Optimal. Leaf size=209 \[ \frac {i (2-n) n (d \tan (e+f x))^{n+2} \, _2F_1\left (1,\frac {n+2}{2};\frac {n+4}{2};-\tan ^2(e+f x)\right )}{4 a^2 d^2 f (n+2)}+\frac {(1-n)^2 (d \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(e+f x)\right )}{4 a^2 d f (n+1)}+\frac {(2-n) (d \tan (e+f x))^{n+1}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {(d \tan (e+f x))^{n+1}}{4 d f (a+i a \tan (e+f x))^2} \]
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Rubi [A] time = 0.36, 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 = {3559, 3596, 3538, 3476, 364} \[ \frac {i (2-n) n (d \tan (e+f x))^{n+2} \, _2F_1\left (1,\frac {n+2}{2};\frac {n+4}{2};-\tan ^2(e+f x)\right )}{4 a^2 d^2 f (n+2)}+\frac {(1-n)^2 (d \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(e+f x)\right )}{4 a^2 d f (n+1)}+\frac {(2-n) (d \tan (e+f x))^{n+1}}{4 a^2 d f (1+i \tan (e+f x))}+\frac {(d \tan (e+f x))^{n+1}}{4 d f (a+i a \tan (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3538
Rule 3559
Rule 3596
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 ) \operatorname {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) \operatorname {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 = 6.02, size = 0, normalized size = 0.00 \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n} {\left (e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{4 \, a^{2}}, x\right ) \]
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 \[ \int \frac {\left (d \tan \left (f x + e\right )\right )^{n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.98, 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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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 \[ - \frac {\int \frac {\left (d \tan {\left (e + f x \right )}\right )^{n}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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