Optimal. Leaf size=54 \[ \frac{a \tan (e+f x) \, _2F_1(1,n p+1;n p+2;i \tan (e+f x)) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)} \]
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Rubi [A] time = 0.104539, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {12, 6677, 64} \[ \frac{a \tan (e+f x) \, _2F_1(1,n p+1;n p+2;i \tan (e+f x)) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 6677
Rule 64
Rubi steps
\begin{align*} \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx &=\frac{i \operatorname{Subst}\left (\int \frac{a \left (c (d x)^p\right )^n}{i+x} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i a) \operatorname{Subst}\left (\int \frac{\left (c (d x)^p\right )^n}{i+x} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\left (i a (d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \operatorname{Subst}\left (\int \frac{(d x)^{n p}}{i+x} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a \, _2F_1(1,1+n p;2+n p;i \tan (e+f x)) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}\\ \end{align*}
Mathematica [B] time = 0.946716, size = 173, normalized size = 3.2 \[ \frac{a e^{-i e} 2^{-n p-1} \cos (e+f x) (1+i \tan (e+f x)) (\cos (f x)-i \sin (f x)) \left (-\frac{i \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}\right )^{n p+1} \left (1+e^{2 i (e+f x)}\right )^{n p+1} \, _2F_1\left (n p+1,n p+1;n p+2;\frac{1}{2} \left (1-e^{2 i (e+f x)}\right )\right ) \tan ^{-n p}(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f n p+f} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.853, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( d\tan \left ( fx+e \right ) \right ) ^{p} \right ) ^{n} \left ( a+ia\tan \left ( fx+e \right ) \right ) \, 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 )} \left (\left (d \tan \left (f x + e\right )\right )^{p} c\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{2 \, \left (c \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 )^{p}\right )^{n} a e^{\left (2 i \, f x + 2 i \, e\right )}}{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 \left (\int \left (c \left (d \tan{\left (e + f x \right )}\right )^{p}\right )^{n}\, dx + \int i \left (c \left (d \tan{\left (e + f x \right )}\right )^{p}\right )^{n} \tan{\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 )} \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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