Optimal. Leaf size=63 \[ -\frac{\left (a+b \tan ^2(e+f x)\right )^{p+1} \text{Hypergeometric2F1}\left (1,p+1,p+2,\frac{a+b \tan ^2(e+f x)}{a-b}\right )}{2 f (p+1) (a-b)} \]
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Rubi [A] time = 0.0669837, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3670, 444, 68} \[ -\frac{\left (a+b \tan ^2(e+f x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right )}{2 f (p+1) (a-b)} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 444
Rule 68
Rubi steps
\begin{align*} \int \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x \left (a+b x^2\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^p}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac{\, _2F_1\left (1,1+p;2+p;\frac{a+b \tan ^2(e+f x)}{a-b}\right ) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 (a-b) f (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0808765, size = 63, normalized size = 1. \[ -\frac{\left (a+b \tan ^2(e+f x)\right )^{p+1} \text{Hypergeometric2F1}\left (1,p+1,p+2,\frac{a+b \tan ^2(e+f x)}{a-b}\right )}{2 f (p+1) (a-b)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.294, size = 0, normalized size = 0. \begin{align*} \int \tan \left ( fx+e \right ) \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{p}\, 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 (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )\,{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 ({\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right ), 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*} \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{p} \tan{\left (e + f x \right )}\, dx \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 (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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