Integrand size = 31, antiderivative size = 292 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=-\frac {(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac {A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}-\frac {(i A+B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (i a+b) d (1+n)}-\frac {(A+i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (a+i b) d (1+n)}+\frac {\left (2 a^2 A-2 a b B n+A b^2 (1-n) n\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^{1+n}}{2 a^3 d (1+n)} \]
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Time = 0.90 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {3690, 3730, 3734, 3620, 3618, 70, 3715, 67} \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=-\frac {\cot (c+d x) (2 a B-A b (1-n)) (a+b \tan (c+d x))^{n+1}}{2 a^2 d}+\frac {\left (2 a^2 A-2 a b B n+A b^2 (1-n) n\right ) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b \tan (c+d x)}{a}+1\right )}{2 a^3 d (n+1)}-\frac {(B+i A) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (b+i a)}-\frac {(A+i B) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (a+i b)}-\frac {A \cot ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{2 a d} \]
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Rule 67
Rule 70
Rule 3618
Rule 3620
Rule 3690
Rule 3715
Rule 3730
Rule 3734
Rubi steps \begin{align*} \text {integral}& = -\frac {A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}-\frac {\int \cot ^2(c+d x) (a+b \tan (c+d x))^n \left (-2 a B+A (b-b n)+2 a A \tan (c+d x)+A b (1-n) \tan ^2(c+d x)\right ) \, dx}{2 a} \\ & = -\frac {(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac {A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}+\frac {\int \cot (c+d x) (a+b \tan (c+d x))^n \left (-2 a^2 A+2 a b B n-A b^2 (1-n) n-2 a^2 B \tan (c+d x)+b n (2 a B-A (b-b n)) \tan ^2(c+d x)\right ) \, dx}{2 a^2} \\ & = -\frac {(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac {A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}+\frac {\int \left (-2 a^2 B+2 a^2 A \tan (c+d x)\right ) (a+b \tan (c+d x))^n \, dx}{2 a^2}+\frac {\left (-2 a^2 A+2 a b B n-A b^2 (1-n) n\right ) \int \cot (c+d x) (a+b \tan (c+d x))^n \left (1+\tan ^2(c+d x)\right ) \, dx}{2 a^2} \\ & = -\frac {(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac {A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}+\frac {1}{2} (-i A-B) \int (1+i \tan (c+d x)) (a+b \tan (c+d x))^n \, dx+\frac {1}{2} (i A-B) \int (1-i \tan (c+d x)) (a+b \tan (c+d x))^n \, dx-\frac {\left (2 a^2 A-2 a b B n+A b^2 (1-n) n\right ) \text {Subst}\left (\int \frac {(a+b x)^n}{x} \, dx,x,\tan (c+d x)\right )}{2 a^2 d} \\ & = -\frac {(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac {A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}+\frac {\left (2 a^2 A-2 a b B n+A b^2 (1-n) n\right ) \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}}{2 a^3 d (1+n)}+\frac {(A-i 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+i B) \text {Subst}\left (\int \frac {(a+i b x)^n}{-1+x} \, dx,x,-i \tan (c+d x)\right )}{2 d} \\ & = -\frac {(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac {A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 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 (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 (a+i b) d (1+n)}+\frac {\left (2 a^2 A-2 a b B n+A b^2 (1-n) n\right ) \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}}{2 a^3 d (1+n)} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.79 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=-\frac {\left (a^3 (a+i b) (A-i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-i b}\right )+(a-i b) \left (a^3 (A+i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+i b}\right )-2 (a+i b) \left (a^2 A \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right )+b \left (a B \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right )-A b \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right )\right )\right )\right )\right ) (a+b \tan (c+d x))^{1+n}}{2 a^3 (a-i b) (a+i b) d (1+n)} \]
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\[\int \cot \left (d x +c \right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )d x\]
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\[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3} \,d x } \]
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\[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{n} \cot ^{3}{\left (c + d x \right )}\, dx \]
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\[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3} \,d x } \]
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\[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^3\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]
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