Optimal. Leaf size=158 \[ \frac {(1-n) \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{2 a d f (1+n)}-\frac {i n \, _2F_1\left (1,\frac {2+n}{2};\frac {4+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}}{2 a d^2 f (2+n)}+\frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))} \]
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Rubi [A]
time = 0.12, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3633, 3619,
3557, 371} \begin {gather*} -\frac {i n (d \tan (e+f x))^{n+2} \, _2F_1\left (1,\frac {n+2}{2};\frac {n+4}{2};-\tan ^2(e+f x)\right )}{2 a d^2 f (n+2)}+\frac {(1-n) (d \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(e+f x)\right )}{2 a d f (n+1)}+\frac {(d \tan (e+f x))^{n+1}}{2 d f (a-i a \tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3619
Rule 3633
Rubi steps
\begin {align*} \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx &=\frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))}-\frac {\int (d \tan (e+f x))^n (-a d (1-n)+i a d n \tan (e+f x)) \, dx}{2 a^2 d}\\ &=\frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))}+\frac {(1-n) \int (d \tan (e+f x))^n \, dx}{2 a}-\frac {(i n) \int (d \tan (e+f x))^{1+n} \, dx}{2 a d}\\ &=\frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))}+\frac {(d (1-n)) \text {Subst}\left (\int \frac {x^n}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{2 a f}-\frac {(i n) \text {Subst}\left (\int \frac {x^{1+n}}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{2 a f}\\ &=\frac {(1-n) \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{2 a d f (1+n)}-\frac {i n \, _2F_1\left (1,\frac {2+n}{2};\frac {4+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}}{2 a d^2 f (2+n)}+\frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 123, normalized size = 0.78 \begin {gather*} \frac {\tan (e+f x) (d \tan (e+f x))^n \left (-\frac {(-1+n) \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\tan ^2(e+f x)\right )}{a (1+n)}-\frac {i n \, _2F_1\left (1,\frac {2+n}{2};\frac {4+n}{2};-\tan ^2(e+f x)\right ) \tan (e+f x)}{a (2+n)}+\frac {1}{a-i a \tan (e+f x)}\right )}{2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.62, size = 0, normalized size = 0.00 \[\int \frac {\left (d \tan \left (f x +e \right )\right )^{n}}{a -i a \tan \left (f x +e \right )}\, dx\]
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 \int \frac {\left (d \tan {\left (e + f x \right )}\right )^{n}}{\tan {\left (e + f x \right )} + i}\, dx}{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 (d\,\mathrm {tan}\left (e+f\,x\right )\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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