Optimal. Leaf size=61 \[ \frac {a \, _2F_1\left (1,1+n;2+n;\frac {c+d \tan (e+f x)}{c-i d}\right ) (c+d \tan (e+f x))^{1+n}}{(i c+d) f (1+n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3618, 70}
\begin {gather*} \frac {a (c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c+d \tan (e+f x)}{c-i d}\right )}{f (n+1) (d+i c)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 3618
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^n \, dx &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {\left (c-\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\left (1,1+n;2+n;\frac {c+d \tan (e+f x)}{c-i d}\right ) (c+d \tan (e+f x))^{1+n}}{(i c+d) f (1+n)}\\ \end {align*}
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Mathematica [F]
time = 1.75, size = 0, normalized size = 0.00 \begin {gather*} \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^n \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.51, size = 0, normalized size = 0.00 \[\int \left (a +i a \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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 (c + d \tan {\left (e + f x \right )}\right )^{n}\right )\, dx + \int \left (c + 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 (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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