Optimal. Leaf size=54 \[ -\frac {\left (a+b \sec ^2(e+f x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sec ^2(e+f x)}{a}+1\right )}{2 a f (p+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4139, 266, 65} \[ -\frac {\left (a+b \sec ^2(e+f x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sec ^2(e+f x)}{a}+1\right )}{2 a f (p+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rule 4139
Rubi steps
\begin {align*} \int \left (a+b \sec ^2(e+f x)\right )^p \tan (e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{x} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=-\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*}
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Mathematica [A] time = 0.05, size = 54, normalized size = 1.00 \[ -\frac {\left (a+b \sec ^2(e+f x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sec ^2(e+f x)}{a}+1\right )}{2 a f (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right ), x\right ) \]
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 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.31, size = 0, normalized size = 0.00 \[ \int \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{p} \tan \left (f x +e \right )\, dx \]
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 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {tan}\left (e+f\,x\right )\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p \,d x \]
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 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{p} \tan {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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