Integrand size = 31, antiderivative size = 228 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=-\frac {A \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{a d}-\frac {(A-i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (i a+b) d (1+n)}+\frac {(A+i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (i a-b) d (1+n)}-\frac {(a B+A b n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^{1+n}}{a^2 d (1+n)} \]
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Time = 0.51 (sec) , antiderivative size = 228, 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, 3620, 3618, 70, 3715, 67} \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=-\frac {(a B+A b n) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b \tan (c+d x)}{a}+1\right )}{a^2 d (n+1)}-\frac {(A-i B) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (b+i a)}+\frac {(A+i B) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (-b+i a)}-\frac {A \cot (c+d x) (a+b \tan (c+d x))^{n+1}}{a d} \]
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Rule 67
Rule 70
Rule 3618
Rule 3620
Rule 3690
Rule 3715
Rule 3734
Rubi steps \begin{align*} \text {integral}& = -\frac {A \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{a d}-\frac {\int \cot (c+d x) (a+b \tan (c+d x))^n \left (-a B-A b n+a A \tan (c+d x)-A b n \tan ^2(c+d x)\right ) \, dx}{a} \\ & = -\frac {A \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{a d}-\frac {\int (a+b \tan (c+d x))^n (a A+a B \tan (c+d x)) \, dx}{a}+\frac {(a B+A b n) \int \cot (c+d x) (a+b \tan (c+d x))^n \left (1+\tan ^2(c+d x)\right ) \, dx}{a} \\ & = -\frac {A \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{a d}-\frac {1}{2} (A-i B) \int (1+i \tan (c+d x)) (a+b \tan (c+d x))^n \, dx-\frac {1}{2} (A+i B) \int (1-i \tan (c+d x)) (a+b \tan (c+d x))^n \, dx+\frac {(a B+A b n) \text {Subst}\left (\int \frac {(a+b x)^n}{x} \, dx,x,\tan (c+d x)\right )}{a d} \\ & = -\frac {A \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{a d}-\frac {(a B+A b n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^{1+n}}{a^2 d (1+n)}+\frac {(i A-B) \text {Subst}\left (\int \frac {(a+i b x)^n}{-1+x} \, dx,x,-i \tan (c+d x)\right )}{2 d}-\frac {(i A+B) \text {Subst}\left (\int \frac {(a-i b x)^n}{-1+x} \, dx,x,i \tan (c+d x)\right )}{2 d} \\ & = -\frac {A \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{a d}+\frac {(i A+B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (a-i b) d (1+n)}+\frac {(A+i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (i a-b) d (1+n)}-\frac {(a B+A b n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^{1+n}}{a^2 d (1+n)} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.89 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\frac {\left (a^2 (a+i b) (A-i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-i b}\right )-(a-i b) \left (a^2 (A+i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+i b}\right )+2 (-i a+b) \left (a B \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right )-A b \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right )\right )\right )\right ) (a+b \tan (c+d x))^{1+n}}{2 a^2 (a-i b) (-i a+b) d (1+n)} \]
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\[\int \cot \left (d x +c \right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )d x\]
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\[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2} \,d x } \]
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\[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{n} \cot ^{2}{\left (c + d x \right )}\, dx \]
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\[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2} \,d x } \]
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\[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^2\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]
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