Optimal. Leaf size=209 \[ \frac {(1-n)^2 \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+n)}+\frac {(2-n) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {i (2-n) n \, _2F_1\left (1,\frac {2+n}{2};\frac {4+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}}{4 a^2 d^2 f (2+n)}+\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2} \]
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Rubi [A]
time = 0.26, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3640, 3677,
3619, 3557, 371} \begin {gather*} \frac {i (2-n) 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 )}{4 a^2 d^2 f (n+2)}+\frac {(1-n)^2 (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 )}{4 a^2 d f (n+1)}+\frac {(2-n) (d \tan (e+f x))^{n+1}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {(d \tan (e+f x))^{n+1}}{4 d f (a+i a \tan (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3619
Rule 3640
Rule 3677
Rubi steps
\begin {align*} \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx &=\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2}+\frac {\int \frac {(d \tan (e+f x))^n (a d (3-n)-i a d (1-n) \tan (e+f x))}{a+i a \tan (e+f x)} \, dx}{4 a^2 d}\\ &=\frac {(2-n) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2}+\frac {\int (d \tan (e+f x))^n \left (2 a^2 d^2 (1-n)^2+2 i a^2 d^2 (2-n) n \tan (e+f x)\right ) \, dx}{8 a^4 d^2}\\ &=\frac {(2-n) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2}+\frac {(1-n)^2 \int (d \tan (e+f x))^n \, dx}{4 a^2}+\frac {(i (2-n) n) \int (d \tan (e+f x))^{1+n} \, dx}{4 a^2 d}\\ &=\frac {(2-n) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2}+\frac {\left (d (1-n)^2\right ) \text {Subst}\left (\int \frac {x^n}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{4 a^2 f}+\frac {(i (2-n) n) \text {Subst}\left (\int \frac {x^{1+n}}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{4 a^2 f}\\ &=\frac {(1-n)^2 \, _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}}{4 a^2 d f (1+n)}+\frac {(2-n) (d \tan (e+f x))^{1+n}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {i (2-n) 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}}{4 a^2 d^2 f (2+n)}+\frac {(d \tan (e+f x))^{1+n}}{4 d f (a+i a \tan (e+f x))^2}\\ \end {align*}
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Mathematica [F]
time = 5.88, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 1.23, size = 0, normalized size = 0.00 \[\int \frac {\left (d \tan \left (f x +e \right )\right )^{n}}{\left (a +i a \tan \left (f x +e \right )\right )^{2}}\, dx\]
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 {\int \frac {\left (d \tan {\left (e + f x \right )}\right )^{n}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx}{a^{2}} \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 = -0.00 \begin {gather*} \int \frac {{\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 )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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