Optimal. Leaf size=193 \[ -\frac {F_1\left (1-n;-m,1;2-n;-\frac {b \cot (e+f x)}{a},-i \cot (e+f x)\right ) \cot (e+f x) (a+b \cot (e+f x))^m \left (1+\frac {b \cot (e+f x)}{a}\right )^{-m} (d \tan (e+f x))^n}{2 f (1-n)}-\frac {F_1\left (1-n;-m,1;2-n;-\frac {b \cot (e+f x)}{a},i \cot (e+f x)\right ) \cot (e+f x) (a+b \cot (e+f x))^m \left (1+\frac {b \cot (e+f x)}{a}\right )^{-m} (d \tan (e+f x))^n}{2 f (1-n)} \]
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Rubi [A]
time = 0.20, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4327, 3656,
926, 140, 138} \begin {gather*} -\frac {\cot (e+f x) (d \tan (e+f x))^n (a+b \cot (e+f x))^m \left (\frac {b \cot (e+f x)}{a}+1\right )^{-m} F_1\left (1-n;-m,1;2-n;-\frac {b \cot (e+f x)}{a},-i \cot (e+f x)\right )}{2 f (1-n)}-\frac {\cot (e+f x) (d \tan (e+f x))^n (a+b \cot (e+f x))^m \left (\frac {b \cot (e+f x)}{a}+1\right )^{-m} F_1\left (1-n;-m,1;2-n;-\frac {b \cot (e+f x)}{a},i \cot (e+f x)\right )}{2 f (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rule 926
Rule 3656
Rule 4327
Rubi steps
\begin {align*} \int (a+b \cot (e+f x))^m (d \tan (e+f x))^n \, dx &=\left ((d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \int (d \cot (e+f x))^{-n} (a+b \cot (e+f x))^m \, dx\\ &=-\frac {\left ((d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \text {Subst}\left (\int \frac {(d x)^{-n} (a+b x)^m}{1+x^2} \, dx,x,\cot (e+f x)\right )}{f}\\ &=-\frac {\left ((d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \text {Subst}\left (\int \left (\frac {i (d x)^{-n} (a+b x)^m}{2 (i-x)}+\frac {i (d x)^{-n} (a+b x)^m}{2 (i+x)}\right ) \, dx,x,\cot (e+f x)\right )}{f}\\ &=-\frac {\left (i (d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \text {Subst}\left (\int \frac {(d x)^{-n} (a+b x)^m}{i-x} \, dx,x,\cot (e+f x)\right )}{2 f}-\frac {\left (i (d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \text {Subst}\left (\int \frac {(d x)^{-n} (a+b x)^m}{i+x} \, dx,x,\cot (e+f x)\right )}{2 f}\\ &=-\frac {\left (i (d \cot (e+f x))^n (a+b \cot (e+f x))^m \left (1+\frac {b \cot (e+f x)}{a}\right )^{-m} (d \tan (e+f x))^n\right ) \text {Subst}\left (\int \frac {(d x)^{-n} \left (1+\frac {b x}{a}\right )^m}{i-x} \, dx,x,\cot (e+f x)\right )}{2 f}-\frac {\left (i (d \cot (e+f x))^n (a+b \cot (e+f x))^m \left (1+\frac {b \cot (e+f x)}{a}\right )^{-m} (d \tan (e+f x))^n\right ) \text {Subst}\left (\int \frac {(d x)^{-n} \left (1+\frac {b x}{a}\right )^m}{i+x} \, dx,x,\cot (e+f x)\right )}{2 f}\\ &=-\frac {F_1\left (1-n;-m,1;2-n;-\frac {b \cot (e+f x)}{a},-i \cot (e+f x)\right ) \cot (e+f x) (a+b \cot (e+f x))^m \left (1+\frac {b \cot (e+f x)}{a}\right )^{-m} (d \tan (e+f x))^n}{2 f (1-n)}-\frac {F_1\left (1-n;-m,1;2-n;-\frac {b \cot (e+f x)}{a},i \cot (e+f x)\right ) \cot (e+f x) (a+b \cot (e+f x))^m \left (1+\frac {b \cot (e+f x)}{a}\right )^{-m} (d \tan (e+f x))^n}{2 f (1-n)}\\ \end {align*}
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Mathematica [F]
time = 3.30, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b \cot (e+f x))^m (d \tan (e+f x))^n \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.77, size = 0, normalized size = 0.00 \[\int \left (a +b \cot \left (f x +e \right )\right )^{m} \left (d \tan \left (f x +e \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \tan {\left (e + f x \right )}\right )^{n} \left (a + b \cot {\left (e + f x \right )}\right )^{m}\, dx \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 {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,{\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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