Optimal. Leaf size=43 \[ \frac {a \, _2F_1(1,1+n;2+n;i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3618, 66}
\begin {gather*} \frac {a (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 3618
Rubi steps
\begin {align*} \int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {\left (-\frac {i d x}{a}\right )^n}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {a \, _2F_1(1,1+n;2+n;i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(159\) vs. \(2(43)=86\).
time = 0.81, size = 159, normalized size = 3.70 \begin {gather*} \frac {2^{-1-n} a e^{-i e} \left (-\frac {i \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}\right )^{1+n} \left (1+e^{2 i (e+f x)}\right )^{1+n} \cos (e+f x) \, _2F_1\left (1+n,1+n;2+n;\frac {1}{2} \left (1-e^{2 i (e+f x)}\right )\right ) (1+i \tan (e+f x)) \tan ^{-n}(e+f x) (d \tan (e+f x))^n}{f (1+n) (\cos (f x)+i \sin (f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.38, size = 0, normalized size = 0.00 \[\int \left (d \tan \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} i a \left (\int \left (- i \left (d \tan {\left (e + f x \right )}\right )^{n}\right )\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\, dx\right ) \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 = -0.02 \begin {gather*} \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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