Integrand size = 31, antiderivative size = 194 \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {b (A b (2+m)+a B (3+m)) \tan ^{1+m}(c+d x)}{d (1+m) (2+m)}+\frac {\left (a^2 A-A b^2-2 a b B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+m)}+\frac {\left (2 a A b+a^2 B-b^2 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (2+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))}{d (2+m)} \]
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Time = 0.40 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3688, 3711, 3619, 3557, 371} \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {\left (a^2 A-2 a b B-A b^2\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(c+d x)\right )}{d (m+1)}+\frac {\left (a^2 B+2 a A b-b^2 B\right ) \tan ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\tan ^2(c+d x)\right )}{d (m+2)}+\frac {b (a B (m+3)+A b (m+2)) \tan ^{m+1}(c+d x)}{d (m+1) (m+2)}+\frac {b B \tan ^{m+1}(c+d x) (a+b \tan (c+d x))}{d (m+2)} \]
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Rule 371
Rule 3557
Rule 3619
Rule 3688
Rule 3711
Rubi steps \begin{align*} \text {integral}& = \frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))}{d (2+m)}+\frac {\int \tan ^m(c+d x) \left (-a (b B (1+m)-a A (2+m))+\left (2 a A b+a^2 B-b^2 B\right ) (2+m) \tan (c+d x)+b (A b (2+m)+a B (3+m)) \tan ^2(c+d x)\right ) \, dx}{2+m} \\ & = \frac {b (A b (2+m)+a B (3+m)) \tan ^{1+m}(c+d x)}{d (1+m) (2+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))}{d (2+m)}+\frac {\int \tan ^m(c+d x) \left (\left (a^2 A-A b^2-2 a b B\right ) (2+m)+\left (2 a A b+a^2 B-b^2 B\right ) (2+m) \tan (c+d x)\right ) \, dx}{2+m} \\ & = \frac {b (A b (2+m)+a B (3+m)) \tan ^{1+m}(c+d x)}{d (1+m) (2+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))}{d (2+m)}+\left (a^2 A-A b^2-2 a b B\right ) \int \tan ^m(c+d x) \, dx+\left (2 a A b+a^2 B-b^2 B\right ) \int \tan ^{1+m}(c+d x) \, dx \\ & = \frac {b (A b (2+m)+a B (3+m)) \tan ^{1+m}(c+d x)}{d (1+m) (2+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))}{d (2+m)}+\frac {\left (a^2 A-A b^2-2 a b B\right ) \text {Subst}\left (\int \frac {x^m}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (2 a A b+a^2 B-b^2 B\right ) \text {Subst}\left (\int \frac {x^{1+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {b (A b (2+m)+a B (3+m)) \tan ^{1+m}(c+d x)}{d (1+m) (2+m)}+\frac {\left (a^2 A-A b^2-2 a b 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 {\left (2 a A b+a^2 B-b^2 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 B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))}{d (2+m)} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.80 \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {\tan ^{1+m}(c+d x) \left (\frac {b (A b (2+m)+a B (3+m))}{1+m}+\frac {\left (a^2 A-A b^2-2 a b B\right ) (2+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right )}{1+m}+\left (2 a A b+a^2 B-b^2 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan (c+d x)+b B (a+b \tan (c+d x))\right )}{d (2+m)} \]
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\[\int \tan \left (d x +c \right )^{m} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (A +B \tan \left (d x +c \right )\right )d x\]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} \tan \left (d x + c\right )^{m} \,d x } \]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{2} \tan ^{m}{\left (c + d x \right )}\, dx \]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} \tan \left (d x + c\right )^{m} \,d x } \]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} \tan \left (d x + c\right )^{m} \,d x } \]
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Timed out. \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2 \,d x \]
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