Optimal. Leaf size=76 \[ \frac{\sec (a+b x) \cos ^2(a+b x)^{\frac{n+2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.0239285, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2617} \[ \frac{\sec (a+b x) \cos ^2(a+b x)^{\frac{n+2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int \sec (a+b x) (d \tan (a+b x))^n \, dx &=\frac{\cos ^2(a+b x)^{\frac{2+n}{2}} \, _2F_1\left (\frac{1+n}{2},\frac{2+n}{2};\frac{3+n}{2};\sin ^2(a+b x)\right ) \sec (a+b x) (d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0665986, size = 64, normalized size = 0.84 \[ \frac{\csc (a+b x) \left (-\tan ^2(a+b x)\right )^{\frac{1-n}{2}} (d \tan (a+b x))^n \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3}{2};\sec ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.223, size = 0, normalized size = 0. \begin{align*} \int \sec \left ( bx+a \right ) \left ( d\tan \left ( bx+a \right ) \right ) ^{n}\, 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 (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right )\,{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 (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan{\left (a + b x \right )}\right )^{n} \sec{\left (a + b x \right )}\, dx \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 (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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