Optimal. Leaf size=244 \[ \frac {\tan ^5(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac {b (c \tan (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {5}{n},-p;\frac {n+5}{n};-\frac {b (c \tan (e+f x))^n}{a}\right )}{5 f}+\frac {2 \tan ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac {b (c \tan (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{n},-p;\frac {n+3}{n};-\frac {b (c \tan (e+f x))^n}{a}\right )}{3 f}+\frac {\tan (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac {b (c \tan (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b (c \tan (e+f x))^n}{a}\right )}{f} \]
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Rubi [A] time = 0.19, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3675, 1893, 246, 245, 365, 364} \[ \frac {\tan ^5(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac {b (c \tan (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {5}{n},-p;\frac {n+5}{n};-\frac {b (c \tan (e+f x))^n}{a}\right )}{5 f}+\frac {2 \tan ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac {b (c \tan (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{n},-p;\frac {n+3}{n};-\frac {b (c \tan (e+f x))^n}{a}\right )}{3 f}+\frac {\tan (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac {b (c \tan (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b (c \tan (e+f x))^n}{a}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 364
Rule 365
Rule 1893
Rule 3675
Rubi steps
\begin {align*} \int \sec ^6(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 )^2 \left (a+b x^n\right )^p \, dx,x,c \tan (e+f x)\right )}{c^5 f}\\ &=\frac {\operatorname {Subst}\left (\int \left (c^4 \left (a+b x^n\right )^p+2 c^2 x^2 \left (a+b x^n\right )^p+x^4 \left (a+b x^n\right )^p\right ) \, dx,x,c \tan (e+f x)\right )}{c^5 f}\\ &=\frac {\operatorname {Subst}\left (\int x^4 \left (a+b x^n\right )^p \, dx,x,c \tan (e+f x)\right )}{c^5 f}+\frac {2 \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^4 \left (1+\frac {b x^n}{a}\right )^p \, dx,x,c \tan (e+f x)\right )}{c^5 f}+\frac {\left (2 \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 {2 \, _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}+\frac {\, _2F_1\left (\frac {5}{n},-p;\frac {5+n}{n};-\frac {b (c \tan (e+f x))^n}{a}\right ) \tan ^5(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}}{5 f}\\ \end {align*}
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Mathematica [A] time = 3.13, size = 165, normalized size = 0.68 \[ \frac {\tan (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \left (\frac {b (c \tan (e+f x))^n}{a}+1\right )^{-p} \left (3 \tan ^4(e+f x) \, _2F_1\left (\frac {5}{n},-p;\frac {n+5}{n};-\frac {b (c \tan (e+f x))^n}{a}\right )+10 \tan ^2(e+f x) \, _2F_1\left (\frac {3}{n},-p;\frac {n+3}{n};-\frac {b (c \tan (e+f x))^n}{a}\right )+15 \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b (c \tan (e+f x))^n}{a}\right )\right )}{15 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \sec \left (f x + e\right )^{6}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.80, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{6}\left (f x +e \right )\right ) \left (a +b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \sec \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p}{{\cos \left (e+f\,x\right )}^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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