Optimal. Leaf size=127 \[ \frac{4 a^3 (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)}-\frac{a^3 (2 n+5) (d \tan (e+f x))^{n+1}}{d f (n+1) (n+2)}-\frac{\left (a^3+i a^3 \tan (e+f x)\right ) (d \tan (e+f x))^{n+1}}{d f (n+2)} \]
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Rubi [A] time = 0.259031, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3556, 3592, 3537, 12, 64} \[ \frac{4 a^3 (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)}-\frac{a^3 (2 n+5) (d \tan (e+f x))^{n+1}}{d f (n+1) (n+2)}-\frac{\left (a^3+i a^3 \tan (e+f x)\right ) (d \tan (e+f x))^{n+1}}{d f (n+2)} \]
Antiderivative was successfully verified.
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Rule 3556
Rule 3592
Rule 3537
Rule 12
Rule 64
Rubi steps
\begin{align*} \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^3 \, dx &=-\frac{(d \tan (e+f x))^{1+n} \left (a^3+i a^3 \tan (e+f x)\right )}{d f (2+n)}+\frac{a \int (d \tan (e+f x))^n (a+i a \tan (e+f x)) (a d (3+2 n)+i a d (5+2 n) \tan (e+f x)) \, dx}{d (2+n)}\\ &=-\frac{a^3 (5+2 n) (d \tan (e+f x))^{1+n}}{d f (1+n) (2+n)}-\frac{(d \tan (e+f x))^{1+n} \left (a^3+i a^3 \tan (e+f x)\right )}{d f (2+n)}+\frac{a \int (d \tan (e+f x))^n \left (4 a^2 d (2+n)+4 i a^2 d (2+n) \tan (e+f x)\right ) \, dx}{d (2+n)}\\ &=-\frac{a^3 (5+2 n) (d \tan (e+f x))^{1+n}}{d f (1+n) (2+n)}-\frac{(d \tan (e+f x))^{1+n} \left (a^3+i a^3 \tan (e+f x)\right )}{d f (2+n)}+\frac{\left (16 i a^5 d (2+n)\right ) \operatorname{Subst}\left (\int \frac{4^{-n} \left (-\frac{i x}{a^2 (2+n)}\right )^n}{-16 a^4 d^2 (2+n)^2+4 a^2 d (2+n) x} \, dx,x,4 i a^2 d (2+n) \tan (e+f x)\right )}{f}\\ &=-\frac{a^3 (5+2 n) (d \tan (e+f x))^{1+n}}{d f (1+n) (2+n)}-\frac{(d \tan (e+f x))^{1+n} \left (a^3+i a^3 \tan (e+f x)\right )}{d f (2+n)}+\frac{\left (i 4^{2-n} a^5 d (2+n)\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{i x}{a^2 (2+n)}\right )^n}{-16 a^4 d^2 (2+n)^2+4 a^2 d (2+n) x} \, dx,x,4 i a^2 d (2+n) \tan (e+f x)\right )}{f}\\ &=-\frac{a^3 (5+2 n) (d \tan (e+f x))^{1+n}}{d f (1+n) (2+n)}+\frac{4 a^3 \, _2F_1(1,1+n;2+n;i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)}-\frac{(d \tan (e+f x))^{1+n} \left (a^3+i a^3 \tan (e+f x)\right )}{d f (2+n)}\\ \end{align*}
Mathematica [A] time = 3.3599, size = 214, normalized size = 1.69 \[ \frac{e^{-3 i e} 2^{-n} \left (-\frac{i \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}\right )^{n+1} \cos ^3(e+f x) (a+i a \tan (e+f x))^3 \left (2 (n+2) \left (1+e^{2 i (e+f x)}\right )^{n+2} \, _2F_1\left (n+1,n+1;n+2;\frac{1}{2} \left (1-e^{2 i (e+f x)}\right )\right )-2^n \left ((4 n+7) e^{2 i (e+f x)}+2 n+5\right )\right ) \tan ^{-n}(e+f x) (d \tan (e+f x))^n}{f (n+1) (n+2) \left (1+e^{2 i (e+f x)}\right ) (\cos (f x)+i \sin (f x))^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.24, size = 0, normalized size = 0. \begin{align*} \int \left ( d\tan \left ( fx+e \right ) \right ) ^{n} \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{8 \, a^{3} \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 (6 i \, f x + 6 i \, e\right )}}{e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int \left (d \tan{\left (e + f x \right )}\right )^{n}\, dx + \int - 3 \left (d \tan{\left (e + f x \right )}\right )^{n} \tan ^{2}{\left (e + f x \right )}\, dx + \int 3 i \left (d \tan{\left (e + f x \right )}\right )^{n} \tan{\left (e + f x \right )}\, dx + \int - i \left (d \tan{\left (e + f x \right )}\right )^{n} \tan ^{3}{\left (e + f x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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