Optimal. Leaf size=160 \[ \frac{\tan ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac{b (c \tan (e+f x))^n}{a}+1\right )^{-p} \text{Hypergeometric2F1}\left (\frac{3}{n},-p,\frac{n+3}{n},-\frac{b (c \tan (e+f x))^n}{a}\right )}{3 f}+\frac{\tan (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac{b (c \tan (e+f x))^n}{a}+1\right )^{-p} \text{Hypergeometric2F1}\left (\frac{1}{n},-p,\frac{1}{n}+1,-\frac{b (c \tan (e+f x))^n}{a}\right )}{f} \]
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Rubi [A] time = 0.130309, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3675, 1893, 246, 245, 365, 364} \[ \frac{\tan ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac{b (c \tan (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c \tan (e+f x))^n}{a}\right )}{3 f}+\frac{\tan (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac{b (c \tan (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 1893
Rule 246
Rule 245
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \sec ^4(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (c^2+x^2\right ) \left (a+b x^n\right )^p \, dx,x,c \tan (e+f x)\right )}{c^3 f}\\ &=\frac{\operatorname{Subst}\left (\int \left (c^2 \left (a+b x^n\right )^p+x^2 \left (a+b x^n\right )^p\right ) \, dx,x,c \tan (e+f x)\right )}{c^3 f}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \left (a+b x^n\right )^p \, dx,x,c \tan (e+f x)\right )}{c^3 f}+\frac{\operatorname{Subst}\left (\int \left (a+b x^n\right )^p \, dx,x,c \tan (e+f x)\right )}{c f}\\ &=\frac{\left (\left (a+b (c \tan (e+f x))^n\right )^p \left (1+\frac{b (c \tan (e+f x))^n}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int x^2 \left (1+\frac{b x^n}{a}\right )^p \, dx,x,c \tan (e+f x)\right )}{c^3 f}+\frac{\left (\left (a+b (c \tan (e+f x))^n\right )^p \left (1+\frac{b (c \tan (e+f x))^n}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \left (1+\frac{b x^n}{a}\right )^p \, dx,x,c \tan (e+f x)\right )}{c f}\\ &=\frac{\, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right ) \tan (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (1+\frac{b (c \tan (e+f x))^n}{a}\right )^{-p}}{f}+\frac{\, _2F_1\left (\frac{3}{n},-p;\frac{3+n}{n};-\frac{b (c \tan (e+f x))^n}{a}\right ) \tan ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (1+\frac{b (c \tan (e+f x))^n}{a}\right )^{-p}}{3 f}\\ \end{align*}
Mathematica [A] time = 1.61931, size = 122, normalized size = 0.76 \[ \frac{\tan (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac{b (c \tan (e+f x))^n}{a}+1\right )^{-p} \left (\tan ^2(e+f x) \text{Hypergeometric2F1}\left (\frac{3}{n},-p,\frac{n+3}{n},-\frac{b (c \tan (e+f x))^n}{a}\right )+3 \text{Hypergeometric2F1}\left (\frac{1}{n},-p,\frac{1}{n}+1,-\frac{b (c \tan (e+f x))^n}{a}\right )\right )}{3 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.378, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( fx+e \right ) \right ) ^{4} \left ( a+b \left ( c\tan \left ( fx+e \right ) \right ) ^{n} \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 (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \sec \left (f x + e\right )^{4}\,{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 (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \sec \left (f x + e\right )^{4}, 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 (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \sec \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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