Optimal. Leaf size=129 \[ \frac {a (d \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(e+f x)\right )}{d f (n+1)}+\frac {b \sec (e+f x) \cos ^2(e+f x)^{\frac {n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+2}{2};\frac {n+3}{2};\sin ^2(e+f x)\right )}{d f (n+1)} \]
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Rubi [A] time = 0.09, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3884, 3476, 364, 2617} \[ \frac {a (d \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(e+f x)\right )}{d f (n+1)}+\frac {b \sec (e+f x) \cos ^2(e+f x)^{\frac {n+2}{2}} (d \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+2}{2};\frac {n+3}{2};\sin ^2(e+f x)\right )}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2617
Rule 3476
Rule 3884
Rubi steps
\begin {align*} \int (a+b \sec (e+f x)) (d \tan (e+f x))^n \, dx &=a \int (d \tan (e+f x))^n \, dx+b \int \sec (e+f x) (d \tan (e+f x))^n \, dx\\ &=\frac {b \cos ^2(e+f x)^{\frac {2+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {2+n}{2};\frac {3+n}{2};\sin ^2(e+f x)\right ) \sec (e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {(a d) \operatorname {Subst}\left (\int \frac {x^n}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{f}\\ &=\frac {a \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {b \cos ^2(e+f x)^{\frac {2+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {2+n}{2};\frac {3+n}{2};\sin ^2(e+f x)\right ) \sec (e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 106, normalized size = 0.82 \[ \frac {(d \tan (e+f x))^n \left (\frac {a \tan (e+f x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(e+f x)\right )}{n+1}+b \csc (e+f x) \left (-\tan ^2(e+f x)\right )^{\frac {1-n}{2}} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3}{2};\sec ^2(e+f x)\right )\right )}{f} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int {\left (b \sec \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.54, size = 0, normalized size = 0.00 \[ \int \left (a +b \sec \left (f x +e \right )\right ) \left (d \tan \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int {\left (b \sec \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \tan {\left (e + f x \right )}\right )^{n} \left (a + b \sec {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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