Optimal. Leaf size=75 \[ -\frac {a^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac {2 a^2 (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)} \]
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Rubi [A] time = 0.11, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3543, 3537, 12, 64} \[ -\frac {a^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac {2 a^2 (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 64
Rule 3537
Rule 3543
Rubi steps
\begin {align*} \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^2 \, dx &=-\frac {a^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\int (d \tan (e+f x))^n \left (2 a^2+2 i a^2 \tan (e+f x)\right ) \, dx\\ &=-\frac {a^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {\left (4 i a^4\right ) \operatorname {Subst}\left (\int \frac {2^{-n} \left (-\frac {i d x}{a^2}\right )^n}{-4 a^4+2 a^2 x} \, dx,x,2 i a^2 \tan (e+f x)\right )}{f}\\ &=-\frac {a^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {\left (i 2^{2-n} a^4\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {i d x}{a^2}\right )^n}{-4 a^4+2 a^2 x} \, dx,x,2 i a^2 \tan (e+f x)\right )}{f}\\ &=-\frac {a^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {2 a^2 \, _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] time = 2.11, size = 168, normalized size = 2.24 \[ \frac {e^{-2 i e} 2^{-n} \left (-\frac {i \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}\right )^{n+1} \cos ^2(e+f x) (a+i a \tan (e+f x))^2 \left (-2^n+\left (1+e^{2 i (e+f x)}\right )^{n+1} \, _2F_1\left (n+1,n+1;n+2;\frac {1}{2} \left (1-e^{2 i (e+f x)}\right )\right )\right ) \tan ^{-n}(e+f x) (d \tan (e+f x))^n}{f (n+1) (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {4 \, a^{2} \left (\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n} e^{\left (4 i \, f x + 4 i \, e\right )}}{e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.48, size = 0, normalized size = 0.00 \[ \int \left (d \tan \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )^{2}\, dx \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \]
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 \[ \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]
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 \[ - a^{2} \left (\int \left (- \left (d \tan {\left (e + f x \right )}\right )^{n}\right )\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{n} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \left (d \tan {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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