Optimal. Leaf size=129 \[ -\frac{\left (a+b \tan ^2(e+f x)\right )^{p+1} \text{Hypergeometric2F1}\left (1,p+1,p+2,\frac{a+b \tan ^2(e+f x)}{a-b}\right )}{2 f (p+1) (a-b)}-\frac{(a+b) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{2 b^2 f (p+1)}+\frac{\left (a+b \tan ^2(e+f x)\right )^{p+2}}{2 b^2 f (p+2)} \]
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Rubi [A] time = 0.171291, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3670, 446, 88, 68} \[ -\frac{(a+b) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{2 b^2 f (p+1)}+\frac{\left (a+b \tan ^2(e+f x)\right )^{p+2}}{2 b^2 f (p+2)}-\frac{\left (a+b \tan ^2(e+f x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right )}{2 f (p+1) (a-b)} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 88
Rule 68
Rubi steps
\begin{align*} \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5 \left (a+b x^2\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2 (a+b x)^p}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{(-a-b) (a+b x)^p}{b}+\frac{(a+b x)^p}{1+x}+\frac{(a+b x)^{1+p}}{b}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac{(a+b) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 b^2 f (1+p)}+\frac{\left (a+b \tan ^2(e+f x)\right )^{2+p}}{2 b^2 f (2+p)}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^p}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac{(a+b) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 b^2 f (1+p)}-\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)}+\frac{\left (a+b \tan ^2(e+f x)\right )^{2+p}}{2 b^2 f (2+p)}\\ \end{align*}
Mathematica [A] time = 0.735409, size = 106, normalized size = 0.82 \[ \frac{\left (a+b \tan ^2(e+f x)\right )^{p+1} \left (b^2 (p+2) \text{Hypergeometric2F1}\left (1,p+1,p+2,\frac{a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \left (a-b (p+1) \tan ^2(e+f x)+b (p+2)\right )\right )}{2 b^2 f (p+1) (p+2) (b-a)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.398, size = 0, normalized size = 0. \begin{align*} \int \left ( \tan \left ( fx+e \right ) \right ) ^{5} \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 )^{5}\,{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 )^{5}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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