Optimal. Leaf size=115 \[ \frac {(i A (1-n)+B (1+n)) \, _2F_1\left (1,n;1+n;\frac {1}{2} (1-i \tan (e+f x))\right ) (c-i c \tan (e+f x))^n}{4 a f n}+\frac {(i A-B) (c-i c \tan (e+f x))^n}{2 a f (1+i \tan (e+f x))} \]
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Rubi [A]
time = 0.13, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {3669, 79, 70}
\begin {gather*} \frac {(B (n+1)+i A (1-n)) (c-i c \tan (e+f x))^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (1-i \tan (e+f x))\right )}{4 a f n}+\frac {(-B+i A) (c-i c \tan (e+f x))^n}{2 a f (1+i \tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 79
Rule 3669
Rubi steps
\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^n}{a+i a \tan (e+f x)} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(A+B x) (c-i c x)^{-1+n}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(i A-B) (c-i c \tan (e+f x))^n}{2 a f (1+i \tan (e+f x))}+\frac {(c (A (1-n)-i B (1+n))) \text {Subst}\left (\int \frac {(c-i c x)^{-1+n}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac {(i A (1-n)+B (1+n)) \, _2F_1\left (1,n;1+n;\frac {1}{2} (1-i \tan (e+f x))\right ) (c-i c \tan (e+f x))^n}{4 a f n}+\frac {(i A-B) (c-i c \tan (e+f x))^n}{2 a f (1+i \tan (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 9.53, size = 111, normalized size = 0.97 \begin {gather*} \frac {2^{-1+n} \left (\frac {c}{1+e^{2 i (e+f x)}}\right )^n \left ((A+i B) (-1+n)+e^{2 i (e+f x)} (A (-1+n)+i B (1+n)) \, _2F_1\left (1,1-n;2-n;1+e^{2 i (e+f x)}\right )\right )}{a f (-1+n) (-i+\tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.08, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \tan \left (f x +e \right )\right ) \left (c -i c \tan \left (f x +e \right )\right )^{n}}{a +i a \tan \left (f x +e \right )}\, dx\]
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*} - \frac {i \left (\int \frac {A \left (- i c \tan {\left (e + f x \right )} + c\right )^{n}}{\tan {\left (e + f x \right )} - i}\, dx + \int \frac {B \left (- i c \tan {\left (e + f x \right )} + c\right )^{n} \tan {\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\, dx\right )}{a} \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 \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^n}{a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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