Optimal. Leaf size=95 \[ \frac {F_1(1-n;1-m,1;2-n;-i \tan (e+f x),i \tan (e+f x)) (d \cot (e+f x))^n (1+i \tan (e+f x))^{-m} \tan (e+f x) (a+i a \tan (e+f x))^m}{f (1-n)} \]
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Rubi [A]
time = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4326, 3645,
140, 138} \begin {gather*} \frac {\tan (e+f x) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (d \cot (e+f x))^n F_1(1-n;1-m,1;2-n;-i \tan (e+f x),i \tan (e+f x))}{f (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rule 3645
Rule 4326
Rubi steps
\begin {align*} \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^m \, dx &=\left ((d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \int (d \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m \, dx\\ &=\frac {\left (i a^2 (d \cot (e+f x))^n (d \tan (e+f x))^n\right ) \text {Subst}\left (\int \frac {\left (-\frac {i d x}{a}\right )^{-n} (a+x)^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {\left (i a (d \cot (e+f x))^n (1+i \tan (e+f x))^{-m} (d \tan (e+f x))^n (a+i a \tan (e+f x))^m\right ) \text {Subst}\left (\int \frac {\left (-\frac {i d x}{a}\right )^{-n} \left (1+\frac {x}{a}\right )^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {F_1(1-n;1-m,1;2-n;-i \tan (e+f x),i \tan (e+f x)) (d \cot (e+f x))^n (1+i \tan (e+f x))^{-m} \tan (e+f x) (a+i a \tan (e+f x))^m}{f (1-n)}\\ \end {align*}
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Mathematica [F]
time = 5.62, size = 0, normalized size = 0.00 \begin {gather*} \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.68, size = 0, normalized size = 0.00 \[\int \left (d \cot \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )^{m}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \cot {\left (e + f x \right )}\right )^{n} \left (i a \left (\tan {\left (e + f x \right )} - i\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 {cot}\left (e+f\,x\right )\right )}^n\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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