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}+\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \text {Subst}\left (\int \frac {x^m}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \text {Subst}\left (\int \frac {x^{1+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {b \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^{1+m}(c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+m)}+\frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac {\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (2+m)}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)} \\ \end{align*}
Time = 4.83 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.88 \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {\tan ^{1+m}(c+d x) \left (-b (2+m) \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right )+\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) (2+m) (3+m) (4+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right )+b^2 (1+m) \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan (c+d x)+\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) (1+m) (3+m) (4+m) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan (c+d x)+b (1+m) (2+m) (A b (4+m)+a B (7+m)) (a+b \tan (c+d x))^2+b B (1+m) (2+m) (3+m) (a+b \tan (c+d x))^3\right )}{d (1+m) (2+m) (3+m) (4+m)} \]
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\[\int \tan \left (d x +c \right )^{m} \left (a +b \tan \left (d x +c \right )\right )^{4} \left (A +B \tan \left (d x +c \right )\right )d x\]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{4} \tan \left (d x + c\right )^{m} \,d x } \]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{4} \tan ^{m}{\left (c + d x \right )}\, dx \]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{4} \tan \left (d x + c\right )^{m} \,d x } \]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{4} \tan \left (d x + c\right )^{m} \,d x } \]
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Timed out. \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^4 \,d x \]
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