Integrand size = 45, antiderivative size = 403 \[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\frac {(A-i B-C) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) (c-i d)^2 f (1+m)}+\frac {(i A-B-i C) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) (c+i d)^2 f (1+m)}-\frac {\left (a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b \left (A d^2 \left (c^2 (2-m)-d^2 m\right )-B c d \left (c^2 (1-m)-d^2 (1+m)\right )-c^2 C \left (c^2 m+d^2 (2+m)\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (1+m)}+\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))} \]
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Time = 1.32 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3730, 3734, 3620, 3618, 70, 3715} \[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\frac {\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {(a+b \tan (e+f x))^{m+1} \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b \left (A c^2 d^2 (2-m)-A d^4 m-B \left (c^3 d (1-m)-c d^3 (m+1)\right )+c^4 (-C) m-c^2 C d^2 (m+2)\right )\right ) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right )}{f (m+1) \left (c^2+d^2\right )^2 (b c-a d)^2}+\frac {(A-i B-C) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a) (c-i d)^2}+\frac {(i A-B-i C) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b) (c+i d)^2} \]
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Rule 70
Rule 3618
Rule 3620
Rule 3715
Rule 3730
Rule 3734
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac {\int \frac {(a+b \tan (e+f x))^m \left ((c C-B d) (a d-b c (1+m))-A \left (a c d-b \left (c^2-d^2 m\right )\right )+(b c-a d) (B c-(A-C) d) \tan (e+f x)-b \left (c^2 C-B c d+A d^2\right ) m \tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{(b c-a d) \left (c^2+d^2\right )} \\ & = \frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac {\int (a+b \tan (e+f x))^m \left (-\left ((b c-a d) \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )\right )-(b c-a d) \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) \tan (e+f x)\right ) \, dx}{(b c-a d) \left (c^2+d^2\right )^2}+\frac {\left (-c d (b c-a d) (B c-(A-C) d)-b c^2 \left (c^2 C-B c d+A d^2\right ) m+d^2 \left ((c C-B d) (a d-b c (1+m))-A \left (a c d-b \left (c^2-d^2 m\right )\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (1+\tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{(b c-a d) \left (c^2+d^2\right )^2} \\ & = \frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac {(A-i B-C) \int (1+i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx}{2 (c-i d)^2}+\frac {(A+i B-C) \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx}{2 (c+i d)^2}+\frac {\left (-c d (b c-a d) (B c-(A-C) d)-b c^2 \left (c^2 C-B c d+A d^2\right ) m+d^2 \left ((c C-B d) (a d-b c (1+m))-A \left (a c d-b \left (c^2-d^2 m\right )\right )\right )\right ) \text {Subst}\left (\int \frac {(a+b x)^m}{c+d x} \, dx,x,\tan (e+f x)\right )}{(b c-a d) \left (c^2+d^2\right )^2 f} \\ & = -\frac {\left (a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b \left (A c^2 d^2 (2-m)-c^4 C m-A d^4 m-c^2 C d^2 (2+m)-B \left (c^3 d (1-m)-c d^3 (1+m)\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (1+m)}+\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac {(i A+B-i C) \text {Subst}\left (\int \frac {(a-i b x)^m}{-1+x} \, dx,x,i \tan (e+f x)\right )}{2 (c-i d)^2 f}-\frac {(i (A+i B-C)) \text {Subst}\left (\int \frac {(a+i b x)^m}{-1+x} \, dx,x,-i \tan (e+f x)\right )}{2 (c+i d)^2 f} \\ & = -\frac {(i A+B-i C) \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) (c-i d)^2 f (1+m)}-\frac {(A+i B-C) \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) (c+i d)^2 f (1+m)}-\frac {\left (a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b \left (A c^2 d^2 (2-m)-c^4 C m-A d^4 m-c^2 C d^2 (2+m)-B \left (c^3 d (1-m)-c d^3 (1+m)\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (1+m)}+\frac {\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))} \\ \end{align*}
Time = 6.24 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=-\frac {\left (A d^2-c (-c C+B d)\right ) (a+b \tan (e+f x))^{1+m}}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {-\frac {\left (-c d (b c-a d) (B c-(A-C) d)-b c^2 \left (c^2 C-B c d+A d^2\right ) m+d^2 \left ((c C-B d) (a d-b c (1+m))-A \left (a c d-b \left (c^2-d^2 m\right )\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (a+b \tan (e+f x))}{-b c+a d}\right ) (a+b \tan (e+f x))^{1+m}}{(-b c+a d) \left (c^2+d^2\right ) f (1+m)}+\frac {\frac {i \left (-\left ((b c-a d) \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )\right )-i (b c-a d) \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {-i a-i b \tan (e+f x)}{-i a+b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) f (1+m)}-\frac {i \left (-\left ((b c-a d) \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )\right )+i (b c-a d) \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {i a+i b \tan (e+f x)}{-i a-b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a-i b) f (1+m)}}{c^2+d^2}}{(-b c+a d) \left (c^2+d^2\right )} \]
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\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{m} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (c +d \tan \left (f x +e \right )\right )^{2}}d x\]
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\[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\int { \frac {{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\int { \frac {{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{2}} \,d x } \]
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\[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\int { \frac {{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right )}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]
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