Optimal. Leaf size=226 \[ \frac {\sec ^4(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \left (\frac {b (c \sec (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {4}{n},-p;\frac {n+4}{n};-\frac {b (c \sec (e+f x))^n}{a}\right )}{4 f}-\frac {\sec ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \left (\frac {b (c \sec (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {2}{n},-p;\frac {n+2}{n};-\frac {b (c \sec (e+f x))^n}{a}\right )}{f}-\frac {\left (a+b (c \sec (e+f x))^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b (c \sec (e+f x))^n}{a}+1\right )}{a f n (p+1)} \]
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Rubi [A] time = 0.52, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4139, 6742, 367, 12, 266, 65, 365, 364} \[ \frac {\sec ^4(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \left (\frac {b (c \sec (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {4}{n},-p;\frac {n+4}{n};-\frac {b (c \sec (e+f x))^n}{a}\right )}{4 f}-\frac {\sec ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \left (\frac {b (c \sec (e+f x))^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {2}{n},-p;\frac {n+2}{n};-\frac {b (c \sec (e+f x))^n}{a}\right )}{f}-\frac {\left (a+b (c \sec (e+f x))^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b (c \sec (e+f x))^n}{a}+1\right )}{a f n (p+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 266
Rule 364
Rule 365
Rule 367
Rule 4139
Rule 6742
Rubi steps
\begin {align*} \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^5(e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^2 \left (a+b (c x)^n\right )^p}{x} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\left (a+b (c x)^n\right )^p}{x}-2 x \left (a+b (c x)^n\right )^p+x^3 \left (a+b (c x)^n\right )^p\right ) \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx,x,\sec (e+f x)\right )}{f}+\frac {\operatorname {Subst}\left (\int x^3 \left (a+b (c x)^n\right )^p \, dx,x,\sec (e+f x)\right )}{f}-\frac {2 \operatorname {Subst}\left (\int x \left (a+b (c x)^n\right )^p \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {c \left (a+b x^n\right )^p}{x} \, dx,x,c \sec (e+f x)\right )}{c f}+\frac {\operatorname {Subst}\left (\int \frac {x^3 \left (a+b x^n\right )^p}{c^3} \, dx,x,c \sec (e+f x)\right )}{c f}-\frac {2 \operatorname {Subst}\left (\int \frac {x \left (a+b x^n\right )^p}{c} \, dx,x,c \sec (e+f x)\right )}{c f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^n\right )^p}{x} \, dx,x,c \sec (e+f x)\right )}{f}+\frac {\operatorname {Subst}\left (\int x^3 \left (a+b x^n\right )^p \, dx,x,c \sec (e+f x)\right )}{c^4 f}-\frac {2 \operatorname {Subst}\left (\int x \left (a+b x^n\right )^p \, dx,x,c \sec (e+f x)\right )}{c^2 f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,(c \sec (e+f x))^n\right )}{f n}+\frac {\left (\left (a+b (c \sec (e+f x))^n\right )^p \left (1+\frac {b (c \sec (e+f x))^n}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int x^3 \left (1+\frac {b x^n}{a}\right )^p \, dx,x,c \sec (e+f x)\right )}{c^4 f}-\frac {\left (2 \left (a+b (c \sec (e+f x))^n\right )^p \left (1+\frac {b (c \sec (e+f x))^n}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int x \left (1+\frac {b x^n}{a}\right )^p \, dx,x,c \sec (e+f x)\right )}{c^2 f}\\ &=-\frac {\, _2F_1\left (1,1+p;2+p;1+\frac {b (c \sec (e+f x))^n}{a}\right ) \left (a+b (c \sec (e+f x))^n\right )^{1+p}}{a f n (1+p)}-\frac {\, _2F_1\left (\frac {2}{n},-p;\frac {2+n}{n};-\frac {b (c \sec (e+f x))^n}{a}\right ) \sec ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \left (1+\frac {b (c \sec (e+f x))^n}{a}\right )^{-p}}{f}+\frac {\, _2F_1\left (\frac {4}{n},-p;\frac {4+n}{n};-\frac {b (c \sec (e+f x))^n}{a}\right ) \sec ^4(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \left (1+\frac {b (c \sec (e+f x))^n}{a}\right )^{-p}}{4 f}\\ \end {align*}
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Mathematica [A] time = 10.12, size = 245, normalized size = 1.08 \[ \frac {\left (a+b (c \sec (e+f x))^n\right )^p \left (\frac {b \left (c \sqrt {\sec ^2(e+f x)}\right )^n}{a}+1\right )^{-p} \left (-4 \left (a+b \left (c \sqrt {\sec ^2(e+f x)}\right )^n\right ) \left (\frac {b \left (c \sqrt {\sec ^2(e+f x)}\right )^n}{a}+1\right )^p \, _2F_1\left (1,p+1;p+2;\frac {b \left (c \sqrt {\sec ^2(e+f x)}\right )^n}{a}+1\right )-4 a n (p+1) \sec ^2(e+f x) \, _2F_1\left (\frac {2}{n},-p;\frac {n+2}{n};-\frac {b \left (c \sqrt {\sec ^2(e+f x)}\right )^n}{a}\right )+a n (p+1) \sec ^4(e+f x) \, _2F_1\left (\frac {4}{n},-p;\frac {n+4}{n};-\frac {b \left (c \sqrt {\sec ^2(e+f x)}\right )^n}{a}\right )\right )}{4 a f n (p+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{5}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.92, size = 0, normalized size = 0.00 \[ \int \left (a +b \left (c \sec \left (f x +e \right )\right )^{n}\right )^{p} \left (\tan ^{5}\left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{5}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^5\,{\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \]
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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