Optimal. Leaf size=86 \[ \frac{\left (a+b \sec ^2(e+f x)\right )^{p+1} \text{Hypergeometric2F1}\left (1,p+1,p+2,\frac{b \sec ^2(e+f x)}{a}+1\right )}{2 a f (p+1)}+\frac{\left (a+b \sec ^2(e+f x)\right )^{p+1}}{2 b f (p+1)} \]
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Rubi [A] time = 0.0906664, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {4139, 446, 80, 65} \[ \frac{\left (a+b \sec ^2(e+f x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sec ^2(e+f x)}{a}+1\right )}{2 a f (p+1)}+\frac{\left (a+b \sec ^2(e+f x)\right )^{p+1}}{2 b f (p+1)} \]
Antiderivative was successfully verified.
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Rule 4139
Rule 446
Rule 80
Rule 65
Rubi steps
\begin{align*} \int \left (a+b \sec ^2(e+f x)\right )^p \tan ^3(e+f x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right ) \left (a+b x^2\right )^p}{x} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-1+x) (a+b x)^p}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{\left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 b f (1+p)}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^p}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{\left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 b f (1+p)}+\frac{\, _2F_1\left (1,1+p;2+p;1+\frac{b \sec ^2(e+f x)}{a}\right ) \left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 a f (1+p)}\\ \end{align*}
Mathematica [A] time = 0.166738, size = 61, normalized size = 0.71 \[ \frac{\left (a+b \sec ^2(e+f x)\right )^{p+1} \left (b \text{Hypergeometric2F1}\left (1,p+1,p+2,\frac{b \sec ^2(e+f x)}{a}+1\right )+a\right )}{2 a b f (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.396, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( \sec \left ( fx+e \right ) \right ) ^{2} \right ) ^{p} \left ( \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 (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{3}\,{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 \sec \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{3}, 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 \sec \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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