Optimal. Leaf size=195 \[ \frac{(A+i B) \cot ^{m-1}(c+d x) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (1-m;-n,1;2-m;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 d (1-m)}+\frac{(A-i B) \cot ^{m-1}(c+d x) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (1-m;-n,1;2-m;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 d (1-m)} \]
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Rubi [A] time = 0.440167, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {4241, 3603, 3602, 135, 133} \[ \frac{(A+i B) \cot ^{m-1}(c+d x) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (1-m;-n,1;2-m;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 d (1-m)}+\frac{(A-i B) \cot ^{m-1}(c+d x) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (1-m;-n,1;2-m;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 d (1-m)} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3603
Rule 3602
Rule 135
Rule 133
Rubi steps
\begin{align*} \int \cot ^m(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx &=\left (\cot ^m(c+d x) \tan ^m(c+d x)\right ) \int \tan ^{-m}(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx\\ &=\frac{1}{2} \left ((A-i B) \cot ^m(c+d x) \tan ^m(c+d x)\right ) \int (1+i \tan (c+d x)) \tan ^{-m}(c+d x) (a+b \tan (c+d x))^n \, dx+\frac{1}{2} \left ((A+i B) \cot ^m(c+d x) \tan ^m(c+d x)\right ) \int (1-i \tan (c+d x)) \tan ^{-m}(c+d x) (a+b \tan (c+d x))^n \, dx\\ &=\frac{\left ((A-i B) \cot ^m(c+d x) \tan ^m(c+d x)\right ) \operatorname{Subst}\left (\int \frac{x^{-m} (a+b x)^n}{1-i x} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\left ((A+i B) \cot ^m(c+d x) \tan ^m(c+d x)\right ) \operatorname{Subst}\left (\int \frac{x^{-m} (a+b x)^n}{1+i x} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{\left ((A-i B) \cot ^m(c+d x) \tan ^m(c+d x) (a+b \tan (c+d x))^n \left (1+\frac{b \tan (c+d x)}{a}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{x^{-m} \left (1+\frac{b x}{a}\right )^n}{1-i x} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\left ((A+i B) \cot ^m(c+d x) \tan ^m(c+d x) (a+b \tan (c+d x))^n \left (1+\frac{b \tan (c+d x)}{a}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{x^{-m} \left (1+\frac{b x}{a}\right )^n}{1+i x} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{(A+i B) F_1\left (1-m;-n,1;2-m;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \cot ^{-1+m}(c+d x) (a+b \tan (c+d x))^n \left (1+\frac{b \tan (c+d x)}{a}\right )^{-n}}{2 d (1-m)}+\frac{(A-i B) F_1\left (1-m;-n,1;2-m;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right ) \cot ^{-1+m}(c+d x) (a+b \tan (c+d x))^n \left (1+\frac{b \tan (c+d x)}{a}\right )^{-n}}{2 d (1-m)}\\ \end{align*}
Mathematica [F] time = 6.2417, size = 0, normalized size = 0. \[ \int \cot ^m(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.397, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{m} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{n} \left ( A+B\tan \left ( dx+c \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\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 )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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