Integrand size = 31, antiderivative size = 267 \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {b \left (3 a A b (3+m)-b^2 B (3+m)+2 a^2 B (4+m)\right ) \tan ^{1+m}(c+d x)}{d (1+m) (3+m)}+\frac {\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+m)}+\frac {b^2 (A b (3+m)+a B (5+m)) \tan ^{2+m}(c+d x)}{d (2+m) (3+m)}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (2+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m)} \]
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Time = 0.85 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3688, 3718, 3711, 3619, 3557, 371} \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {b \left (2 a^2 B (m+4)+3 a A b (m+3)-b^2 B (m+3)\right ) \tan ^{m+1}(c+d x)}{d (m+1) (m+3)}+\frac {\left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(c+d x)\right )}{d (m+1)}+\frac {\left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \tan ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\tan ^2(c+d x)\right )}{d (m+2)}+\frac {b^2 (a B (m+5)+A b (m+3)) \tan ^{m+2}(c+d x)}{d (m+2) (m+3)}+\frac {b B \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^2}{d (m+3)} \]
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Rule 371
Rule 3557
Rule 3619
Rule 3688
Rule 3711
Rule 3718
Rubi steps \begin{align*} \text {integral}& = \frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m)}+\frac {\int \tan ^m(c+d x) (a+b \tan (c+d x)) \left (-a (b B (1+m)-a A (3+m))+\left (2 a A b+a^2 B-b^2 B\right ) (3+m) \tan (c+d x)+b (A b (3+m)+a B (5+m)) \tan ^2(c+d x)\right ) \, dx}{3+m} \\ & = \frac {b^2 (A b (3+m)+a B (5+m)) \tan ^{2+m}(c+d x)}{d (2+m) (3+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m)}-\frac {\int \tan ^m(c+d x) \left (a^2 (2+m) (b B (1+m)-a A (3+m))-\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) (2+m) (3+m) \tan (c+d x)-b (2+m) \left (3 a A b (3+m)-b^2 B (3+m)+2 a^2 B (4+m)\right ) \tan ^2(c+d x)\right ) \, dx}{6+5 m+m^2} \\ & = \frac {b \left (3 a A b (3+m)-b^2 B (3+m)+2 a^2 B (4+m)\right ) \tan ^{1+m}(c+d x)}{d (1+m) (3+m)}+\frac {b^2 (A b (3+m)+a B (5+m)) \tan ^{2+m}(c+d x)}{d (2+m) (3+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m)}-\frac {\int \tan ^m(c+d x) \left (-\left (\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) (2+m) (3+m)\right )-\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) (2+m) (3+m) \tan (c+d x)\right ) \, dx}{6+5 m+m^2} \\ & = \frac {b \left (3 a A b (3+m)-b^2 B (3+m)+2 a^2 B (4+m)\right ) \tan ^{1+m}(c+d x)}{d (1+m) (3+m)}+\frac {b^2 (A b (3+m)+a B (5+m)) \tan ^{2+m}(c+d x)}{d (2+m) (3+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m)}+\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \int \tan ^{1+m}(c+d x) \, dx+\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \int \tan ^m(c+d x) \, dx \\ & = \frac {b \left (3 a A b (3+m)-b^2 B (3+m)+2 a^2 B (4+m)\right ) \tan ^{1+m}(c+d x)}{d (1+m) (3+m)}+\frac {b^2 (A b (3+m)+a B (5+m)) \tan ^{2+m}(c+d x)}{d (2+m) (3+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m)}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \text {Subst}\left (\int \frac {x^{1+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \text {Subst}\left (\int \frac {x^m}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {b \left (3 a A b (3+m)-b^2 B (3+m)+2 a^2 B (4+m)\right ) \tan ^{1+m}(c+d x)}{d (1+m) (3+m)}+\frac {\left (a^3 A-3 a A b^2-3 a^2 b B+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 (A b (3+m)+a B (5+m)) \tan ^{2+m}(c+d x)}{d (2+m) (3+m)}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a 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))^2}{d (3+m)} \\ \end{align*}
Time = 2.52 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.87 \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {\tan ^{1+m}(c+d x) \left (b (2+m) \left (3 a A b (3+m)-b^2 B (3+m)+2 a^2 B (4+m)\right )+\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) (2+m) (3+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right )+b^2 (1+m) (A b (3+m)+a B (5+m)) \tan (c+d x)+\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) (1+m) (3+m) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan (c+d x)+b B (1+m) (2+m) (a+b \tan (c+d x))^2\right )}{d (1+m) (2+m) (3+m)} \]
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\[\int \tan \left (d x +c \right )^{m} \left (a +b \tan \left (d x +c \right )\right )^{3} \left (A +B \tan \left (d x +c \right )\right )d x\]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \tan \left (d x + c\right )^{m} \,d x } \]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{3} \tan ^{m}{\left (c + d x \right )}\, dx \]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \tan \left (d x + c\right )^{m} \,d x } \]
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\[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \tan \left (d x + c\right )^{m} \,d x } \]
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Timed out. \[ \int \tan ^m(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]
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