Integrand size = 31, antiderivative size = 282 \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2 A-A b^2+2 a b B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{\left (a^2+b^2\right )^2 d (1+m)}+\frac {b \left (a^2 A b (2-m)-A b^3 m+a b^2 B (1+m)-a^3 (B-B m)\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b \tan (c+d x)}{a}\right ) \tan ^{1+m}(c+d x)}{a^2 \left (a^2+b^2\right )^2 d (1+m)}-\frac {\left (2 a A b-a^2 B+b^2 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{\left (a^2+b^2\right )^2 d (2+m)}+\frac {b (A b-a B) \tan ^{1+m}(c+d x)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]
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Time = 0.79 (sec) , antiderivative size = 282, 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 = {3690, 3734, 3619, 3557, 371, 3715, 66} \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2 A+2 a b B-A b^2\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(c+d x)\right )}{d (m+1) \left (a^2+b^2\right )^2}-\frac {\left (a^2 (-B)+2 a A b+b^2 B\right ) \tan ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\tan ^2(c+d x)\right )}{d (m+2) \left (a^2+b^2\right )^2}+\frac {b (A b-a B) \tan ^{m+1}(c+d x)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {b \left (-\left (a^3 (B-B m)\right )+a^2 A b (2-m)+a b^2 B (m+1)-A b^3 m\right ) \tan ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b \tan (c+d x)}{a}\right )}{a^2 d (m+1) \left (a^2+b^2\right )^2} \]
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Rule 66
Rule 371
Rule 3557
Rule 3619
Rule 3690
Rule 3715
Rule 3734
Rubi steps \begin{align*} \text {integral}& = \frac {b (A b-a B) \tan ^{1+m}(c+d x)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\tan ^m(c+d x) \left (a^2 A-A b^2 m+a b B (1+m)-a (A b-a B) \tan (c+d x)-b (A b-a B) m \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{a \left (a^2+b^2\right )} \\ & = \frac {b (A b-a B) \tan ^{1+m}(c+d x)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \tan ^m(c+d x) \left (a \left (a^2 A-A b^2+2 a b B\right )-a \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )^2}+\frac {\left (a^2 b (A b-a B)-a^2 b (A b-a B) m+b^2 \left (a^2 A-A b^2 m+a b B (1+m)\right )\right ) \int \frac {\tan ^m(c+d x) \left (1+\tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{a \left (a^2+b^2\right )^2} \\ & = \frac {b (A b-a B) \tan ^{1+m}(c+d x)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2 A-A b^2+2 a b B\right ) \int \tan ^m(c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (2 a A b-a^2 B+b^2 B\right ) \int \tan ^{1+m}(c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 b (A b-a B)-a^2 b (A b-a B) m+b^2 \left (a^2 A-A b^2 m+a b B (1+m)\right )\right ) \text {Subst}\left (\int \frac {x^m}{a+b x} \, dx,x,\tan (c+d x)\right )}{a \left (a^2+b^2\right )^2 d} \\ & = -\frac {b \left (a^3 B (1-m)-a^2 A b (2-m)+A b^3 m-a b^2 B (1+m)\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b \tan (c+d x)}{a}\right ) \tan ^{1+m}(c+d x)}{a^2 \left (a^2+b^2\right )^2 d (1+m)}+\frac {b (A b-a B) \tan ^{1+m}(c+d x)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\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 )}{\left (a^2+b^2\right )^2 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 )}{\left (a^2+b^2\right )^2 d} \\ & = \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)}{\left (a^2+b^2\right )^2 d (1+m)}-\frac {b \left (a^3 B (1-m)-a^2 A b (2-m)+A b^3 m-a b^2 B (1+m)\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b \tan (c+d x)}{a}\right ) \tan ^{1+m}(c+d x)}{a^2 \left (a^2+b^2\right )^2 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)}{\left (a^2+b^2\right )^2 d (2+m)}+\frac {b (A b-a B) \tan ^{1+m}(c+d x)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \\ \end{align*}
Time = 3.33 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.85 \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {\tan ^{1+m}(c+d x) \left (\frac {b \left (-a^2 A b (-2+m)+a^3 B (-1+m)-A b^3 m+a b^2 B (1+m)\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b \tan (c+d x)}{a}\right )}{a \left (a^2+b^2\right ) (1+m)}+\frac {b (A b-a B)}{a+b \tan (c+d x)}+\frac {a \left (\frac {\left (a^2 A-A b^2+2 a b B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\tan ^2(c+d x)\right )}{1+m}+\frac {\left (-2 a A b+a^2 B-b^2 B\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\tan ^2(c+d x)\right ) \tan (c+d x)}{2+m}\right )}{a^2+b^2}\right )}{a \left (a^2+b^2\right ) d} \]
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\[\int \frac {\tan \left (d x +c \right )^{m} \left (A +B \tan \left (d x +c \right )\right )}{\left (a +b \tan \left (d x +c \right )\right )^{2}}d x\]
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\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \tan ^{m}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \]
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