Optimal. Leaf size=195 \[ \frac {(A+i B) F_1\left (1-m;-n,1;2-m;-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \cot ^{-1+m}(c+d x) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{2 d (1-m)}+\frac {(A-i B) F_1\left (1-m;-n,1;2-m;-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right ) \cot ^{-1+m}(c+d x) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{2 d (1-m)} \]
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Rubi [A]
time = 0.30, antiderivative size = 195, normalized size of antiderivative = 1.00, 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 = {4326, 3684,
3683, 140, 138} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rule 3683
Rule 3684
Rule 4326
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [F]
time = 4.85, size = 0, normalized size = 0.00 \begin {gather*} \int \cot ^m(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.52, size = 0, normalized size = 0.00 \[\int \left (\cot ^{m}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cot}\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 )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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